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\begin{document}

\vspace{2in}

\thispagestyle{empty}
\Huge 
\begin{center}
{\bf CAMTP}\\
\end{center}

\vspace{2.5in}

\Large

\begin{center}
{\bf LET'S FACE CHAOS}\\ {\bf through}\\
{\bf NONLINEAR DYNAMICS}\\
$\bullet$  \\
$\bullet$  $\bullet$  $\bullet$\\ 
$\bullet$  \\
{\bf 4th International Summer School/Conference}\\
{\bf at the University of Maribor}\\
{\bf 27 June - 11 July 1999}\\
\end{center}


\newpage

\thispagestyle{empty}
\noindent
{\fontsize{9pt}{11pt}
\selectfont
\begin{tabular}[c]{|p{2cm}||p{2.3cm}|p{2cm}|p{2cm}|p{2cm}|p{2.3cm}
|p{2cm}|}
\hline 
1st Week & Mon & Tue & Wed & Thu & Fri & Sat\\
         & 28 June & 29 June & 30 June & 1 July & 2 July & 3 July\\
\hline \hline
Chairman & Robnik & St\" ockmann & Bountis & Prosen & Hasegawa & Casati\\
\hline
09:00-10:00 & HAKEN & HAKEN & HAKEN & HAKEN & 09:00-09:50 AIZAWA & 09:00-09:20
ALONSO\\ \hline
10:00-11:00 & CVITANOVI\' C & CASATI & CVITANOVI\' C & CASATI & 09:50-10:40
PROSEN & 09:20-09:40 CLERC\\ \hline
11:00-11:30 & \multicolumn{4}{c|}{--- COFFEE \& TEA ---} & 10:40-11:10
\linebreak -- COFFEE --&
09:40-10:00 DAMGOV\\ \hline
11:30-12:30 & BOUNTIS & AIZAWA & GROSSMANN & GROSSMANN & 11:10-12:00 \linebreak
KURAMOTO & 10:00-10:20 GARCIA-RUIZ\\ \hline
12:30-13:30 & KURAMOTO & HASEGAWA & ST\" OCKMANN & GALLAVOTTI & 12:00-12:50
ROBNIK & 10:20-10:40 GORIN\\ \hline
13:30-15:00 & \multicolumn{4}{c|}{--- LUNCH ---} & 12:50-15:00\linebreak --  
LUNCH --&
10:40-11:00 GYERGYEK\\ \hline
Chairman & Cvitanovi\' c & Aizawa & Grossmann & Stefanovska & 
Robnik & 11:00-11:30
\linebreak -- COFFEE --\\ \hline
15:00-16:00 & MATSUSHITA & KURAMOTO & GALLAVOTTI & HAKEN & 15:00-15:40 
HASEGAWA & Chairman: Cvitanovi\' c\\ \cline{2-5} \hline
16:00-16:30 & \multicolumn{4}{c|}{--- COFFEE \& TEA ---} & 15:40-16:30 
GRAVNER & 11:30-11:50 HARAYAMA\\  \hline
16:30-17:30 & McCLINTOCK & McCLINTOCK & TASS & ST\" OCKMANN & 16:30-17:00
\linebreak -- COFFEE -- & 11:50-12:10 KRYLOV\\ \hline
17:30-18:30 & NAKAMURA & NAKAMURA & GRAVNER & MATSUSHITA & 17:00-17:50
MATSUSHITA & 12:10-12:30 LAULICHT\\ \hline
 & 18:30-18:50 ZYCK\nolinebreak 
OWSKI & 18:30-19:30 DINNER & 18:30-18:50 SALASNICH &
18:30-19:30 JIRSA & 17:50-18:40 \linebreak 
STE\nolinebreak FANOVSKA & 12:30-12:50 RINKEVICH\\
\hline
 & 18:50-19:10 LOPAC 
&19:30-21:00 CONCERT & 18:50-19:10 BERGLUND & 19:30-21:00 DINNER & 
18:40-20:10 DINNER & 12:50-13:10 RUFFING\\
\hline
 & 19:10-20:40 DINNER & 
 21:00-23:00 RECEPTION & 19:10-20:40 DINNER& & &  13:10-13:30 VEBLE\\ \hline
 & & & & & & 13:30-15:00 \linebreak -- LUNCH --\\ \hline
\end{tabular}
\vfill

\noindent
\begin{tabular}[c]{|p{2cm}||p{2.3cm}|p{2cm}|p{2cm}|p{2cm}|p{2.3cm}|p{2cm}|}
\hline
2nd Week & Mon & Tue & Wed & Thu & Fri & Sat\\ 
 & 5 July & 6 July & 7 July & 8 July & 9 July & 10 July\\ \hline \hline
Chairman & Fujisaka & Eckhardt & Coullet & Mosekilde & Sonis & Stefanovska\\
\hline
09:00-10:00 & GROSSMANN & CVITANOVI\' C &
 CVITANOVI\' C & COULLET & CVITANOVI\' C
& MOSEKILDE \\ \hline
10:00-11:00 & BOUNTIS & BISHOP & BISHOP & ECKHARDT & COULLET & PROSEN \\
\hline
11:00-11:30 & \multicolumn{6}{c |}{ --- COFFEE \& TEA --- }\\ \hline
11:30-12:30 & BOUNTIS & GEISEL & FUJISAKA & JIRSA & SONIS & SELIGMAN \\ \hline
12:30-13:30 & AIZAWA & SELIGMAN & KRINSKY & SHUDO & RAPISARDA & ECKHARDT \\
\hline
13:30-15:00 & \multicolumn{6}{c|} {--- LUNCH ---} \\ \hline
Chairman & Shudo & Mosekilde & Krinsky & Richter & Pichard & Kuvshinov
 \\ \hline
15:00-16:00 & GLASS & GEISEL & GLASS & GLASS & MOSEKILDE & ROBNIK \\ \hline
16:00-16:30 & \multicolumn{6}{c |} {--- COFFEE \& TEA ---} \\ \hline
16:30-17:30 & TASS & STARMER & SHUDO & PICHARD & RICHTER & SONIS \\ \hline
17:30-18:30 & FUJISAKA & PICHARD & STARMER & PICHARD & STEFANOVSKA &
MOSEKILDE \\ \hline
 & 18:30-18:50 RO\nolinebreak 
MANOVSKY & 18:30-19:15 CHIAN & 18:30-19:15 PRELOV\v SEK &
18:30-19:15 KU\nolinebreak VSHINOV & 18:30-18:50 MARCQ & 18:30-20:00 DINNER
\\ \hline
 & 18:50-19:10 LEYVRAZ & 19:15-20:45 DINNER & 20:30-23:00 FESTIVE DINNER `SLOW
\nolinebreak FOOD' & 19:15-20:45 DINNER & 18:50-20:20 DINNER & \\ \hline
 & 19:10-20:40 DINNER & & & & & \\ \hline
\end{tabular} \normalsize
}


\newpage

\thispagestyle{empty}
\vspace{2.0in}

\Huge
\begin{center}
{\bf CAMTP}\\
\end{center}

\normalsize

\vspace{0.2in}

\begin{center}
%\epsfig{file=camtp.eps,width=17cm}\\[2mm]
{\sc Center for Applied Mathematics and Theoretical Physics}\\
{\sc Univerza v Mariboru  $\bullet$ University of Maribor}\\
{\sc  Krekova 2 $\bullet$ SI-2000 Maribor $\bullet$ Slovenia}\\
Phone +386 62 213151 and 2296440 $\bullet$ Fax +386 62 213151 and 213541\\
Robnik at uni-mb.si $\bullet$ www.camtp.uni-mb.si\\
{\sc Director: Professor Dr. Marko Robnik}\\
\end{center}


\vspace{0.3in}

\begin{center}
{\bf The Supporting Scientific Institution of}
\end{center}

\vspace{0.3in}


\begin{center}
{\bf The 4th International Summer School/Conference\\
"Let's Face Chaos through Nonlinear Dynamics"\\
University of Maribor, Maribor, Slovenia\\
27 June - 11 July 1999\\
e-mail ss99 at uni-mb.si\\
http://www.uni-mb.si/\~{ }ucamtpssn/}\\
{\em The mailing and contact address otherwise as for CAMTP above}\\
\end{center}

\vspace{0.3in}

\begin{center}
{\bf thanks the Patron and the General Sponsor}\\
\end{center}

\vspace{0.3in}

\begin{center}
{\bf MINISTRY OF SCIENCE AND TECHNOLOGY}\\
{\bf of the Republic of Slovenia}\\
\end{center}

\newpage


\vspace{2.0in}

\Large
\begin{center}
{\bf THE ORGANIZING COMMITTEE}\\
\end{center}

\vspace{1.0in}

\begin{center}
{\bf Director and Chairman:}\\
$\bullet$  \\
{\bf Marko Robnik (CAMTP, Maribor, Slovenia)}\\ 
$\bullet$   $\bullet$   $\bullet$   \\
{\bf Honorary Directors:}\\
$\bullet$   \\
{\bf Giulio Casati (Como and Milano, Italy)}\\
{\bf Predrag Cvitanovi\'c (Copenhagen and Chicago, USA)}\\
{\bf Siegfried Grossmann (Marburg, Germany)}\\
{\bf Hermann Haken (Stuttgart, Germany)}\\ 
$\bullet$   $\bullet$   $\bullet$   \\
{\bf Members:}\\
$\bullet$   \\
{\bf Tassos Bountis (Patras, Greece)}\\ 
{\bf Hiroshi Hasegawa (Tokyo, Japan)}\\
{\bf Jean-Louis Pichard (Saclay, France)}\\
{\bf Toma\v z Prosen (Ljubljana, Slovenia)}\\
{\bf Aneta Stefanovska (Ljubljana, Slovenia)}\\
$\bullet$   $\bullet$   $\bullet$   \\
\end{center}


\newpage


\vspace{2.0in}

\Large
\begin{center}
{\bf INVITED LECTURERS AND SPEAKERS}\\
\end{center}

\vspace{0.2in}

\normalsize
\begin{center}
\bmp
{\bf Professor Dr. Yoram Alhassid}\\
{\em Yale University, USA}\\
Center for Theoretical Physics\\
Sloane Physics Laboratory\\
Yale University\\
New Haven, Connecticut 06511\\
U. S. A.\\
Telefax +(1) (203) 432 5741\\
Telephone  +(1) (203) 432 6922\\
e-mail yoram at nst.physics.yale.edu\\
$\bullet$\\
\emp

\bmp
{\bf Professor Yoji Aizawa}\\                   
{\em          Waseda University, Japan}\\
              Department of Applied Physics\\
              School of Science and Engineering\\
              Waseda University\\
              Okubo 3-4-1,  Shinjuku-ku,  Tokyo 1690072\\
              Japan\\
Telefax +(81) 3 3200 2457\\
Telephone  +(81) 3 3203 4141  ext.  3562\\
e-mail aizawa at aizawa.phys.waseda.ac.jp\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Steven Bishop}\\
{\em University College London, UK}\\
Centre for Nonlinear Dynamics and Applications\\
University College London\\
London  WC1E 6BT\\
England, U.K.\\
Telefax +(44) (171) 380 0986\\
Telephone +(44) (171) 380 7729\\
e-mail s.bishop at ucl.ac.uk\\             
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Tassos Bountis}\\
{\em University of Patras, Greece}\\
Department of Mathematics\\
Division of Applicable Analysis\\
University of Patras\\
GR-26110 Patras\\
Greece\\
Telefax +(30) (61) 997 163\\
Telephone +(30) (61) 997 381\\
e-mail bountis at math.upatras.gr\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Giulio Casati}\\
{\em Universit\' a degli Studi dell' Insubria, Italy}\\
International Center for the Study of Dynamical Systems\\
Universit\' a degli Studi dell' Insubria\\
Via Lucini, 3\\
I-22100 Como\\
Italy\\
Telefax  +(39) (31) 326 230\\
Telephone +(39)  (31) 326 229\\
e-mails  casati at milano.infn.it\\
        casati at fis.unico.it\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Abraham C.-L. Chian}\\
{\em University of Adelaide, Australia}\\
Center for Subatomic Structure of Matter\\
University of Adelaide\\
10 Pulteney Street, 4th floor\\
Adelaide, SA 5005\\
Australia\\
Telefax  +(61) (8) 8303 3551\\
Telephone +(61) (8) 8303 3424 \\
e-mails  achian at physics.adelaide.edu.au\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Pierre Coullet}\\ 
{\em Institut Non-Lineaire de Nice, France}\\
Institut Non-Lineaire de Nice\\
U.M.R. C.N.R.S. 6618\\
Sophia Antipolis\\
1361, Route des Lucioles\\
F-06560 Valbonne\\ 
France\\
Telefax  +(33) 4 93 65 25 17\\
Telephone +(33) 4 92 96 73 51\\ 
e-mail  coullet at inln.cnrs.fr\\
$\bullet$\\
\emp

\bmp
{\bf Predrag Cvitanovi\'c}\\
{\em            Niels Bohr Institute, Denmark}\\
{\em            Northwestern University, USA}\\
        Center for Chaos and Turbulence Studies\\
                Niels Bohr Institute\\
                Blegdamsvej 17\\
                dk-2100 Copenhagen O\\
                Denmark\\       
                Telefax +(45) 353 25 425\\ 
                Telephone +(45) 353 25 346 office\\      
                e-mail   predrag at nbi.dk\\[0.3cm]
 
                Department of Physics \& Astronomy\\
                Northwestern University\\
                2145 Sheridan Road\\
                Evanston, Illinois 60208-3112\\
                USA\\
                Telefax   +(1) (847) 491 9982\\
                Telephone +(1) (847) 491 3235\\    
                e-mail         predrag at snowmass.phys.nwu.edu\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Bruno Eckhardt}\\
{\em University of Marburg(Lahn), Germany}\\
Fachbereich Physik\\
Philipps Universit\"at Marburg\\
D-35032 Marburg\\
Germany\\                          
Telefax +(49) 6421 28 4511\\       
Telephone +(49) 6421 28 1316\\     
e-mail bruno.eckhardt at physik.uni-marburg.de\\  
$\bullet$\\
\emp

\bmp
{\bf  Professor Dr. Hirokazu Fujisaka }\\
{\em Kyoto university, Japan}\\
  Department of Applied Analysis and Complex Dynamical Systems\\
  Graduate School of Informatics\\
  Kyoto University\\
  Kyoto 606-8501\\
  Japan\\
  Telefax +(81) 75 753 3391\\
  Telephone +(81) 75 753 3387\\
 e-mail fujisaka at acs.i.kyoto-u.ac.jp\\ 
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Giovanni Gallavotti}\\
{\em University La Sapienza, Italy}\\
Fisica\\
Universita di Roma la Sapienza\\
P. le Moro 2\\
I-00185 Roma\\
Italy\\
Telefax +(39) (6) 4957 697 \\
Telephone +(39) (6) 4991 4370 \\
e-mail giovanni at ipparco.roma1.infn.it \\
$\bullet$\\
\emp

\bmp                                        
{\bf   Professor Dr. Theo Geisel}\\
{\em University of G\"ottingen, Germany}\\
   Max-Planck-Institut f\"ur Stromungsforschung\\
   and Fakult\"at Physik, Universitat G\"ottingen\\
   Bunsenstrasse 10\\
   D-37073 Gottingen\\
   Germany\\                      
  Telephone  +(49) (551) 5176 400\\
  Telefax    +(49) (551) 5176 402\\
  e-mail geisel at chaos.gwdg.de\\
$\bullet$\\
\emp

\bmp 
{\bf Professor Dr. Leon Glass}\\
{\em McGill University, Canada}\\
Department of Physiology\\
3655 Drummond Street\\                  
McGill University\\
Montreal Quebec Canada H3G 1Y6\\      
Telefax    +(1)  514 398 7452 \\                                        
Telephone  +(1)  514 398 4338 \\
e-mail    glass at cnd.mcgill.ca \\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Janko Gravner}\\
{\em University of California, USA}\\
Mathematics Department\\
University of California\\
Davis, CA 95616\\
USA\\                            
Telefax  +(1) 530 752 6635\\               
Telephone +(1) 530 752 0825\\               
e-mail gravner at math.ucdavis.edu\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Siegfried Grossmann}\\
{\em University of Marburg(Lahn), Germany}\\
Fachbereich Physik der Philipps-Universit\"at\\
Renthof 6\\
D-35032 Marburg(Lahn)\\
Germany\\
Telephone +(49) (0) 6421 28 2949\\
Telefax   +(49) (0) 6421 28 4110\\
e-mail      grossmann\_s at physik.uni-marburg.de\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Hermann Haken}\\
{\em University of Stuttgart, Germany}\\
Institut f\"ur Theoretische Physik und Synergetik\\
Fakult\"at f\"ur Physik\\
Pfaffenwaldring 57/IV\\
D-70550 Stuttgart (Vaihingen)\\
Germany\\
Telefax +(49) (711) 685 4909\\
Telephone        +(49) (711)   685 4990\\
e-mail bund at theo.physik.uni-stuttgart.de\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Hiroshi Hasegawa}\\
{\em Nihon University, Japan}\\
Atomic Energy Research Institute\\
Nihon University\\
Kanda-Surugadai\\
Tokyo 101-8301\\
Japan\\
Telefax  +(81) (3) 3795 9587\\
Telephone  +(81) (3) 3795 9587\\
e-mail h-hase at mxj.meshnet.or.jp \\
$\bullet$\\
\emp

\bmp
{\bf Dr. Viktor Jirsa}\\
{\em Florida Atlantic University, USA}\\
Center for Complex Systems\\
Florida Atlantic University\\
777 Glades Road, P.O. Box 3091\\
Boca Raton, Florida 33431-0991\\
USA\\                          
Telefax +(1)  561 297 3634\\   
Telephone +(1)  561 297 2230\\ 
e-mail jirsa at walt.ccs.fau.edu\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Valentin Krinsky}\\
{\em Institut Non-Lineaire de Nice, France}\\
 Institut Non-Lineaire de Nice\\        
U.M.R. C.N.R.S. 6618\\
Sophia Antipolis\\
1361, Route des Lucioles\\
F-06560 Valbonne\\
 France\\                          
   Telefax +(33) 4 93 65 25 17\\  
   Telephone +(33) 4 92 96 73 45\\
e-mail  krinsky at inln.cnrs.fr\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Yoshiki Kuramoto}\\
{\em Kyoto University, Japan}\\
        Department of Physics\\
        Graduate School of Sciences\\
        Kyoto University\\
        Kyoto 606\\
        Japan\\
Telefax +(81) (75) 753 3819\\
Telephone +(81) (75) 753 3740\\
e-mail kuramoto at ton.scphys.kyoto-u.ac.jp\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Viatcheslav Ivanovich Kuvshinov}\\
{\em Stepanov Institute of Physics, Minsk, Belarus}\\
Stepanov Institute of Physics\\
Belarus Academy of Sciences\\
Scoryna Avenue 68\\
220072 Minsk\\
Belarus\\
Telefax +(375) (172) 840879\\
Telephone +(375) (172) 841628\\
e-mail kuvshino at dragon.bas-net.by\\
$\bullet$\\
\emp


\bmp  
{\bf Professor Mitsugu Matsushita}\\
{\em Chuo University, Japan}\\
Department of Physics\\
Chuo University\\
1-13-27 Kasuga, Bunkyo-ku\\
Tokyo 112-8551\\
Japan\\
Telephone +(81) 3 3817 1787\\
Telefax +(81) 3 3817 1792\\      
e-mail matusita at phys.chuo-u.ac.jp\\
$\bullet$\\
\emp

\bmp                        
{\bf Professor Dr.
Peter V.E. McClintock}\\
{\em Lancaster University, UK}\\          
Department of Physics\\           
Lancaster University\\            
Lancaster LA1 4YB\\                
England, UK\\          
Telefax +(44) 1524 844037\\   
Telephone +(44) 1524 593073\\
e-mail p.v.e.mcclintock at lancaster.ac.uk\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Erik Mosekilde}\\
{\em Technical University of Denmark}\\ 
Dept. of Physics, Bldg. 309\\
Technical University of Denmark\\
2800 Lyngby\\
Denmark\\
Telefax +(45) 45 93 16 69\\
Telephone +(45) 45 25 25 25\\ 
e-mail ellen at chaos.fys.dtu.dk\\
$\bullet$\\   
\emp

\bmp                            
{\bf           Professor Dr. Katsuhiro Nakamura}\\
{\em Osaka City University, Japan}\\
          Dept. of Applied Physics\\
           Faculty of Engineering \\
           Osaka City University\\
           Osaka 558\\
           Japan\\
           Telefax  +(81) 6 605 2768\\
           Telephone  +(81) 6 605 2768\\
           e-mail    nakamura at a-phys.eng.osaka-cu.ac.jp\\
$\bullet$\\
\emp

\bmp
{\bf            Professor Dr. Jean-Louis  Pichard}\\
{\em     Service de Physique de l'Etat Condens\'e, France} \\
          Service de Physique de l'Etat Condens\'e\\
                CEA-Saclay\\
               91191 Gif sur Yvette cedex\\
                  France\\
           Telefax +(33) 169088786\\
           Telephone +(33) 169087236\\
           e-mail pichard at spec.saclay.cea.fr\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Peter Prelov\v sek}\\
{\em University of Ljubljana, Slovenia}\\
Physics Department\\
Faculty of Mathematics and Physics\\
 University of Ljubljana\\
 Jadranska 19\\
 1111 Ljubljana\\
 Slovenia\\
 Telefax +(386) 61 217 281\\
 Telephone +(386) 61 1766 500\\
 e-mail peter.prelovsek at ijs.si\\
$\bullet$\\
\emp

\bmp
{\bf Dr. Toma\v z Prosen}\\
{\em University of Ljubljana, Slovenia}\\
Physics Department\\
Faculty of Mathematics and Physics\\
 University of Ljubljana\\
 Jadranska 19\\
 1111 Ljubljana\\
 Slovenia\\
 Telefax +(386) 61 217 281\\
 Telephone +(386) 61 1766 623\\
 e-mail prosen at fiz.uni-lj.si\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Andrea Rapisarda}\\      
{\em University of Catania, Italy}\\
INFN - Sez.di Catania\\                
Dipartimento di Fisica\\               
Universita' di Catania\\                                          
Corso Italia, 57\\                    
I-95129 Catania\\
Italy\\    
Telefax    +(39) 095 383023\\  
Telephone  +(39) 095 7195214\\   
e-mail rapisarda at ct.infn.it\\
$\bullet$\\
\emp

\bmp
{\bf        Dr. Klaus Richter}\\        
{\em        Max Planck-Institut f\"ur Physik komplexer Systeme, Germany}\\
            Max Planck-Institut f\"ur Physik komplexer Systeme\\
        Noethnitzer Str. 38\\
        D-01187 Dresden\\
        Germany\\
        Telefax               +(49) 351 871 1199\\
        Telephone               +(49) 351 871 2210\\
        e-mail  richter at mpipks-dresden.mpg.de\\
$\bullet$\\
\emp

\bmp
{\bf                               Professor Dr. Marko Robnik}\\
{\em CAMTP, University of Maribor, Slovenia}\\
                               Center for Applied Mathematics\\
                                and Theoretical Physics\\
                               University of Maribor\\
                               Krekova 2\\
                               SI-2000 Maribor\\
                               Slovenia\\               
                               Telefax +(386) (62) 213151 and 213541\\
                               Telephone +(386) (62) 213151 and 2296440\\
                               e-mail Robnik at uni-mb.si\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Thomas Seligman}\\
{\em University of Mexico, Mexico}\\
Centro de Ciencias F\'isicas\\
University of Mexico (UNAM)\\
Cuernavaca/Morelos\\
Instituto de f\'isica\\
Apdo. post. 20364\\
01000 Mexico 20 DF\\         
Telefax +(52) (73) 173077\\
Telephone +(52) (73) 291733\\
e-mails seligman at fenix.ifisicacu.unam.mx\\
seligman at ubaclu.unibas.ch\\
$\bullet$\\
\emp
 
\bmp
{\bf Professor Dr. Akira Shudo}\\
{\em Tokyo Metropolitan University, Japan}\\
Department of Physics\\
Tokyo Metropolitan University\\
Minami-Ohsawa 1-1\\
Hachioji-shi\\
Tokyo 192-0397\\
Japan\\                          
Telefax +(81) 426 77 2483\\        
Telephone +(81) 426 77 2503\\      
e-mail shudo at phys.metro-u.ac.jp\\
$\bullet$\\
\emp
    
\bmp                  
{\bf Professor Dr. Michael Sonis}\\
{\em Bar-Ilan University, Israel}\\
Department of Geography\\
Bar-Ilan University\\
Ramat-Gan 52900\\
Israel\\                    
Telefax +(972) 3 53 444 30\\      
Telephone +(972) 3 531 7856\\ 
e-mail sonism at mail.biu.ac.il\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. C. Frank Starmer}\\
{\em Medical University of South Carolina, USA}\\
629C Strom Thurmond Building\\
Medical University of South Carolina\\
Charleston, SC 29425\\
USA\\
Telephone +(1) 843 953 6681\\
Telefax   +(1)  843 953 6655\\
email  starmerf at musc.edu\\     
$\bullet$\\
\emp

\bmp
{\bf    Doc. Dr. Aneta Stefanovska }\\
{\em University of Ljubljana, Slovenia}\\
    Group of Nonlinear Dynamics and Synergetics\\
    Faculty of Electrical Engineering\\
    University of Ljubljana\\
    1000 Ljubljana\\
    Slovenia\\                        
    Telefax  +(386) 61 1264 630\\    
    Telephone   +(386) 61 1768 246\\ 
    e-mail aneta at osc.fe.uni-lj.si\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Ian Stewart}\\
{\em University of Warwick, UK}\\
Mathematics Institute\\
University of Warwick\\
Coventry CV4 7AL\\
England, UK\\
Telefax     +(44) (1203) 514182\\
Telephone   +(44) (1203) 523740\\
e-mail  ins at maths.warwick.ac.uk\\
$\bullet$\\
\emp

\bmp
{\bf Professor Dr. Hans-J\"urgen St\"ockmann}\\
{\em University of Marburg(Lahn), Germany}\\
Fachbereich Physik der Philipps-Universit\"at Marburg\\
Renthof 5\\
D-35032 Marburg\\
Germany\\                 
Telephone +(49) 6421 28 4137\\
Telefax +(49) 6421 28 6535\\
e-mail stoeckmann at physik.uni-marburg.de\\
$\bullet$\\
\emp

\bmp
{\bf Dr. Peter Tass}\\
{\em Heinrich Heine University, Germany}\\
Neurologische Klinik\\
 Heinrich-Heine-Universitaet\\
Moorenstr. 5\\
 D-40225 D\"usseldorf\\
 Germany\\                     
Telefax +(49)  211 8118469\\   
Telephone +(49) 211 8117079\\  
e-mail tass at neurologie.uni-duesseldorf.de\\
\emp

\bmp
$\bullet$\\
{\bf Professor Dr. Andre Voros}\\
{\em Service de Physique Theorique, France}\\
CEA-Service de Physique Theorique de Saclay\\
F-91191 Gif sur Yvette CEDEX\\
France\\
Telefax     +(33) 01 6908 8120\\            
Telephone  +(33) 01 6908 7566\\
e-mail voros at spht.saclay.cea.fr\\        
$\bullet$$\bullet$$\bullet$\\
\emp

\end{center}

\newpage

\vspace{3.0in}

\Large
\begin{center}
{\bf THE SCIENTIFIC PROGRAMME - THE INVITED LECTURES}\\
\end{center}
\normalsize


\vspace{3.0in}

The following are extended abstracts of all invited lectures,
each one 60 minutes long. The ordering is alphabetical according to
the surname of the lecturer, whilst his or her lectures are 
ordered temporaly in accordance with appearance in the programme.
We expect only a few minor updates which will be done in due course.

On Friday, 2 July 1999, the whole daily programme will be devoted
to the {\bf Japan-Slovenia Seminar on Chaos Science}, which
is fully and separately displayed on the next page. {\bf The Organizing
Committee thanks the Japan Society for the Promotion of Science}
for the full financial support of this Seminar, which is a part
of the scientific programme of the School/Conference.

\newpage

\vspace{3.0in}

\begin{center}
{\large {\bf JAPAN-SLOVENIA SEMINAR ON CHAOS SCIENCE}} \\
(A Special Session at Maribor, Slovenia, 2 July 1999)
\end{center}
\vspace{2mm}
{\bf Main Talks} (50 minutes for each)
\begin{description}
\item[1. Yoji Aizawa ] {\it Ergodic Properties of \, Non-stationary
Chaos}
\item[2. Toma\v z Prosen ] {\it Quantum Poincar\'e Mapping}
\item[3. Yoshiki Kuramoto ] {\it Turbulence with Multiscaling in Large
Assemblies of Oscillators}
\item[4. Marko Robnik ] {\it Topics in Quantum Chaos of Generic Systems}

\begin{center}  
Intermission for Lunch
\end{center}

\item[5. Hiroshi Hasegawa ] {\it Information Theoretical Basis of Random
Matrix Distributions}
\item[6. Janko Gravner ] {\it Growth Phenomena in Cellular Automata}
\item[7. Mitsugu Matsushita ] \, \, {\it Formation of Colony Patterns by a
Bacterial Cell Population}
\item[8. Aneta Stefanovska ]{\it Topics in Nonlinear Dynamics in
the Human Cardiovascular \\
\hspace*{37mm}   System}

\end{description}

{\bf The Organizing
Committee thanks the Japan Society for the Promotion of Science}
for the full financial support of this Seminar, which is a special programme
embedded into the scientific programme of the School/Conference.





\newpage

\begin{center}
\Large
{\bf Long time tails in N-body Hamiltonian systems}
\normalsize

{\bf Yoji Aizawa}
  
{\em Department of Applied Physics, Faculty of Science and
Engineering,\\Waseda University, Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
One of the most striking phenomena in chaotic dynamics is the
appearance of the long time tails such as the $1/f$
fluctuations [1,2]. In the nearly integrable Hamiltonian systems, the 
long time tails are universally generated due to the stagnant
motions near the invariant KAM tori;
\setcounter{equation}{0}
\beq
H(I,\theta) = H_0(I) + \varepsilon H_1 (I,\theta)
\eeq
Here the parameter $\varepsilon$ stands for the perturbation to
the integrable Hamiltonian $H_0$. An important theorem
(Nekhoroshev,1977) explained that the residence time $T$ in the
stagnant layer obeys [3],
\beq
T\simeq \frac{1}{\varepsilon} \exp[\varepsilon ^{-b}] 
 \ , \ (\varepsilon \ll 1)
\eeq
where $b$ is a positive constant determined by the unperturbed
Hamiltonian $H_0$ in Eq.(1).\\
The significant point in Eq.(2) is that the divergence of $T$
does not obey the inverse power law but exhibits an essential
singularity when $\varepsilon$ goes to zero. In 1980's, the
origin of such singularity was studied in terms of the scaling
theory for the stagnant layers mentioned above, where the
hierarchical structure of resonant tori (islands around island) 
plays an essential role to induce the long time tails in
dynamical quantities. The stagnant layer theory (Aizawa , 1989)
demonstrated that the distribution of the residence time, say
P(T), obeys a universal law [2], 
\beq
P(T) \simeq \frac{1}{T[\log T]^c} \ , \ (T \gg 1)
\eeq
where $c$ is a positive constant larger than unity. Equation(3)
has been confirmed by simulations (Aizawa {\it et al}, 1989). The 
essential singularity in Eq.(2) reflects the onset of $1/T$
divergence in Eq.(3). The point is that the distribution is not
normalizable,i.e., a typical infinite measure.\\ 
Firstly, my lecture will be directed to the review of the
stagnant layer theory and some numerical evidence in many body
systems. Secondly, the onset of a new type of long time tails
will be discussed carrying out with the clustering motions in
N-body systems with short range attractive forces, where the
distribution of the trapping time $T$ obeys another universal law,
\beq
P(T) \simeq T^{- \beta -1}\exp[-aT^{- \beta}]
\eeq
where $a$ is a positive constant which depends on the size of
cluster. The regularly varying part of Eq.(4) denotes the tail with 
$T^{- \beta -1} (T \gg 1)$ , and the parameter $\beta$ depends 
on the dimension of the cluster under consideration. The stability of the
clustering motions will be explained based on the long time tail
of Eq.(4).
\\\\
\noindent{\bf References}\\
$[1]$ Aizawa Y 1989a {\em Prog.Theor.Phys.Suppl.} {\bf 99}149; Aizawa Y
1999 {\em Chaos, Soliton and Fractals} (in press); Tanaka K and
Aizawa Y 1993 {\em Prog.Theor.Phys.} {\bf 90(3)}
\\
$[2]$ Aizawa Y 1989b {\em Prog.Theor.Phys.} {\bf 81(2)} 249; Aizawa Y
{\it et al} 1989c {\em Prog.Theor.Phys.Suppl.} {\bf 98}
37; Aizawa Y 1995 {\em J.Korean Phys.Soc.} {\bf 28} 310; Aizawa Y
1991 {\em Dynamical Theory of $f^{- \nu}$ spectral
chaos}, eds. Musha.T {\it et al},pp483-487
\\
$[3]$ Nekhoroshev N N 1977 {\em Russ.Math.Surveys} {\bf 32} 1

\newpage

\begin{center}
\Large
{\bf Ergodic properties of  non-stationary chaos}
\normalsize

{\bf Yoji Aizawa}
  
{\em Department of Applied Physics, Faculty of Science and
Engineering,\\Waseda University,
 Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
Non-stationary chaos is a universal phenomenon in non-hyperbolic
dynamical systems.
%
Basic problems regarding the non-stationarity are discussed from
ergodic-theoretical viewpoints.
%
By use of a simple system, it is shown that ``the law of large
number'' as well as ``the law of small number'' break down in the
non-stationary regime.
%
The non-stationarity in dynamical systems proposes a crucial problem
underlying in the transitional region between chance and
necessity(onceness and recurrence), where non-observable processes
behind reality interplay with observable ones.
%
The incompleteness of statistical ensembles is discussed from the
Karamata's theory.
%
Finally, the significance of the stationary/non-stationary interface
is emphasized in relation to the universality of $1/f$ fluctuations.

\noindent
{\bf References}\\
Aizawa Y 1989 {\em Prog.Theor.Phys.} {\bf 81} pp249-253\\
Aizawa Y et al 1989 {\em Prog.Theor.Phys. Suppl.} {\bf 98}\\
Aizawa Y 1995 {\em J.Korean Phys.Soc.} {\bf 28} pp310-314\\
Aizawa Y 1991 {\em ICNF},eds Musha T et al pp483-487\\
Kurosaki S and Aizawa Y 1997{\em Prog.Theor.Phys.} {\bf 98}
pp783-793\\
Yuri M 1995 {\em Indag.Math.N.S.} {\bf 6} pp355-383\\
Yuri M 1996 {\em Nonlinearity} {\bf 9} pp1439-1461\\
Yuri M 1997 {\em Erg.Th.and Dynam.Sys.} {\bf 17} pp977-1000\\
Aizawa Y 1989 {\em Studies of Dynamical Systems},eds Aoki N(World
sci.) pp182-191\\
Kikuchi Y and Aizawa Y 1990 {\em Prog.Theor.Phys.} {\bf 84}
pp563-567\\
 Kikuchi Y and Aizawa Y 1990 {\em Prog.Theor.Phys} {\bf 84}
pp1014-1018\\
Aizawa Y 1984 {\em Prog.Theor.Phys.} {\bf 72} pp659-661\\
Aizawa Y and Kohyama T 1984 {\em Prog.Theor.Phys.} {\bf 71}
pp847-850\\
Aizawa Y 1989 {\em Prog.Theor.Phys.Suppl.} {\bf 99} pp149-164\\
Aizawa Y 1993 {\em Prog.Theor.Phys.} {\bf 90} pp547-567\\
Aizawa Y 1998 {\em International Journal of Computing Anticipatory
Systems} {\bf 2} pp235-249\\


\newpage

\begin{center}
\Large
{\bf Universal aspects in level statistics of oval billiards}
\normalsize

{\bf Yoji Aizawa, Hironori Makino and Mayuko Morita}
  
{\em Department of Applied Physics, Faculty of Science and
Engineering,\\Waseda University, Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
We studied the energy level statistics for one parameter
family of oval billiards whose classical phase space consists
of some regular and some irregular components.  As the parameter
is varied, a transition from an integrable system to a
strongly chaotic one occurs with successive bifurcations.  We
applied the Berry-Robnik formulae to the level-spacing
distributions for a variety of shapes of quantum oval
billiards and found some striking effects of bifurcations in
the classical mechanical systems on the level-spacing
distributions.  The validity of the Berry-Robnik formula is
also checked for those shapes of the oval billiard for which
there exist two separated chaotic components in the phase
space [1].  However, the Berry-Robnik formula is not 
available for the 
entire energy range, especially in the lower energy region the Brody-like
behaviors have been often observed until now [2,3].  In the latter 
part of our talk, we will discuss that the transition from the Brody
distribution
to the Berry-Robnik formula universally occurs for all parameter regime
of our model, and the detailed process of the transition near the bifurcation
point will be reported.

\noindent
{\bf References}\\
{[1]}Makino H, Harayama T and Aizawa Y 1999 {\em Phys.Rev.E} {\bf 59} 4026\\
{[2]}Robnik M and Prosen T 1997 {\em J. Phys. A:Math.Gen.} {\bf 30}
8787\\
{[3]}Prosen T 1998 {\em J. Phys. A:Math.Gen.} {\bf 31} 7023\\

\newpage

\begin{center}
\Large
{\bf Complex dynamics of a ``simple'' mechanical system:\\the parametrically excited pendulum}
\normalsize

{\bf Steven R. Bishop}

{\em Center for Nonlinear Dynamics and its Applications,\\
University College London, UK}
\end{center}

\normalsize
\noindent
A planar pendulum is perhaps the simplest and most quoted example of a
dynamical system, yet when driven its simplicity of description belies a
range of complex dynamical motions. If driven vertically at the pivot, a
pendulum which can freely move in the plane exhibits equilibrium states,
periodic solutions, as well as chaotic motions, all of which can easily be
seen in a mechanical experiment. Miles (1989) produced an excellent overview
of the basic, qualitative dynamics but research interest is not yet
exhausted with some recent results investigating the topological structure
of phase space (Clifford and Bishop 1994) and the stability of the inverted 
state (Clifford and Bishop 1996).

The dynamics of the pendulum can be envisaged a particle moving in a
cosinusoidal potential energy function. Small oscillations correspond to
periodic motions within the well while rotations lead to escape from the
local potential well between $(+\pi,-\pi)$. For small driving amplitudes the
downward hanging state forms a stable equilibrium of the model system which
becomes unstable in a series of zones. Numerically we can follow the stable
solutions  and follow bifurcations to plot the zones in parameter space in
which the various solutions exist where typically the boundary between
solutions is fractal and as a consequence the resulting dynamics is complicated
even for this 'simple' system.

In addition, combining a mixture of methods to locate unstable orbits, and
developing robust methods for control even in the presence of noise, means
that now we are able to select a desired solution from a large selection of
unstable motions onto which the system can be controlled (Bishop {\it et.al.} 
1996) without globally changing the system parameters.

\noindent{\bf References}
\\
Bishop, S.R., Xu, D. and Clifford, M.J. 1996 Flexible control of
the parametric pendulum, Proc. Roy. Soc. Lond. A452, 1-18.
\\
Capecchi, D. and Bishop, S.R. 1994 Periodic oscillations and
attracting basins for a parametrically excited pendulum, Dynamics and
Stability of Systems 9, No.2, 123-143.
\\
Clifford, M.J. and Bishop, S.R. 1994 Bifurcational precedences for
parametric escape from a symmetric potential well, International J.
Bifurcation \& Chaos 4, No.3, 623-630.
\\
Clifford, M.J. and Bishop, S.R. 1996 Locating oscillatory orbits of
the parametrically excited pendulum, J. Austral. Math. Soc. Series B 37,
309-319.
\\
Clifford, M.J. and Bishop, S.R. 1998 Inverted oscillations of a
driven pendulum, Proc. Roy. Soc. Lond. A 454, 2811-2817.
\\
Miles, J. (1989) The pendulum from Huygens' Horologium to symmetry
breaking and chaos, in Theoretical and Applied Mechanics, edited by P.
Germain, M. Piau and D. Caillerie, Elsevier Science: North Holland, pp
193-215.

\newpage

\begin{center}
\Large
{\bf The nonlinear dynamics of ship roll and capsize}
\normalsize

{\bf Steven R. Bishop}

{\em Center for Nonlinear Dynamics and its Applications,\\
University College London, UK}
\end{center}

\normalsize
\noindent
Ensuring that a ship can resist capsize when confronted with steep waves is
an important consideration during design. Unfortunately, current ship
design practices, due their reliance on ad hoc, empirical and static-type
criteria of stability, are grossly inadequate to address the dynamic
effects incurred on the ship due to wave action where the capsize process
is both nonlinear and transient. The need for suitable dynamic capsize
criteria is accentuated by the proximity of typical ocean wave frequencies
to the natural roll frequency of many ships, which makes the possibility of
beam-sea resonance a key feature of any worst-case scenario.

Due to the nonlinearity of the accompanying restoring force multiple
steady-state roll responses can emerge, which can undergo further
complicated qualitative changes of their character sometimes even becoming
chaotic. More complex still can be the transient motions near to capsize
limits. Recent applications of the global geometrical techniques of
nonlinear dynamics and the associated  problem of escape of a driven
oscillator from a potential well, have provided new possibilities for
advancing the state-of-the-art by achieving deeper understanding of the
phenomena that precede capsize. An interesting characteristic of these
phenomena is their remarkable robustness against gross changes in the forms
of stiffness and damping functions, which can yield simple and useful
design criteria against transient capsize.

We review here novel studies on ship capsize in beam seas, which have taken
place at the Centre for Nonlinear Dynamics and its Applications of
University College London.  Large-amplitude ship rolling has been
investigated both theoretically and experimentally, under steady-state as
well as under transient conditions.

\noindent{\bf References}
\\
Thompson, J.M.T. 1989. Chaotic phenomena triggering the escape from a potential
well, Proc. R. Soc. Lond. A421,195-225.
\\
Thompson, J.M.T. 1997. Designing against capsize in beam seas: Recent advances and new insights, Applied Mechanics Reviews 50,307-325.

\newpage


\begin{center}
\Large
{\bf Bifurcations, symmetry-breaking and pattern formation in
nonlinear systems}

\normalsize

{\bf Tassos Bountis}

{\em Department of Mathematics and \\
Center for Research and Applications of Nonlinear Systems, \\
University of Patras, Greece}

\end{center}

\noindent

\begin{center}

\vspace{0.5cm}

{\bf Lecture 1 : Bifurcations and Symmetry-Breaking in Nature}
\end{center}

In these 3 lectures we shall review the basic concepts of stability analysis
and bifurcation theory for nonlinear systems, with emphasis on the
applications of these concepts to physics, chemistry and biology. We shall
begin with a discussion of the meaning of {\bf{bifurcation}} and its
connection with {\bf symmetry-breaking} in finite dimensional and
{\bf spatially extended} (infinite dimensional) nonlinear systems.
As an example of the former, we examine chemical oscillations in closed, 
continuously stirred reactors, while for the latter, we use the
Rayleigh-Benard experiment and reaction-diffusion systems from chemistry and
biology.

Concentrating then on the topic of bifurcations in finite dimensional systems
we will list the {\bf normal forms} of the most commonly encountered types
of bifurcations : saddle-node, transcritical, pitchfork and Hopf.
Introducing the notions of {\bf codimension} and {\bf transversality} we will
consider which of these types are {\bf structurally stable}, when additional
parameters and lower order terms are included in the dynamics.
The importance of nonlinear stability analysis will be emphasized in connection
with the study of {\bf center manifolds} at bifurcation points.


\begin{center}
{\bf Lecture 2 : Linear Stability of Spatially Extended Systems}
\end{center}

In this lecture we review the main features of linear stability analysis
of steady states of spatially distributed systems, occurring in problems of
hydrodynamics and reaction-diffusion models of chemical and biological 
systems. More specifically, we  consider the Rayleigh-Benard problem
of a thin fluid layer heated from below and describe how one
can estimate the critical Rayleigh parameter $R_c$, at which the symmetry
of the spatially uniform state is broken by the formation of cylindrical
rolls.

Next, we discuss the onset of diffusion-driven or {\bf Turing instability}
in reaction-diffusion systems and compare it with the onset of other types of
instability (caused e.g. by a Hopf bifurcation) on the example of the
Brusselator model. Symmetry-breaking and {\bf pattern formation} will be
examined on diffusive, 2-variable models of competing species.
The dependence of the critical parameter value $\lambda_c$ on a
characteristic length scale in such systems will be compared to the
absence of such a dependence in hydrodynamics.

\vspace{2cm}
\begin{center}
{\bf Lecture 3 : Wave Propagation in Excitable Media}
\end{center}


In the last lecture, we turn our attention to the study of
reaction-diffusion equations describing {\bf excitable media}. These are
systems composed of active units, like cells, possessing a certain
threshold of excitability, beyond which they can "fire" and transmit
their action potential to neighboring cells. They are described by
Fitzhugh-Nagumo (FHN) 2-variable partial differential equations in one or
more space dimensions.

Depending on the parameters of the FHN equations, action potentials can
travel in the form of localized waves, delivering their "message" along
the medium. At other parameters (and places) however, on the medium,
{\bf stationary pulses} can either collapse or grow, forming a possible
"obstacle", hindering the propagation of action potential waves. We derive
analytical expressions for these stationary pulses and examine ways by
which their effect on the travelling waves can be controlled. The
connection of these phenomena with the occurrence of {\bf cardiac arrhythmia}
is discussed, when the excitable medium under consideration is the
myocardium.

\noindent{\bf References}\\
J. Guckenheimer and P. Holmes : {\em Nonlinear Oscillations, Dynamical
Systems and Bifurcations of Vector Fields}, Springer, Berlin, 1983.\\
S. Wiggins : {\em Introduction to Applied Nonlinear Dynamical Systems
and Chaos}, Springer, Berlin, 1990.\\
J. Murray : {\em Mathematical Biology}, Springer, Berlin, 1989.\\
A. S. Mikhailov : {\em Foundations of Synergetics I : Distributed Active
Sytems}, Springer, Berlin, 1990.\\
G. Nicolis and I. Prigogine : {\em Exploring Complexity}, W. H. Freeman,
New York, 1989.\\
G. Nicolis : {\em Introduction to Nonlinear Science}, Cambridge University
Press, 1997.\\
T. Bountis : {\it Fundamental Concepts of Classical Chaos : Part I}, Open
Syst. Inf. Dyn {\bf 3(1)}, 23 - 95, 1995.\\
T. Bountis, C. F. Starmer and A. Bezerianos : {\it Wave Front Formation
 and Stationary Pulses in Excitable Media}, preprint, 1999.\\
\newpage

\begin{center}
\Large
{\bf  Quantum localization and cantori in the stadium billiard}
\normalsize

{\bf Giulio Casati}
  
{\em International Center for the study of Dynamical Systems,
Universit\'a degli Studi della Insubria, Como, Italy}
\end{center}                            

\normalsize
\noindent



We discuss the quantum behaviour of chaotic billiards which exhibit
classically diffusive behaviour. More precisely we consider the stadium
billiard and discuss how the interplay between quantum localization and
the rich structure of the classical phase space influences the quantum
dynamics. In particular we show how the presence of cantori in the
classical phase space affects the structure of the eigenfunctions and
the statistical properties of the eigenvalues. The analysis of this model
leads to new insight in the understanding of quantum properties of
classically chaotic systems.

\noindent{\bf References}
\\
Casati G and Prosen T 1999 {\em Phys. Rev. E} {\bf
59} R2516 and references therein.
\\

\newpage

\begin{center}
\Large
{\bf  Quantum Poincar\'e recurrences}
\normalsize

{\bf Giulio Casati}
  
{\em International Center for the study of Dynamical Systems,
Universit\'a degli Studi della Insubria, Como, Italy}
\end{center}                            

\normalsize
\noindent




We study quantum chaos in open dynamical systems and show that it is
characterized by quantum fractal eigenstates located on the underlying
classical strange repeller. The states with longest life times typically
reveal a scar structure on the classical fractal set.We also show that
quantum effects modify the decay rate of Poincar\'e recurrences $P(t)$ in
classical chaotic systems with mixed hierarchical structure of phase
space. It is shown that $P(t)$ has an algebraic decay with a universal power
law $p=1$,  due to tunnelling and localization effects. Experimental evidence
of such decay should be observable in mesoscopic systems and cold atoms.

\noindent{\bf References}
\\
Casati G, Maspero G  and Shepelyansky D L 1999 {\em Phys. Rev. Lett.} {\bf
82} 524 and references therein.
\\


\newpage


\begin{center}
\Large
{\bf Nonlinear dynamics and chaos in space plasmas}
\end {center}
\vspace{0.2cm}
\begin{center}
{\bf Abraham C.-L. Chian}\\

\vspace{0.2cm}
{\em Center for Subatomic Structure of Matter, University of Adelaide,
Australia\\
and National Institute for Space Research, S\~ao Jos\'e dos Campos,
Brazil}
\end{center} 
\vspace{0.2cm}                           
\noindent
We discuss the theory and observation of nonlinear phonomena in solar-system
plasmas. Plasma dynamics is governed by a variety of complex wave motions
resulting from the collective electrodynamic interactions involving plasma
field and plasma particles. Nonlinear wave-wave coupling in plasmas can occur
due to the growth of parametric instabilities. Rocket and satellite
observations have provided evidence of nonlinear wave interactions in the
solar system, in particular, in relation to the generation and propagation of
radio waves in solar corona, interplanetary medium and Earth's
magnetosphere. These radio waves represent the electromagnetic signature of
solar-terrestrial coupling and are useful for monitoring the
space weather.

\vspace{0.2cm}
\noindent
Nonlinear wave-wave coupling in plasmas can
be modelled as a dynamical system of coupled oscillators involving one, two or
higher-order wave triplets. For a three-wave dissipative system, the transition
from order to chaos may evolve via period doubling or intermittence. For a
two coupled three-wave Hamiltonian system, the transition may evolve via
the route of quasiperiodicity. We show that chaos in a dissipative three-wave
system can be controlled by applying a small sinusoidal wave to the system.

\vspace{0.2cm}
\noindent{\bf References}
\\
Chian A C-L 1999 Order and chaos in nonlinear wave interactions in
astrophysical and space plasmas, {\em Plasma Phys. Contr. Fusion} {\bf 41}, 
A437.
\\
Chian A C-L, Lopes S R and Alves M V 1994 Generation of auroral 
whistler-mode radiation via nonlinear coupling of Langmuir waves and
Alfv\'en waves, {\em Astron. Astrophys.} {\bf 290}, L13.
\\
Chian A C-L, Lopes S R and Abalde J R 1996, Hamiltonian chaos in two coupled 
three-wave parametric interactions with quadratic nonlinearity, {\em Physica D}
{\bf 99}, 269.
\\
Chian A C-L, Abalde J R, Alves M V and Lopes J R 1997, Coherent generation of
narrow-band circularly polarized radio bursts from the sun and flare
stars, {\em Solar Phys.} {\bf 173}, 199.
\\
Chian A C-L, Borotto F A and Gonzalez W D 1998, Alfv\'en intermittent 
turbulence driven by temporal chaos, {\em Astrophys. J.} {\bf 505}, 993.
\\
Chian A C-L and Abalde J R 1999, Nonlinear coupling of Langmuir waves with
whistler waves in the solar wind, {\em Solar Phys.} {\bf 184}, 403.
\\
Chian A C-L, Borotto, F A, Lopes S R and Abalde J R 1999, Chaotic dynamics of  
nonthermal planetary radio emissions, {\em Planetary Space Sci.}, in press.
\\
Lopes S R and Chian A C-L 1996, Controlling chaos in nonlinear
three-wave coupling, {\em Phys. Rev. E} {\bf 54}, 170.
\\
Lopes S R and Rizzato F B 1998, Chaos and energy redistribution in the
nonlinear interaction of two spatio-temporal wave triplets, 
{\em Physica D} {\bf 117}, 13.
\\
Pakter R, Lopes S R and Viana R L 1997, Transition to chaos in the 
conservative four-wave parametric interactions, {\em Physica D} {\bf 110}, 277.

\newpage

\begin{center}
\Large
{\bf  Robust spatiotemporal phenomena in one space dimension (I)}
\normalsize

{\bf Pierre Coullet}

{\em Institut Non-Lin\'eaire de Nice, Sophia-Antipolis, France}
\end{center}

\normalsize



\noindent
 We review some of the robust instabilities which are observed in one
dimensional dissipative systems. More particularly, this lecture is
devoted to the instabilities which are induced by the continuous spatial
coupling of local dynamical systems.

 The first of these instabilities was discovered by A.Turing in 1952. It
occurs already in two components reaction diffusion models when the
diffusion of the two species are very different. The singular nature of
this instability will be discussed. We will show in particular that
close to a particular bifurcation (Bogdanov-Takens codimension two
bifurcation) of the homogeneous reaction system, the instability can
occur with almost equal diffusion constant.

 The Benjamin-Feir-Kuramoto-Newell instability will then be considered.
This "phase" instability of spatially coupled limit cycles will be
described geometrically.

 We eventually discuss the question of the synchronization of coupled
limit cycles close to a homoclinic bifurcation. It will be shown that,
generically an instability always occurs in the vicinity of a homoclinic
bifurcation destroying the synchronization of the individual
oscillators. The nature of the instability depends on the nature of the
local dynamical system and the coupling considered. It can be either a
Benjamin-Feir-Kuramoto-Newell instability or a new amplitude instability
characterized by a period doubling which occurs at a finite wave-number.
A complete study of this new instability will be given.

 The aim of this lecture is to give a qualitative approach to the
phenomena by using their normal form description. Interactive
simulations will be used during the lecture in order to illustrate these
phenomena. Examples from mechanics, chemistry, optics will be
considered.


\noindent{\bf References}
\\
Turing, A.N. {\em Trans. Roy. Soc. Lond.}\textbf{\ B }237, 37
(1952).
\\
J. Guckenheimer, P. Holmes, {Nonlinear Oscillations, Dynamical Systems
and Bifurcation of Vector Fields}. Appl .Math. Sci. 41 , Springer,
New-York (1983).
\\
M. C. Cross and P. C. Hohenberg, {\em Rev. Mod. Phys.}
\textbf{65}, number 3, part II, 1993.
\\
T. B. Benjamin and J. E. Feir, {\em J. Fluid. Mech.}
\textbf{27}, 241, 1967.
\\
Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence",
Springer-Verlag, Berlin, 1984.
\\
A.A Andronov, and L. Pontyagin. {\em Dokl. Akad. Nauk,}
SSSR, \textbf{14}, 247, (1937). See also A. A. Andronov and al,
\textit{%
Theory of Bifurcation of Dynamical Systems on the Plane}, Israel Program
For
Sci. Trans. (1971)
\\
M. Argentina, P. Coullet, {\em  Phys. Rev. E} {\bf 56}, 2359  (1997)
\\
M. Argentina et P. Coullet,  {\em  Physica A}  {\bf 257}, 45 (1998)
\\
M. Argenina, P. Coullet and E. Risler, "Homoclinic instabilities in
spatially extended systems", preprint INLN (1999)
\\

\newpage

\begin{center}
\Large
{\bf  Robust spatiotemporal phenomena in one space dimension (II)}
\normalsize

{\bf Pierre Coullet}

{\em Institut Non-Lin\'eaire de Nice, Sophia-Antipolis, France}
\end{center}

\normalsize
\noindent
This second lecture is devoted to the description of localised
structures which occur in one-dimensional dissipative systems.


In the first part of the talk we study the propagative localised
structures known as excitable waves. Using the language of dynamical
system, we will describe the collision of two pulses. Depending on the
parameters the pulses either cross each other or coalesce. We show that
this phenomenon is actually a global bifurcation for the partial
differential equation. This bifurcation will be analyzed in the frame of
the geometrical theory of dynamical systems. This example provides a
unique case where the qualitative dynamical system theory have been used
to understand a phenomenon which cannot be understood in the frame of a
low dimensional dynamical system.

Static localised structures which can be observed in dissipative
one-dimensional systems have received recently a great deal of
attention. In the context of optics in particular they have been
considered as good candidates for building storage devices.
The question of the existence, the stability and the bifurcations of
these static localized structures will be related to the bistability
between a stable homogeneous state and a spatially periodic one. We will
show that the existence of a persistent front between a homogeneous
solution and a static periodic pattern is actually a consequence of the
transversality of heteroclinic orbits connecting a fixed point and a
limit cycle in reversible dynamical systems. This transversality
property implies the existence and the stability of localised solutions.
It also gives indications on the nature of the bifurcations of those
objects.

Examples from Optics and Chemistry will be considered. Interactive
numerical simulations will be presented during the talk in order to give an
intuition of the phenomena described.


\noindent{\bf References}
\\
M. Argentina, P. Coullet and L. Mahadevan, {\em Phys. Rev. Lett.}
\textbf{79}, 2803, (1997).
\\
M. Argentina, P. Coullet and V. Krinsky, ''Crossing of excitable waves
in the
Fizu-Nagumo model'', preprint (1998), submitted to {\em Trans. Phil.
Soc.}
\\
W. J. Firth and A. J. Scroggie, {\em Phys. Rev. Lett.}, \textbf{76},
1623, (1996).
\\
Y. Pomeau, {\em Physica D}  \textbf{23}, 3, (1985).
\\
C. Riera, P. Coullet and C. Tresser, "Localised structures in one space
dimension", preprint INLN (1999)
\\

\newpage

\begin{center}

\Large
{\bf  Chaos and what to do about it: An overview}
\normalsize

{\bf Predrag Cvitanovi\'c}

{\em Physics {\&} Astronomy, Northwestern Univ.,
Evanston, IL USA\\
and Niels Bohr Institute, Copenhagen, Denmark}

\end{center}                            

\normalsize
\noindent
That deterministic dynamics leads to chaos is no surprise to
anyone who has tried pool, billiards or snooker - that is what the game
is about - so we start our
course about what is chaos and what to do about it by a game of pinball.
This might seem a trifle trivial, but a pinball is to chaotic dynamics
what a pendulum is to integrable systems:
thinking clearly about what
is ``chaos'' in a pinball will help us tackle more difficult problems, such
as computing diffusion constants in deterministic gases, or
computing the Helium spectrum.

We all have an intuitive feeling for what a pinball does as it bounces
between the pinball machine disks, and only highschool level
Euclidean geometry is needed to describe the trajectory.
Turning this intuition into calculation will lead us,
in clear physically motivated steps, to almost everything one
needs to know about deterministic chaos:
from unstable dynamical flows, Poincar\'e sections,
Smale horseshoes, symbolic dynamics, pruning, 
discrete symmetries, periodic orbits,
averaging over chaotic sets, evolution operators, 
dynamical zeta functions, Fredholm determinants,
cycle expansions, quantum trace formulas and zeta functions,
and to the semiclassical quantization of helium.

\noindent{\bf Reference}
\\
Read chapter 1 and  appendix A of
P. Cvitanovi\'c, R. Artuso, R.~Mainieri, G. Vattay et al.,
{\em Classical and Quantum Chaos},
        {\tt http://www.nbi.dk/ChaosBook/}.

\newpage

\begin{center}

\Large
{\bf  Dynamics, qualitative}

\normalsize

{\bf Predrag Cvitanovi\'c}

{\em Physics {\&} Astronomy, Northwestern Univ.,
Evanston, IL USA\\
and Niels Bohr Institute, Copenhagen, Denmark}

\end{center}                            

\normalsize
\noindent
Confronted with a potentially chaotic dynamical system,
we analyze it through a sequence of three distinct stages; diagnose,
count, measure. First, we
determine the intrinsic {\em dimension} of the system - the minimum number of
degrees of freedom necessary to capture its essential dynamics.
If the system is
very turbulent (its attractor is of high dimension) we are, at present, out
of luck. We know only how to deal with the transitional regime between
regular motions and weak turbulence. In this regime the chaotic dynamics
is restricted to a space of
low dimension, the number of relevant parameters is small, and we can
proceed to the second step; we {\em count} and {\em classify}
all possible topologically distinct trajectories of the system.
If successful, we can proceed with the third step: investigate
the {\em weights} of the different pieces of the system.

In this lecture qualitative
dynamics of simple stretching and mixing flows is used to introduce Smale
horseshoes and symbolic dynamics, and
the topological dynamics is encoded by means of transition
matrices/Markov graphs.

We learn how to count and describe itineraries.
While computing the topological entropy
from transition matrices/Markov graphs, we encounter
our first zeta function.

By now we have covered for the first time the whole distance from diagnosing chaotic dynamics to
computing zeta functions. Historically, these topological zeta functions were the inspiration for
injecting statistical mechanics into computation of dynamical averages; Ruelle's
 zeta functions are a
weighted generalization of the counting zeta functions.
%


\noindent{\bf Reference}
\\
Read chapters 2 and  3 of
P. Cvitanovi\'c, R. Artuso, R.~Mainieri, G. Vattay et al.,
{\em Classical and Quantum Chaos},
        {\tt http://www.nbi.dk/ChaosBook/}.

\newpage

\begin{center}

\Large
{\bf  Global dynamics}

\normalsize

{\bf Predrag Cvitanovi\'c}

{\em Physics {\&} Astronomy, Northwestern Univ.,
Evanston, IL USA\\
and Niels Bohr Institute, Copenhagen, Denmark}

\end{center}                            

\normalsize

This lecture is the core of the course: 
we discuss the necessity of studying the averages of
observables in chaotic dynamics, and cast the formulas for averages
in a multiplicative form that motivates the introduction of
evolution operators.

In chaotic dynamics detailed prediction is impossible, as any finitely specified initial condition, no
matter how precise, will fill out the entire accessible phase space (similarly finitely grained) in finite
time. Hence for chaotic dynamics one does not attempt to follow individual trajectories to
asymptotic times; what is possible (and sensible) is description of the geometry of the set of
possible outcomes, and evaluation of the asymptotic time averages. Examples of such averages are
transport coefficients for chaotic dynamical flows, such as the escape rate, mean drift and the
diffusion rate; power spectra; and a host of mathematical constructs such as the generalized
dimensions, Lyapunov exponents and the Kolmogorov entropy. We shall now set up the formalism
for evaluating such averages within the framework of the periodic orbit theory. The key idea is to
replace the expectation values of observables by the expectation values of generating functionals.
This associates an evolution operator with a given observable, and leads to formulas for its
dynamical averages. 

If there is one idea that you should learn about dynamics, it happens in this
lecture(s) and it is this: there is a fundamental local - global duality which
says that (global) eigenstates are dual to the (local) periodic geodesics. For
dynamics on the circle, this is called Fourier analysis; for dynamics on
well-tiled manifolds this is called Selberg trace formulas and zeta functions;
and for generic nonlinear dynamical systems the duality is embodied in trace
formulas, zeta functions and spectral determinants that we will now
introduce. These objects are to dynamics what partition functions are to
statistical mechanics. The bold claim is that once you understand this,
classical ergodicity, wave mechanics and stochastic mechanics are but
special cases, to be worked out at your leisure. 

The strategy is this: Global averages such as escape rates can be extracted
from the eigenvalues of evolution operators. The eigenvalues are given by
the zeros of appropriate determinants. One way to evaluate determinants is
to expand them in terms of traces, log det = tr log. The traces are evaluated
as integrals over Dirac delta functions, and in this way the spectra of
evolution operators become related to periodic orbits. 

The rest of the course is making sense out of this objects and learning how
to apply them to evaluation of physically measurable properties of chaotic
dynamical systems. 

\noindent{\bf Reference}
\\
Read chapters 7, 8 and  9 of
P. Cvitanovi\'c, R. Artuso, R.~Mainieri, G. Vattay et al.,
{\em Classical and Quantum Chaos},
        {\tt http://www.nbi.dk/ChaosBook/}.


\newpage

\begin{center}

\Large
{\bf  
Cycle expansions: 
Semiclassical quantum mechanics
}

\normalsize

{\bf Predrag Cvitanovi\'c}

{\em Physics {\&} Astronomy, Northwestern Univ.,
Evanston, IL USA\\
and Niels Bohr Institute, Copenhagen, Denmark}

\end{center}                            

\normalsize

In last lecture we have derived a plethora of periodic orbit trace
formulas, spectral determinants and zeta functions. Now we learn how to
expand these as cycle expansions, series ordered by increasing
topological cycle length, and evaluate average quantities like escape rates.
These formulas are exact, and, when the winds are kind, highly convergent.
The pleasant surprise is that the terms in such expansions fall off
exponentially or even faster, so that a handful of shortest orbits suffices for
rather accurate estimates of asymptotic averages. 

The course now shifts gear to
recent advances in the periodic orbit theory of chaotic, non-integrable
systems, and the modern generalization of the De Broglie - Bohr quantization
of hydrogen atom. 

Instead of quantizing by suspending standing-wave configurations
on stable Keplerian orbits, one suspends the standing-wave configurations
on the infinity of unstable orbits. Such unstable periodic orbits are observed
experimentally in the helium atom, the hydrogen in strong external fields,
and other systems. 

This is what could have been done with the old quantum mechanics if
physicists of 1910's were as familiar with chaos as you by now are. 
The Gutzwiller trace formula together with the 
corresponding spectral determinant, the
central results of the semiclassical periodic orbit theory, are derived.

The helium atom spectrum can then be computed via spectral determinants.

\noindent{\bf Reference}
\\
Read chapters 9, 15 and 17 of
P. Cvitanovi\'c, R. Artuso, R.~Mainieri, G. Vattay et al.,
{\em Classical and Quantum Chaos},
        {\tt http://www.nbi.dk/ChaosBook/}.

\newpage

\begin{center}

\Large
{\bf  
Trace formulas for stochastic evolution operators
}

\normalsize

{\bf Predrag Cvitanovi\'c}

{\em Physics {\&} Astronomy, Northwestern Univ.,
Evanston, IL USA\\
and Niels Bohr Institute, Copenhagen, Denmark}

\end{center}                            

\normalsize


Intuitively, the noise inherent in any realistic system washes out
fine details and makes chaotic averages more robust. Quantum mechanical
$\hbar$ resolution of phase space implies that in semi-classical approaches
no orbits longer than the Heisenberg time need be taken into account.
We explore these ideas in some detail by casting stochastic dynamics
into path integral form and developing perturbative and nonperturbative
methods for evaluating such integrals. In the weak noise case the
standard perturbation theory is expansion in terms of Feynman diagrams.
Now the surprise; we can compute the same corrections faster and to a 
higher order in perturbation theory by integrating over the neighborhood 
of a given saddlepoint exactly by means of a nonlinear change of
field variables.  The new perturbative expansion appears more compact
than the standard Feynman diagram perturbation theory; whether it is
better than traditional loop expansions for computing field-theoretic
saddlepoint expansions remains to be seen, but for a simple system we study
the result is a stochastic analog of the Gutzwiller trace formula
with  the $\hbar$ corrections so far computed to five orders higher than
what has been attainable in the quantum-mechanical applications.
\\\\
{\bf Resume} 

A motion on a strange attractor can be approximated by
shadowing the orbit by a sequence of nearby periodic orbits of finite
length. This notion is here made precise by approximating orbits by
primitive cycles, and evaluating associated curvatures.  A curvature
measures the deviation of a longer cycle from its approximation by shorter
cycles;  the smoothness of the dynamical system implies exponential 
(or faster) fall-off for (almost) all curvatures.  
The technical prerequisite for implementing
this shadowing is a good understanding of the symbolic
dynamics of the classical dynamical system. 
The resulting cycle expansions offer an
efficient method for evaluating classical and
quantum periodic orbit sums; accurate estimates can be obtained by 
using as input the lengths and eigenvalues of
a few prime cycles.

To keep exposition simple we have here illustrated the utility of cycles
and their curvatures by a pinball game. 
Glancing back, we see that the formalism is very general, and
should work for any average over any  chaotic set which satisfies
two conditions: 
1. the weight associated with the observable under 
   consideration is multiplicative along the trajectory; 
2. the set is organized in such a way that the nearby points 
in the symbolic dynamics have nearby   weights. 

\noindent{\bf Reference}
\\
Read chapter 22 and the
take-home problem set for the next millennium in
P. Cvitanovi\'c, R. Artuso, R.~Mainieri, G. Vattay et al.,
{\em Classical and Quantum Chaos},
        {\tt http://www.nbi.dk/ChaosBook/}.
\\

\newpage

\begin{center}
\Large
{\bf Semiclassical matrix elements and cross sections}
\normalsize

{\bf Bruno Eckhardt}
  
{\em Fachbereich Physik, Philipps Universit\"at Marburg,
 Germany}
\end{center}                            

\normalsize
\noindent
The semiclassical trace formula of Gutzwiller can be extended 
to include diagonal matrix elements. With this it becomes possible 
to describe cross sections, e.g. for excitations of molecules. 
In the lectures I will discuss how this can be used to  
describe both universal features which otherwise could  
be discussed within random matrix theory and non-universal 
features connected with classical periodic orbits and their 
bifurcations. 
 
{\bf Lecture 1}:\\ 
Transitions in molecules are described by Franck-Condon 
factors, which are matrix elements between eigenstates 
of an operator and projectors on the initial state.  
In order to calculate them within periodic orbit theory 
the Gutzwiller trace formula has to be extended. 
For operators which are smooth in the classical limit 
this is easy, but for singular ones like projection 
operators higher order corrections have to be taken into  
account. This extension is particularly easy to achieve 
in the time domain, both for the calculation of 
the smooth average part and for the contributions 
from periodic orbits. \\
%
Given this expression for the matrix element weighted 
density of states one can now turn to a calculation of  
the correlation function or, more generally, the two point 
form factor. One of the aims of the lecture will be to  
describe how the random matrix results of Alhassid and  
Fyodorov can be obtained. 
 
{\bf Lecture 2}:\\ 
One of the distinguishing features of semiclassical 
periodic orbit theory is its potential to describe 
non-universal properties, specific to the system in 
question and beyond the reach of random matrix theory. 
The most prominent effects arise in the neighborhood 
of bifurcations where the semiclassical amplitudes 
diverge. Catastrophe theory can be used to calculate 
uniformized amplitudes (as shown in a number of 
recent works). The lecture will focus on two 
aspects of our work: \\
% 
i) In order to analyze the collective effects of  
many bifurcations we have studied  
the statistical behaviour in the standard map  
as a function of the control parameter $k$.  
This contributes to our understanding of the  
importance of bifurcations in long periodic orbits. \\
% 
ii) The divergent classical weights have a pronounced effect 
also on wave functions. One measure of localization in 
wave functions is the inverse participation ratio. 
In particular, for saddle node bifurcations, we find  
prelocalized states below the bifurcation and oscillations  
in the inverse participation ratio above the bifurcation. 
The oscillations can be described by Airy functions, 
which can be motivated (but not fully justified)  
by the semiclassical matrix element theory discussed in 
lecture 1. 
 

\noindent{\bf References}
\\
B. Eckhardt, S.~Fishman, K.~M\"uller and D.~Wintgen,
{Phys. Rev. A} {\bf 45}\rm, (1992) 3531--3539\\
%
B. Eckhardt, O. Agam, S. Fishman, J. Keating, J. Main und K. M\"uller
{Phys. Rev. E \bf 52}, (1995) 5893--5903 \\
%
B. Eckhardt and J. Main, {Phys. Rev. Lett. \bf 75}, (1995) 2300--2303 \\
%
B. Eckhardt, {Physica D \bf 109}, (1997) 53--58\\
%
B. H\"upper and B. Eckhardt, {Phys. Rev. A \bf 57}, (1998) 1536--1547\\

\newpage


\begin{center}
\Large
{\bf  Chaotic Huygens phenomenon \\
and on-off intermittency}
\normalsize

{\bf Hirokazu Fujisaka}

{\em Department of Applied Analysis and Complex Dynamical Systems,\\
Graduate School of Informatics, Kyoto University,
 Japan}
\end{center}
\normalsize
\noindent


Intermittency phenomena is quite ubiquitous in nonlinear dynamics. The most
famous one is observed in fluid turbulence.  Nowadays it is
believed that the intermittency is one of key concepts for characterization
or
analyzing various phenomena in nonlinear dynamical systems.  The most
familiar one is the Pomeau-Manneville intermittency, which is observed when
a periodic motion is either destroyed or undergoes the instability when an
external control parameter is changed.  On the other hand, when a particular
chaotic motion undergoes an instability, an intermittency with statistical
characteristics different from that of PM comes to appear.  This is called
the
modulational intermittency or
on-off intermittency.  In particular, under certain condition two identical
chaotic oscillators synchronize, (chaotic Huygens phenomenon). When the
synchronization becomes unstable, we typically observe the on-off
intermittency, (Fujisaka-Yamada 1985).
\par
On-off intermittency is one of typical evolutions in nonlinear dynamics.
In the present lecture, I will give several different mathematical
models with both small and large number of degrees of freedom and also discuss
physical system. Then I will review general criteria on the observability of
on-off intermittency and give statistical laws.  Introducing a stochastic
model, I will show that these statistical laws can be derived.
Furthermore the possibility of the construction of solvable models of on-off
intermittency will be addressed.

\noindent{\bf References}
\\
Fujisaka H and Yamada T 1985 {\em Prog. Theor. Phys.} {\bf 74} 918
\\
Fujisaka H and Yamada T 1986 {\em Prog. Theor. Phys.} {\bf 75} 1087
\\
Fujisaka H and Yamada T 1983 {\em Prog. Theor. Phys.} {\bf 69} 32
\\
Yamada T and Fujisaka H 1983 {\em Prog. Theor. Phys.} {\bf 70} 1240
\\
Platt N, Spiegel E.-A. and Tresser C 1993 {\em Phys. Rev. Lett.} {\bf 70}
279
\\
Heagy J.-F., Platt N and Hammel S.-M. 1994 {\em Phys. Rev. E} {\bf 49} 1140
\\
Ott E and Sommerer J.-C. 1994 {\em Phys. Lett. A} {\bf 188} 39
\\
Lai Y.-C. and Grebogi C 1995 {\em Phys. Rev. E} {\bf 52} R3313
\\
Harada T, Hata H and Fujisaka H 1999 {\em J. Phys. A} {\bf 32}
\\
Fujisaka H, Matsushita S and Yamada T 1997 {\em J. Phys. A} {\bf 30} 5697
\\
Fujisaka H, Ouchi K, Hata H, Masaoka B and Miyazaki S 1998 {\em Physica D}
{\bf 114} 237
\\
Yamada T, Fukushima K and Yazaki T 1989 {\em Prog. Theor. Phys.
Suppl.} No.99 120
\\
Cenys A, Namajunas A, Tamserius A and Schneider T 1996 {\em Phys. Lett. A}
{\bf 213} 259
\\
Rodelsperger F, Cenys A and Benner H 1995 {\em Phys. Rev. Lett.} {\bf 75}
2594
\\
Yamada T and Fujisaka H 1986 {\em Prog. Theor. Phys.} {\bf 76} 582
\\
Hata H and Miyazaki S 1997 {\em Phys. Rev. E} {\bf 55} 5311
\\
Miyazaki S and Hata H 1998 {\em Phys. Rev. E} {\bf 58} 7172


\newpage

\begin{center}
\Large
{\bf  ``Thermodynamics" approach \\
to on-off intermittency }
\normalsize

{\bf Hirokazu Fujisaka}

{\em Department of Applied Analysis and Complex Dynamical Systems,\\
Graduate School of Informatics, Kyoto University, Japan}
\end{center}

\normalsize
\noindent
Large deviation theory (LDT) in the probability theory is the
straightforward extension of the
central limit theorem which has played a very important role in statistical
mechanics. Since LDT addresses the large fluctuations observed in
nonlinear dynamics, typically for intermittency dynamics, it is natural to
expect that LDT can also be a powerful theoretical framework to analyze
chaotic dynamics.
\par
In the lecture I first review LDT from the physicist point of view, and
discuss how the statistical quantities characterizing LDT are determined
both experimentally and theoretically.  Then I will apply the LDT
approach to examples in both chaotic and stochastic systems including
the stochastic model of on-off intermittency. It is shown that the LDT
analysis yields many different aspects of fluctuations which cannot be
captured by
the traditional statistical analysis based on two-time correction functions.
\par
The short time dynamics of on-off intermittency shows a characteristic
quite different from the long time dynamics which can be discussed
within LDT.  To analyze it, we introduce the concept of
a self-similar dynamics for on-off fluctuations.  Although the conventional
LDT cannot be applied for such short time dynamics, I will show that
LDT is ``again" applicable.


\noindent{\bf References}
\\
Fujisaka H and Inoue M 1987 {\em Prog. Theor. Phys.} {\bf 77} 1334
\\
Fujisaka H and Shibata H 1991 {\em Prog. Theor. Phys.} {\bf 85} 187
\\
Fujisaka H 1992 in {\em From Phase Transitions to Chaos } eds. G Gy\"{o}rgyi
et al (Singapore: World Scientific) pp 34-48
\\
Beck C and Schl\"{o}gl F 1993 {\em Thermodynamics of Chaotic Systems - An
Introduction -} (Cambridge: Cambridge University Press)
\\
Ott E 1993 {\em Chaos in Dynamical Systems} (Cambridge: Cambridge University
Press)
\\
Fujisaka H and Yamada T 1985 {\em Prog. Theor. Phys.} {\bf 74} 918
\\
Fujisaka H and Yamada T 1986 {\em Prog. Theor. Phys.} {\bf 75} 1087
\\
Platt N, Spiegel E.-A. and Tresser C 1993 {\em Phys. Rev. Lett.} {\bf 70}
279
\\
Heagy J.-F., Platt N and Hammel S.-M. 1994 {\em Phys. Rev. E} {\bf 49} 1140
\\
Ott E and Sommerer J.-C. 1994 {\em Phys. Lett. A} {\bf 188} 39
\\
Lai Y.-C. and Grebogi C 1995 {\em Phys. Rev. E} {\bf 52} R3313
\\
Yamada T and Fujisaka H 1986 {\em Prog. Theor. Phys.} {\bf 76} 582
\\
Yamada T and Fujisaka H 1990 {\em Prog. Theor. Phys.} {\bf 84} 824
\\
Fujisaka H and Yamada T 1993 {\em Prog. Theor. Phys.} {\bf 90} 529
\\
Miyazaki S and Hata H 1998 {\em Phys. Rev. E} {\bf 58} 7172
\\
Fujisaka H and Yamada T 1987 {\em Prog. Theor. Phys.} {\bf 77} 1045

\newpage

\begin{center}
\Large
{\bf  The chaotic hypothesis: a review and some applications}
\normalsize

{\bf Giovanni Gallavotti}
  
{\em Fisica, Universit\'a di Roma, Rome, Italia}
\end{center}                            

\normalsize
\noindent
The chaotic hypothesis is a reformulation of a principle stated by
Ruelle in the early '70s (1973). It is a principle of very ambitious
nature as it is a proposal for the extension of the ergodic hypothesis
to nonequilibrium statistical mechanics systems. The entropy creation
rate in a thermostated system by thermostats of mechanical nature, in
which the energy cannot grow in spite of nonconservative external
forces acting on the system, is identified with the divergence of the
equations of motion. The fluctuations of the latter obey a universal
relation: the fluctuation theorem. Further more heuristic
developments of the chaotic hypothesis suggest introducing the notion
of nonequilibrium ensembles: they offer the new idea that in systems
out of equilibrium it can happen that the equations of motion
themselves may become ``parameters'' of the nonequilibrium
ensembles. And systems obeying different equations of motions may be
equivalent in a sense analogous to the equivalence of the canonical
and microcanonical ensembles, for instance. Similar ideas can be
discussed in the theory of developed turbulence, which is the field in
which the original ideas of Ruelle were developed.

\noindent{\bf References}
\\
{ Bonetto, F., Gallavotti, G.}: {\it Reversibility,
coarse graining and the chaoticity principle, Communications in
Mathematical Physics,} {\bf 189}, 263--276, 1997.
\\
{ Bonetto, F., Gallavotti, G., Garrido, P.}: {\it Chaotic
principle: an experimental test, Physica D,} {\bf 105}, 226--252,
1997.
\\
{ Bonetto, F., Gallavotti, G., Garrido, P.}: {\it
Chaotic principle: an experimental test, Physica D,} {\bf 105},
226--252, 1997.
\\
{ Gallavotti, G.}: {\it Some rigorous results about 
3D Navier Stokes, in "Turbulence in spatially extended systems",} p.45--
74, ed. R. Benzi, C. Basdevant, S. Ciliberto, Nova Science Publishers,
Commack (NY), 1993.
\\
{ Gallavotti, G.}: {\sl Ergodicity, ensembles,
irreversibility in Boltzmann and beyond, Journal of Statistical
Physics.}  {\bf 78}, 1571--1589, 1995.
\\
{ Gallavotti, G.}: {\it Extension of Onsager's
reciprocity to large fields and the chaotic hypothesis,
Physical Review Letters,} {\bf 77}, 4334--4337, 1996.
\\
{Gallavotti, G.}: {\it New methods in nonequilibrium gases and
  fluids}, Proceedings of the conference {\it Let's face chaos through
  nonlinear dynamics}, U. of Maribor, 24 june-- 5 july 1996,
  ed. M. Robnik, in print in Open Systems and Information Dynamics,
  Vol. 6, 1999.
\\
{Gallavotti, G., Ruelle, D.}: {\it SRB states and nonequilibrium
   statistical mechanics close to equilibrium, Communications in
   Mathematical Physics,} {\bf 190}, 279--285, 1997.
\\
{ Gallavotti, G.}: {\it Dynamical ensembles equivalence
in fluid mechanics, Physica D,} {\bf 105}, 163--184, 1997.
\\
{ Gallavotti, G.}: {\it Chaotic hypothesis: Onsager reciprocity and 
   fluctuation--dissipation theorem,
   Journal of Statistical Phys.,} {\bf 84}, 899--926, 1996.
\\
{Gallavotti, G.}: {\it
Equivalence of dynamical ensem\-bles and Navier Stokes equations,
Physics Letters,} {\bf 223}, 91--95, 1996.
\\
{ Gallavotti, G.}: {\it SRB states and nonequilibrium statistical
   mechanics close to equilibrium, Communications in Mathematical Physics,}
   {\bf 190}, 279--285, 1997 (D. Ruelle)
\\
{Gallavotti, G.}: {\it Ipotesi per una introduzione alla Meccanica
dei Fluidi, ``Quaderni del CNR-GNFM'',} vol. {\bf 52}, p. 1--428,
Firenze, 1997.
\\
{Gallavotti, G.}: {\it Chaotic principle: some
applications to developed turbulence, Journal of Statistical Physics,}
{\bf 86}, 907--934, 1997.
\\
{ Gallavotti, G.}: {\it Reversible Anosov maps and
large deviations, Mathematical Physics Electronic Journal, MPEJ,}
({\tt http:// mpej.unige.ch}), {\bf 1}, 1--12, 1995.
\\
{Gallavotti, G.}: {\it Breakdown and regeneration of time reversal
  symmetry in nonequilibrium Statistical Mechanics, Physica D,} {\bf 112},
  250--257, 1998.
\\
{ Gallavotti, G., Cohen, E.G.D}: {\it Dynamical
ensembles in nonequi\-li\-brium statistical mechanics, Physical Review
Letters,} {\bf 74}, 2694--2697, 1995.
\\
{Gallavotti, G., Cohen, E.G.D}: {\it Dynamical
ensembles in stationary states, Journal of Statistical Physics,} {\bf
80}, 931--970, 1995.
\\
{Gallavotti, G.}: {\it Extension of Onsager's
reciprocity to large fields and the chaotic hypothesis,
Physical Review Letters,} {\bf 77}, 4334--4337, 1996.
\\
{Cohen, E.G.D., Gallavotti, G.}: {\it Note on Two Theorems in
Nonequilibrium Statistical Mechanics}, mp$\_$arc 99-88, cond-mat
9903418.
\\
{ Gallavotti, G.}: {\it Fluctuation patterns and conditional 
  reversibility in
  nonequilibrium systems}, in print on Annales de l' Institut
  H. Toincar\'e, mp$\_$arc at  math. utexas. edu \#97-124, chao-dyn at 
  xyz. lanl. gov \#9703007.
\\
{ Gallavotti, G.}: 
{\it M\'ecanique statistique hors \'equilibre: l'h\'eritage
  de Boltzmann}, mp$\_$arc at math. utexas. edu: \# 98-54,
  chao-dyn at xxx. lanl. gov: chao-dyn/9802012.
\\
{Gallavotti, G.}: {\it Breakdown and regeneration of time reversal
  symmetry in nonequilibrium Statistical Mechanics, Physica D,} {\bf 112},
  250--257, 1998.
\\
{Gallavotti, G.}: {\it Chaotic dynamics, fluctuations,
non-equilibrium ensembles, Chaos,} {\bf 8}, 384--392, 1998.
\\
{Gallavotti, G.}: {\it A local fluctuation theorem, Physica A,}
qq   {\bf 263}, 39--50, 1999.
\\
{ Gallavotti, G.}: {\it Chaotic Hypothesis and Universal Large Deviations
   Properties, Documenta Mathematica,} extra volume ICM98, vol. {\bf I}, 
   p. 205--233, 1998, also in chao-dyn 9808004.
\\
{ Gallavotti, G.}: {\it Trattatello di Meccanica Statistica, 
``Quaderni del CNR-GNFM'',} vol. {\bf 50}, p. 1--350, Firenze, 1995.
English version available on {\tt http://ipparco.roma1.infn.it.}
\\
{Gallavotti, G.}: {\it Ipotesi per una introduzione alla Meccanica
dei Fluidi, ``Quaderni del CNR-GNFM'',} vol. {\bf 52}, p. 1--428,
Firenze, 1997.
\\

Most of the papers (and books) can be freely downloaded from:
{\tt http://ipparco.roma1.infn.it}

e-mail: giovanni at ipparco.roma1.infn.it

\newpage

\begin{center}
\Large
{\bf  Quantum diffusion and fractal spectra}
\normalsize

{\bf T. Geisel and R. Ketzmerick}
  
{\em Max-Planck-Institut f\"ur Str\"omungsforschung, G\"ottingen,\\
and Fakult\"at Physik, Universit\"at G\"ottingen, 
Germany}
\end{center}                            

\normalsize
\noindent


While in the past the influence of classical chaos on quantum spectra was investigated
intensely in the case of discrete spectra, the cases of uncountable and in particular 
fractal spectra have received attention only recently. We show that the latter are 
associated with specific dynamical properties:
\\
(i) They generate unbounded quantum 
mechanical diffusion processes in contrast to the dynamical localization known from
the kicked rotator and other systems, and
\\
(ii) cause an algebraic decay of correlations 
as a function of time. We show that these properties are governed by generalized
dimensions $D_q$ of the spectrum and eigenfunctions.
\\
In particular, the asymptotic decay of correlations is determined by the dimension $D_2$ of
the spectral measure, while the diffusive spreading of wave packets is related to the
ratio of the dimensions $D_2$ of the spectral measure and of the eigenfunctions,
respectively.

Such fractal spectra show up for Bloch electrons in magnetic fields, in particular in 
artificial superlattices of quantum dots and antidots on semiconductor heterojunctions
(electronic Sinai billiards). Quantum mechanical descriptions based on Harper's equation
lead to a fractal spectrum known as Hofstadter's butterfly. This model has an integrable
classical limit, however, and thus fails in these superlattices, where chaotic trajectories
prevail. More realistic models that allow for chaos in the classical limit exhibit
peculiar metal-insulator transitions and transitions between absolutely continuous and pure
point spectra induced by classical chaos. We show that they can be understood in terms of
avoided band crossings.

\noindent{\bf References}
\\
T. Geisel, R. Ketzmerick, G.Petschel, in {\em Quantum Chaos - 
Between Order and Disorder},\\
eds. G. Casati and B. V. Chirikov (Cambridge University Press, Cambridge 1995) p. 633.
\\
R. Ketzmerick, K. Kruse, S. Kraut, T. Geisel, {\em Phys. Rev. Lett.}, {\bf 79}, 1959 (1997).
\\
R. Ketzmerick, K. Kruse, T. Geisel, {\em Phys. Rev. Lett.},  {\bf 80}, 137 (1998).

\newpage

\begin{center}
\Large
{\bf  Pattern formation in the developing visual cortex}
\normalsize

{\bf T. Geisel and F. Wolf}
  
{\em Max-Planck-Institut f\"ur Str\"omungsforschung, G\"ottingen,\\
and Fakult\"at Physik, Universit\"at G\"ottingen,
Germany}
\end{center}                            

\normalsize
\noindent


Nonlinear dynamics appears to play a role not only in chaos and faces but also in the 
information 
processing in brains and the development of brains. Visual and other external information is 
often represented
in the cortex in the form of neural maps (e.g. retinal space or ocularity into 
cortical space),
which exhibit characteristic patterns varying from species to species. These maps are
known in great detail from neurobiological studies which employ high resolution 
techniques e.g. 
like optical imaging. Some of the best studied examples are the orientation preference
 maps,
cortical neurons exhibiting different preferred orientations of visual stimuli.
\\
\\
It has long been hypothesized that the orientation selectivity of single neurons and
the spatial pattern of orientation preferences in primary visual cortex arise due to 
activity-dependent
self-organization of neuronal circuitry during early life. We demonstrate the relevance
of nonlinear dynamics for the theoretical analysis of this self-organization process.
\\
\\
The talk will present theoretical work applied to recent experiments on strabismic and 
normal cats and will address the following questions: 
\\
Which mechanisms control the formation of neural maps? 
\\
How does experience influence the patterns?
\\
Does the cortex undergo a symmetry breaking bifurcation during development?
\\
Can we explain the different pattern layout in different species from common principles?

\noindent{\bf References}
\\
H.-U. Bauer, T. Geisel, K. Pawelzik, F. Wolf, in {\em From Statistical Physics to Statistical Inference and Back},
eds. J.-P. Nadal and P. Grassberger (Kluwer Academic Publ., 1994) p. 249.
\\
H.-U. Bauer, T. Geisel, K. Pawelzik, F. Wolf,
{\em Spektrum der Wissenschaft} (1996), {\bf 4}, 38.
\\
F. Wolf, T. Geisel, {\em Nature},  {\bf 395}, 73 (1998).

\newpage

\begin{center}
\Large
{\bf  Resetting and entraining biological oscillators}
\normalsize

{\bf Leon Glass}
  
{\em Department of Physiology,\\
McGill University, Montreal, Quebec, Canada}
\end{center}                            

\normalsize
\noindent
Biological oscillators are ubiquitous and underly many important
biological functions including the heartbeat, respiration, and
reproduction. Each of these rhythms is generated and regulated by
complex mechanisms and feedback.  Despite the different mechanisms,
from a mathematical perspective, they might all be associated with a
stable limit cycle oscillations in an appropriate nonlinear equation.
Based on this assumption, it is possible to make a number of different
predictions concerning the effects of single and periodic stimuli
delivered to the oscillator (Winfree, 1980; Glass and Mackey, 1988).
The main results are: single stimuli might either advance or delay the
onset of the next oscillation depending on the amplitude and phase of
the stimulus; plots of the resetting functions as a function of the
phase of the stimuli fall into distinct classes which can be
classified based on continuity considerations; and if the limit cycle
is strongly attracting, then based on the analysis of the effects of
single stimuli on the oscillations it is possible to predict the effects
of periodic stimuli 
(Guevara and Glass, 1982; Keener and Glass, 1984). These theoretical
predictions have succeeded in helping to identify chaotic dynamics
during periodic stimulation delivered to spontaneously beating
aggregates of heart cells (Guevara, Glass, Shrier, 1981).  I also
discuss the limitations of the above theory. In particular, if
relaxation to the limit cycle is not instantaneous, then the analysis
becomes much more difficult (Glass and Sun, 1994). Also the effects of
stimulation can change the properties of the limit cycle oscillator
(Kunysz, Glass, Shrier, 1995).  Although the geometric approach I
describe in this and subsequent lectures is conceptually simple, there
are many subtleties. It is hard to appreciate these without doing
computations. Students who are particularly motivated by this topic,
would benefit by reading Chapters 6 and 7 in Glass and Mackey (1988)
prior to the summer school and attempting Problem 14(B) on p. 208.  I
will treat anyone who makes a decent effort to a free beer (or
coffee)!


\noindent{\bf References}
\\
L. Glass, M.C. Mackey. {\it From Clocks to Chaos: The
Rhythms of Life.} (Princeton University Press, Princeton, 1988).
\\
L. Glass, J. Sun. Periodic forcing of a limit cycle oscillator:
Fixed points, Arnold tongues, and the global organization of
bifurcations. {\it Physical Review E} {\bf 50}, 5077-5084 (1994).
\\
      M.R. Guevera, L. Glass.  Phase-locking, period-doubling
       bifurcations and chaos in a mathematical model of a
       periodically driven biological oscillator: A theory for the
       entrainment of biological oscillators and the generation of
       cardiac dysrhythmias. {\it Journal of Mathematical Biology}
       {\bf 14}, l-23 (1982).
\\
M.R. Guevera, L. Glass, A. Shrier. Phase-locking, period-doubling
       bifurcations and irregular dynamics in periodically stimulated
       cardiac cells. {\it Science} {\bf 214}, 1350-1353 (1981).
\\
J. Keener, L. Glass.  Global
       bifurcations of a periodically forced nonlinear
       oscillator. {\it Journal of Mathematical Biology} {\bf21},
       175-190 (1984).  
\\
A. Kunysz, L. Glass, A. Shrier. Overdrive suppression of
spontaneously beating chick heart cell aggregates: Experiment and
theory. {\it American Journal of Physiology} {\bf 269} ({\it Heart,
Circulatory Physiology} {\bf 38}), H1153-H1164 (1995).  
\\
A. T. Winfree {\it The Geometry of Biological Time} (Springer-Verlag,
New York, 1980). 
\\

\newpage

\begin{center}
\Large
{\bf  Dynamics of cardiac arrhythmias}
\normalsize

{\bf Leon Glass}
  
{\em Department of Physiology,\\
McGill University, Montreal, Quebec, Canada}
\end{center}                            

\normalsize
\noindent
Cardiac arrhythmias are abnormal cardiac rhythms in which there is
abnormal impulse generation, abnormal conduction of excitation, or
both. Since the heart is a nonlinear system, the classifications of
cardiac arrhythmias (by cardiologists) should correspond to
classifications of bifurcations in nonlinear systems (by
mathematicians). This lecture introduces basic cardiac anatomy and
electrophysiology (Goldberger and Goldberger, 1994; Glass, 1996).  Two
arrhythmias that have successfully been analyzed mathematically are
heart block, in which some of the excitations in the upper chambers of
the heart do not conduct successfully to the lower chambers of the
heart (Shrier et al., 1987) and parasystole, in which there is a
competition for control of the heart by two competing pacemakers
(Glass, Goldberger, B\'{e}lair, 1984; Bub and Glass, 1995). These two
arrhythmias are not particularly dangerous. However, there has also been
mathematical analysis of more dangerous arrhythmias such as
tachycardias, in which the heartbeat is abnormally fast.  One
conceptual model of tachycardias is that the excitation is associated
with a pulse that circulates on a one-dimensional ring.  Surprisingly,
this simple model leads to the following predictions: if the
circumference of the ring is too small, either reentry will be
impossible or there will be quasiperiodic fluctuations of cycle time
(Courtemanche, Glass, Keener, 1993); a single stimulus will either
annihilate or reset the rhythm (Glass and Josephson, 1995); the
effects of periodic stimulation can be computed based on the analysis
of the resetting induced by a single stimulus (Nomura and Glass,
1996). I will describe recent efforts to test these predictions
experimentally. Anyone wishing to prepare for this lecture should try
to analyze the following not so hypothetical situation. An electrode
is inserted into the right atrium of a patient's heart. The electrode
is controlled by a computer under your guidance. The electrode delivers
brief shocks that are capable of exciting the heart (this is the
principle of an artificial pacemaker).  What different rhythms could
you elicit from the heart as you change the frequency and amplitude of
the stimulation?

\noindent{\bf References}
\\
G. Bub,
L. Glass.  Bifurcations in a discontinuous circle map: A theory for a
chaotic cardiac arrhythmia. {\it International Journal of Bifurcation
and Chaos} {\bf 5}, 359-371 (1995).  
\\
Courtemanche, L. Glass, J.P. Keener. Instabilities of
a propagating pulse in a ring of excitable media. {\it Physical Review
Letters} {\bf 70}, 2182-2185 (1993).
\\
L. Glass. Dynamics of cardiac arrhythmias. {\it Physics Today} {\bf
49} (Number 8, Part 1) 40-45 (1996).
\\
L. Glass, A. Goldberger, J. B\'{e}lair. Dynamics of pure
       parasystole. {\it American Journal of Physiology} {\bf 251}
       {\it (Heart Circ. Physiol.} {\bf 20}), H841-H847 (1986).
\\
L. Glass,
M.E. Josephson. Resetting and annihilation of reentrant abnormally
rapid heartbeat. {\it Physical Review Letters} {\bf 75}, 2059-2063
(1995).  
\\
A. L. Goldberger, E. Goldberger {\it Clinical Electrocardiology, Fifth
Edition} (Mosby, St. Louis, 1994). 
\\
T. Nomura, L. Glass.  Entrainment and termination of reentrant
wave propagation in a periodically stimulated ring of excitable
media. {\it Physical Review E} {\bf 53}, 6353-6360 (1996).  
\\
A. Shrier,
       H. Dubarsky, M. Rosengarten, M.R. Guevara, S. Nattel,
       L. Glass.  Prediction of complex atrioventricular conduction
       rhythms in humans using the atrioventricular nodal recovery
       curve. {\it Circulation} {\bf 76}, 1196-1205 (1987).  

\newpage

\begin{center}
\Large
{\bf  Controlling cardiac arrhythmias}
\normalsize

{\bf Leon Glass}
  
{\em Department of Physiology,\\
McGill University, Montreal, Quebec, Canada}
\end{center}                            

\normalsize
\noindent
A variety of different approaches have been devised to help control
cardiac arrhythmias.  Therapeutic approaches to controlling cardiac
arrhythmias include drug therapy, electrical therapy such as
pacemakers or implantable defibrillators, and ablation in which a
portion of the heart is destroyed. To the best of my knowledge, none
of the therapies currently used have been developed based on a
mathematical analysis of nonlinear dynamics of cardiac arrhythmias.
This lecture further analyzes dynamics of arrhythmias and demonstrates
approaches to controlling arrhythmias in model systems using
mathematical analyses. The first part of the talk describes repetitive
paroxysmal tachycardias, in which there is an incessant starting and
stopping of arrhythmia (Parkinson and Papp, 1967), and experimental
models displaying similar dynamics. The experimental models are
generated by: stimulation of spontaneously oscillating cardiac cells
with a fixed delay after an activation (Kunysz, Shrier, Glass, 1997);
stimulation of the atria of rabbit hearts at a fixed delay after the
activation of the ventricles (Sun et al., 1995); and culturing heart
cells in a monolayer (Bub, Glass, Shrier, 1998). In all three
examples, the interactions between pacemakers, conduction and
``fatigue'' leads to bursting rhythms. I then describe mathematical
methods being developed to control cardiac arrhythmias. An early
example adopted methods of ``chaos'' control to an experimental model
system (Garfinkel et al., 1992) but see also (Glass and Zeng, 1994).
More recent work tunes the adjustable delay in a reentrant circuit to
control an alternating rhythm (Hall et al., 1997). Finally, analysis
of wave propagation might help cardiologists successfully locate areas
to target for ablation (Hall and Glass, 1999). In view of the
nonlinear nature of the heart, and the complex dynamics of cardiac
arrhythmias it seems inevitable that theoretical analyses based on
nonlinear dynamics will lead to successful new therapies, but we are
still awaiting the first demonstration.




\noindent{\bf References}
\\
G. Bub, L. Glass, A. Shrier. Bursting calcium rotors in cultured cardiac
myocyte monolayers. {\it Proceedings of the National Academy of
Sciences (USA)} {\bf 95}, 10283-10287 (1998). 
\\
A. Garfinkel, M. L. Spano, W. L. Ditto, J.N. Weiss. Controlling
cardiac chaos.  {\it Science} {\bf
257}, 1230-1235 (1992).  
\\
L. Glass, W. Zeng. Bifurcations in flat-topped maps
and the control of cardiac chaos.  {\it International Journal of
Bifurcation and Chaos} {\bf 4}, 1061-1067 (1994).
\\
K. Hall, L. Glass. Locating ectopic foci. {\it Journal of 
Cardiovascular Electrophysiology} {\bf 10} 387-398 (1999). 
\\
K. Hall, D. J. Christini, M. Tremblay, J. J. Collins, L. Glass,
J. Billette. Dynamic control of cardiac alternans. {\it Physical
Review Letters} {\bf 78}, 4518-4521 (1997). 
\\
A. Kunysz, A. Shrier, L. Glass.  Bursting behavior
during fixed delay stimulation of spontaneously beating chick heart
cell aggregates. {\it American Journal of Physiology} {\bf 273} ({\it
Cell Physiology} {\bf 42}), C331-C346 (1997).
\\
J. Parkinson, C. Papp. Repetitive paroxysmal tachycardia {\it Br.
Heart J} {\bf 9}, 241-262 (1947). 
\\
J. Sun, F. Amellal,
L. Glass, J. Billette. Alternans and period-doubling bifurcations in
atrioventricular nodal conduction. {\it Journal of Theoretical
Biology} {\bf 173}, 79-91 (1995). 
\\

\newpage

\begin{center}
\Large
{\bf  Growth phenomena in cellular automata}
\normalsize

{\bf Janko Gravner}
  
{\em Mathematics Department,\\
University of California, Davis}
\end{center}                            

\normalsize
\noindent
A cellular automaton (CA) is a spatially 
distributed dynamical system which evolves via 
local, homogeneous, parallel updating, which 
may be deterministic or random. 
Our focus will be on self--organization: 
a tendency toward large--scale, coherent 
structure from disordered initial states. 
The examples will mostly feature irreversible 
dynamics outside the scope of traditional 
statistical mechanics, which we loosely call
{\it growth models\/}. We will mainly review development
of mathematical theory developed for several classes 
of CA, but will also present insights gathered
from simulations and open problems. 

In the first talk, we will start with the basic definitions and 
a menagerie of illustrative examples. 
We will present an analysis of {\it additive\/} CA, 
discussing particularly replication and generation of
fractals. Another basic example is {\it oriented 
percolation\/}, a prototype for a phase transition
and the fundamental building block in rescaling 
arguments. A model for an excitable medium 
will provide an example of locally periodic CA. 
We will then briefly touch on 
measures of complexity of CA rules and on
the most famous of all CA, the {\it Game of Life\/}. 

The second talk will go into more depth on 
the subject of nucleation and growth properties 
of CA. The first subject will be shape theory 
for monotone growth, reviewing Wulff construction 
for deterministic and random CA. By contrast, 
non-monotone cases can only be rigorously 
handled when a recursive analysis is available. 
We will 
present some examples of recursive solidification 
which lead to fractal growth and are extremely sensitive 
to random perturbations. Next, we will 
mention some growth models based
on random walks that generate isotropic shapes, and other, 
much less tractable, 
which yield random fractals.  
Finally, we will review nucleation theory in both 
supercritical and critical monotone CA. Specifically, we will 
review power laws, density decays and last holes 
problems, emergent shapes, and effects of 
polluted environments. Time permitting, some 
percolation effects in excitable media will 
be discussed. 

\noindent{\bf References}
\\
Aizenman M and Lebowitz J 1988 {\em J. Phys. A: Math. Gen} {\bf 21} 3801. 
\\
Bramson M and Neuhauser C 1994 {\em Annals of Probability} {\bf 22} 244. 
\\
Bramson M, Griffeath D, and Lawler G 1992 {\em Annals of Probability} 
{\bf 20} 2117. 
\\
Chopard B and Droz M 1999 {\it Cellular Automata Modeling 
of Physical Systems\/}. Cambridge University Press. 
\\
Fisch R, Gravner J, and Griffeath D 1993 {\em Annals 
of Applied Probability} {\bf 3} 935. 
\\
Gravner J 1999 {\it Growth phenomena in cellular automata\/}. Preprint. 
\\
Gravner J and Griffeath D 1996 {\em Annals of Probability} {\bf 24}  1752.
\\
Gravner J and Griffeath D 1998 {\em Advances in Applied Mathematics} {\bf 21} 
241. 
\\
Gravner J and Griffeath D 1998 {\it Scaling laws for a class of critical 
cellular automaton growth rules.\/} Preprint.  
\\
Griffeath D 1994. In {\it Probability and Phase Transition\/}, 
G. Grimmett, Editor, Kluwer, pg. 49. 
\\
Packard N and Wolfram S 1985 {\em Journal of Statistical Physics} 
{\bf 38} 901. 
\\
Toffoli T and Margolus N 1987 {\it Cellular Automata Machines\/}. 
MIT Press. 
\\
Vichniac G 1984 {\em Physica D} {\bf 10} 96. 
\\
Willson S 1984 {\em Discrete Appl. Math.} {\bf 8} 1991. 

\newpage

\begin{center}
\Large
{\bf  The onset of turbulence despite stable laminarity}
\normalsize

{\bf Siegfried Grossmann}
  
{\em Fachbereich Physik der Philipps-Universit{\"a}t, Marburg, Germany}
\end{center}                            

\normalsize
\noindent
This lecture is devoted to the recent explanation of the mechanism
which determines the onset of turbulence in laminar shear flows. This 
is different from the routes to turbulence characterized by a sequence
of instabilities or bifurcations cf. Schuster (1988) or Berg\'{e} et
al (1986), showing quasi-periodicity, period-doubling, or temporal
intermittency. In fluid flow these routes can be observed in the onset
range of heat convection in Rayleigh-B\'{e}nard systems (fluid heated
from below) or in Taylor-Couette systems (concentric cylinders, the
inner one rotating), see e.g. Koschmieder (1993). Laminar shear flows
between plane boundaries or through pipes stay linearly stable,
instead, but nevertheless become turbulent, though at larger Reynolds
numbers Re = O(1~000). The mechanism is not an eigenvalue instability
but a subtle interplay between {\bf nonnormal} bunching of the
eigenstates and their {\bf nonlinear} interaction, see Boberg and
Brosa (1988), Gebhardt and Grossmann (1993, 1994), Trefethen et al
(1993), Grossmann (1996, 1999). The most characteristic features are
the double threshold and a large phase space Hausdorff dimension at
onset. The threshold is fractal under a given type of perturbations and
in phase space a chaotic repeller (arising from unstable stationary
states) generates the complex, chaotic dynamics of turbulent flow,
Eckhardt et al (1998).  


\noindent{\bf References}
\\
Berg\'{e} P,~ Pomeau Y and Vidal C 1986 {\em Order within 
Chaos--Towards a deterministic approach to turbulence} (New York:
Wiley)
\\
Boberg L and Brosa U 1988 ~{\em Z. Naturforsch} {\bf 43a}~ 697
\\
Eckhardt B,~ Marzinzik K and~ Schmiegel A 1998 in ~{\em A Perspective
Look at Nonlinear Media} \\eds. J Parisi, S C M{\"u}ller and
W Zimmermann {\em Lecture Notes in Physics} {\bf 503} (Berlin:
Springer) pp 327--338
\\
Gebhardt T and Grossmann S 1993 {\em Z. Phys.} {\bf B90} 475
\\
Gebhardt T and Grossmann S 1994 {\em Phys. Rev.} {\bf E50} 3705
\\
Grossmann S in {\em Nonlinear Physics of Complex Systems} eds. J
Parisi, S C M{\"u}ller and W Zimmermann {\em Lecture Notes in
Physics} {\bf 476} (Berlin: Springer) pp 10--22
\\
Grossmann S ~1999 ~{\em Rev. Mod. Phys.}
\\
Koschmieder E L 1993 {\em B\'{e}nard Cells and Taylor Vortices}
(Cambridge: Cambridge University Press)
\\
Schuster H G 1988 {\em Deterministic Chaos--An Introduction}
(Weinheim: VCH-Wiley)
\\
Trefethen L N, Trefethen A E, Reddy S C and Driscol T A 1993 {\em
Science} {\bf 261} 578

\newpage

\begin{center}
\Large
{\bf  Turbulent heat transport: \\ 
global performance--global scaling}
\normalsize

{\bf Siegfried Grossmann}
  
{\em Fachbereich Physik der Philipps-Universit{\"a}t, Marburg, Germany}
\end{center}                            

\normalsize
\noindent
A temperature difference $\Delta$ between the bottom and the top
plates  enclosing a gas or a liquid at rest imply molecular heat
transport. The heat current $J_{\kappa}$ is determined by, besides 
$\Delta$, the thermal conductivity $\kappa$ and the height $L$ of the
fluid layer, $J_{\kappa}$ being $\kappa \Delta / L$. If the
nondimensionalized temperature difference $Ra = \beta g L^3 \Delta /
\kappa \nu$, the Rayleigh number of this Rayleigh-B\'{e}nard system,
becomes larger, the molecular heat transport is supported by 
convective transport. Here $\nu$ is the kinematic viscosity, g the
gravitational acceleration, and $\beta$ the thermal volume expansion
coefficient. At medium $Ra$ rolls of fluid flow with increasing velocity
$U$ are developed. For large $Ra$ the flow in the bulk of the rolls
becomes turbulent. The heat current $J$ then  can be carried by the
turbulent convection. This is very effective, so $J$ increases
with $Ra$. The heat current, called the Nusselt number $Nu$ when
measured in multiples of the molecular current $J_{\kappa}$, i.e., $Nu =
J/J_{\kappa}$, depends algebraically on $Ra$. It also scales
algebraically with the Prandtl number $Pr$, which characterizes the
importance of the kinematic viscosity relative to the thermal
conductivity, i.e., $Pr = \nu / \kappa$. The scaling exponents of the
heat current $Nu$ as well as of the roll velocity, measured by the roll
Reynolds number $Re = U L / \nu$ as functions of $Ra$ and $Pr$, the
control parameters, are derived and discussed in this lecture. \\ 
\noindent
Since heat current $Nu$ and roll velocity $Re$ are global
responses of the fluid to the driving temperature difference only
global arguments turn out to determine the scaling behaviors with $Ra$
and with $Pr$. It is suggested that the only relevant physical
quantities are the kinematic and the thermal dissipation rates. These
can be determined in the bulk as well as in the boundary layers.
Utilizing exact relations between the dissipation rates and the 
currents $Nu$ and $Re$ allows to derive the various scaling laws valid
in the different regions of the parameter space $Ra-Pr$. We also give
the crossover corrections between regions, which jeopardize pure
scaling. Comparison with several previous and recent experiments
is very encouraging for this unifying theory of scaling in thermal
convection. The reference for this unifying theory is Grossmann and
Lohse (1998). \\
\noindent
Introductory references to thermal convection are Koschmieder (1993),
Castaing et al. (1989); for information about experiments see Wu and
Libchaber (1991), Chavanne et al. (1997), Cioni et al. (1997);
theoretical overviews are given by Shraiman and Siggia (1990), Siggia
(1994); an extended list of references can be found in the cited 
work by Grossmann and Lohse (1998).

\noindent{\bf References}
\\
Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S,
Wu X Z, Zaleski S and Zanetti G 1989 {\em J. Fluid Mech.} {\bf 204} 1
\\
Chavanne X, Chilla F, Castaing B, Hebral B, Chaboud B and Chaussy J
1997 {\em Phys. Rev. Lett.} {\bf 79} 3648
\\
Cioni S, Ciliberto S and Sommeria J 1997 {\em J. Fluid Mech.} {\bf 335}
111
\\
Grossmann S and Lohse D 1998 {\em Scaling in thermal convection: A
unifying theory} submitted to {\em J. Fluid Mech.}
\\
Koschmieder E L 1993 {\em B\'{e}nard Cells and Taylor Vortices}
~(Cambridge University Press)
\\
Shraiman B I and Siggia E D 1990 {\em Phys. Rev.} {\bf A42} 3650
\\
Siggia E D 1994 {\em Annu. Rev. Fluid Mech.} {\bf 26} 137
\\
Wu X Z and Libchaber A 1991 {\em Phys.Rev.} {\bf A43} 2833

\newpage

\begin{center}
\Large
{\bf  Energy dissipation in turbulent flow: \\
rigorous and less rigorous insights}
\normalsize

{\bf Siegfried Grossmann}
  
{\em Fachbereich Physik der Philipps-Universit{\"a}t, Marburg, Germany}
\end{center}                            
\normalsize
\noindent
To maintain fluid flowing one needs stirring by some external force. 
There is no
turbulence without ongoing energy input, which compensates the
permanent energy loss by viscous damping. Therefore the most relevant
quantity to characterize  turbulence is the mean energy flow rate per
mass, $\varepsilon = \nu \langle (\partial_ju_i)(\partial_ju_i)
\rangle$, and its dependence on the Reynolds number $Re = U L / \nu$.
From dimensional argument the energy flow $\varepsilon$ must be
$\propto U^3 L^{-1}$. The dimensionless prefactor, denoted as the
dissipation coefficient $c_{\varepsilon}(Re)$, is $Re^{-1}$ for laminar
flow. Its behavior for turbulent flow and in particular asymptotically
for large
$Re$ is of high interest. \\
In this lecture, after introductory estimates of the dissipation
coefficient $c_{\varepsilon}(Re)$, a variational principle is derived,
leading to a rigorous upper bound of the dissipation rate's coefficient
(Nicodemus, Grossmann and Holthaus 1997). The variational principle is
formulated in terms of a static auxiliary field which is optimized to
carry the dissipation of the real turbulent flow field. It is one of
the few exact results on the nonlinear Navier-Stokes dynamics. This
principle is a generalization of Doering and Constantin's work (1992,
1994). Hopf (1941) was the first to study bounds on energy
dissipation with this idea. Another line of approach is the Busse (1970)
and Howard (1972) variational method; in that the flow field is
decomposed into its mean and the fluctuating deviations. Using our
results it could be shown (Kerswell 1997) that both methods lead to the
same bounds on energy dissipation. Asymptotically $c_{\varepsilon}(Re
\rightarrow \infty) = 0.01087(1)$. From the volume mean one has to
distinguish the bulk dissipation, Sreenivasan (1984), Lohse (1994),
Grossmann(1995). Recently we identified for the first time a mechanism
that leads to a {\em decrease} of
$c_{\varepsilon}$ with $Re
\rightarrow \infty$, Nicodemus, Grossmann and Holthaus (1999). The
lecture finishes with a comparison of the kinetic dissipation with the
thermal dissipation in heat convection (Nicodemus, Grossmann and
Holthaus, unpublished). More details about the mathematical and the
numerical methods for d-dimensional bounds, also for
the $Re$-asymptotics, can be found in Nicodemus, Grossmann and
Holthaus (1997, 1998, 1999).
\\[3mm]
\noindent{\bf References}
\\
Busse F H 1970 {\em J. Fluid Mech.} {\bf 41} 219
\\
Busse F H 1978 {\em Adv. Appl. Mech.} {\bf 18} 77
\\
Doering C R and Constantin P 1992 {\em Phys. Rev. Lett.} {\bf 69} 1648
\\
Doering C R and Constantin P 1994 {\em Phys. Rev.} {\bf E49} 4087 
\\
Grossmann S 1995 {\em Phys.Rev.} {\bf E51} 6275
\\
Hopf E 1941 {\em Math Annalen} {\bf 117} 764
\\
Howard L N 1972 {\em Annu. Rev. Fluid~ Mech.} {\bf 4} 473
\\
Kerswell R R 1997 {\em Physica} {\bf D100} 355
\\
Lohse D 1994 {\em Phys. Rev. Lett.} {\bf 73} 3223
\\
Nicodemus R, Grossmann S and Holthaus M 1997 {\em Physica} {\bf D101}
178
\\
Nicodemus R, Grossmann S and Holthaus M 1997 {\em Phys. Rev. Lett.} {\bf
79} 4170
\\
Nicodemus R, Grossmann S and Holthaus M 1997 {\em Phys. Rev.} {\bf E56}
6774
\\
Nicodemus R, Grossmann S and Holthaus M 1998 {\em J. Fluid Mech.} {\bf
363} 281
\\
Nicodemus R, Grossmann S and Holthaus M 1998 {\em J. Fluid Mech.} {\bf
363} 301
\\
Nicodemus R, Grossmann S and Holthaus M 1999 ~~"Towards lowering
dissipation bounds"~~\\ {\em Eur. Phys. J}~ {\bf B} in press 
\\
Sreenivasan K R 1984 {\em Phys. Fluids} {\bf 27} 1048

\newpage


\begin{center}
\Large
{\bf  Brain dynamics - Part I}
\normalsize

{\bf Hermann Haken}
  
{\em Center of Synergetics, Stuttgart, Germany}
\end{center}                            

\normalsize
\noindent
The human brain is the most mysterious organ we know of. In it perception, 
speech control, thinking, feelings, emotions, and many other things exist.
Seen from the point of view of a physicist, it is the most complex system in
the world. It consists of about a hundred billion of neurons that are 
interconnected in a highly complicated fashion.

In my lectures I will address two main topics:

1. The explicit example of a neural net (the lighthouse model)

2. How do macroscopic features, such as perception or motor-control, arise from
microscopic elements.

The vehicle I shall use to study the second question is synergetics. In my
first lecture I wish to remind the audience of some physical methods to study
brain functions, such as EEG, MEG, PET and MRI. In order to understand the
macroscopic activity patterns of the brain, basic concepts of synergetics
will be used and a reminder will be given of basic concepts, such as control
parameters, instability, order parameters, slaving principle, critical 
fluctuations, and symmetry breaking in open systems far from thermal
equilibrium. These concepts are exemplified by phenomena of visual perception,
such as bistability, hysteresis, and oscillations. The synergetic computer
will serve as a model for pattern recognition and links with Gestalt theory
will be established.

\noindent{\bf References}
\\
H. Haken, {\em Principles of Brain Functioning}, Springer, Berlin (1996)
\\
J.A.S. Kelso, {\em Dynamic Patterns}, MIT Press (1995)

\newpage

\begin{center}
\Large
{\bf  Brain dynamics - Part II: The lighthouse model of a neural net}
\normalsize

{\bf Hermann Haken}
  
{\em Center of Synergetics, Stuttgart, Germany}
\end{center}                            

\normalsize
\noindent
I present some salient features of neurons and their connections, and I discuss
the role of soma, axon, dendrites and synapses. I briefly sketch previous 
models, such as that of McCulloch and Pitts, Wilson and Cowan, Jirsa and Haken,
which neglect phase relations, and models that deal with phase couplings 
(Kuramoto, Tass), with sinusoidal couplings, and integrate and fire models 
(Strogatz et al., Geisel et al.).

I then present the basic concepts of the lighthouse model of a neural net as
developed by the present author, where attention is focused on phase-locking
by means of pulse trains. Basic variables are dendritic currents that are 
generated by axonal pulses and phases describing the timing of the axonal
pulses. The latter are generated by dendritic currents, processed by the soma
of a neuron. While below a threshold, the neuron is, as usually assumed, 
quiescent, according to my model above that threshold its firing rate increases
linearly with the inputs.

\noindent{\bf References}
\\
McCulloch and W. Pitts, {\em Bulletin of Math. Biophysics} {\bf 5}, 115-133 
(1943)
\\
H.R. Wilson and J.D. Cowan, {\em Biophysical Journal} {\bf 12}, 1-24 (1972)
\\
V.K. Jirsa and H.Haken, {\em Phys. Rev. Lett.} (1996); {\em Physica D} (1997)
\\
Y. Kuramoto and I. Nishikawa, {\em J. Stat. Phys.} {\bf 49}, 569 (1987); Y.
Kuramoto, Physica D {\bf 50}, 15 (1991)
\\
P. Tass and H. Haken, {\em Z. Phys. B} {\bf 100}, 303-320 (1996); {\em Biol.
Cybern.} {\bf 74}, 31-39 (1996)
\\
C. Uhl, {\em Analysis of Neurophysiological Brain Functioning}, Springer,
Berlin (1999)
\\
R.E. Mirollo and S.H. Strogatz, {\em SIAM (Soc. Ind. Appl. Math.) J. Appl. 
Math.} {\bf 50}, 1645 (1990)
\\
U. Ernst, K. Pawelzik and T. Geisel, {\em Physical Review E} {\bf 57}, no. 2
(1998)

\newpage

\begin{center}
\Large
{\bf  Brain dynamics - Part III: Steady states, mutual interaction and 
associative memory of the lighthouse model}
\normalsize

{\bf Hermann Haken}
  
{\em Center of Synergetics, Stuttgart, Germany}
\end{center}                            

\normalsize
\noindent 
Though the model is highly nonlinear, because of the occurrence of pulses, it 
can be treated exactly. Especially the case of small damping of the dendritic
currents leads to very elegant results. The elimination of the variables 
describing the dendritic currents leads to phase-equations. By means of the 
example of two neurons, I demonstrate how the steady state can be calculated.
This leads to a mutual activation of neurons so that neurons become active 
even if they have been below threshold because of a too weak sensory input.

I then treat the case of an arbitrary number of neurons that are arbitrarily
coupled with each other and are subject to arbitrary sensory inputs. The
activation of patterns shows up in specific pulse rates, and the action of
the whole net can be interpreted as that of a associative memory. I distinguish
between weak and strong associative memory and explain the application of this
concept to Kaniza figures.

\newpage


\begin{center}
\Large
{\bf  Brain dynamics - Part IV: Phase-locking of two and many neurons, impact
of noise and delayed interactions}
\normalsize

{\bf Hermann Haken}
  
{\em Center of Synergetics, Stuttgart, Germany}
\end{center}                            

\normalsize
\noindent
In this part it is assumed that all sensory inputs are equal, but the coupling
between the neurons may still be arbitrary. I start from a phase-locked state
and derive equations for deviations from that state. The deviations may have
different causes, namely resetting of the individual phases externally, or
internal noise. In this way I demonstrate the stability of the phase-locked 
state against these perturbations. On the other hand, if a pulse is lacking due
to failure of a vesicle, or an additional pulse is emitted due to spontaneous
opening of a vesicle, a phase lag is caused. The results can be extended to an
arbitrary net and delayed interactions, provided the delay-time is the same 
for all neurons.

\newpage

\begin{center}
\Large
{\bf  Brain dynamics - Part V: Phase averaged equations}
\normalsize

{\bf Hermann Haken}
  
{\em Center of Synergetics, Stuttgart, Germany}
\end{center}                            

\normalsize
\noindent
I show how the lighthouse model and phase averaged equations, such as the
Nunez and Jirsa-Haken equations, are related to each other. I will discuss
a number of open problems.

\noindent{\bf References}
\\
P.L. Nunez, Oxford University Press, Oxford (1995)
\\
V.K. Jirsa and H.Haken, {\em Phys. Rev. Lett.} (1996); {\em Physica D} (1997)

\newpage

\begin{center}
{\Large\bf Random matrix theories and the problem of \\
 Anderson localization   } 

  {\bf Hiroshi Hasegawa}

{\it
Research Center of Quantum Communications, Tamagawa University, Macida,
and \\
 Atomic Energy Research Institute, Nihon University, Kanda-Surugadai,
Tokyo} 
\end{center}

\noindent
The problem ``Anderson localization" is one of major subjects in
Condensed Matter Physics, and in recent years it has been explored 
intensively from a viewpoint of {\it quantum level statistics}[1].   
Historically, this emphasis has occurred from a technical development in 
the decade of 80's about Gaussian mean over matrix ensembles by the 
method of {\it supersymmetry} that implemented studies of correlation 
functions[2].  An interest in the subject is the intermediate nature of 
the statistics between the standard Wigner-Dyson (in the metallic phase) 
and the Poisson (in the localized phase): it would be a 
level-statistical evidence of the phase transition between these two 
phases.   Among numerous works on this subject in the past decade, the 
results provided by Al'tshuler and coworkers should not be 
overlooked[4-6][8-11]. In the present lecture, therefore, we focus our 
attention on the most interesting result due to them, namely the {\it 
compressibility of level gas at the mobility edge}.  

{\bf References}\\
 1. B.L.Al'tshuler,{\it Universalities: from Anderson
localization to quantum chaos} in {\it Mesoscopic Quantum Physics} 
LesHouches Session LXI \, ed. E. Akkermans, G. \\ Montambaux, J.-L. 
Pichard, and J.
Zinn-Justin(ELSEVIER, Amsterdam1995).\\
 2. T. Guhr, A. M\"{u}ller-Groeling,and H. A. Weidenm\"{u}ller, 
{\it Random Matrix Theories in Quantum Physics: Common Concepts}, Phys. 
Rep.{\bf 299}(1998),189.\\
 3. M. Toda, R. Kubo,and N. Sait$\hat{o}$ {\it Statistical 
Physics }I
2nd ed. Springer-Verlag Berlin(1992).\\
4. B. L. Al'tshuler and B. I. Shklovskii, Sov. Phys. JETP{\bf 
64}(1986),127.\\
 5. B. L. Al'tshuler, I. Kh. Zharekeshev, S. A. Kotochigova and 
B. I. Shklovskii, Sov. Phys. JETP{\bf 67}(1988),625.\\
 6. V. E. Kravtsov, I. V. Lerner,B. L. Al'tshuler and A. G. 
Aronov, Phys. Rev. Lett.{\bf 72}(1994), 888; A. G. Aronov, V. E. 
Kravtsov and I.V. Lerner, Sov. Phys. JETP Lett. {\bf 59}(1994),39.\\  
 7. M. L. Mehta and F. J. Dyson, J. Math. Phys.{\bf 4}(1963),713.\\
 8. A. G. Aronov, V. E. Kravtsov and I. V. Lerner, Phys. Rev. \\
Lett.{\bf 74}(1995), 1174; V. E. Kravtsov and I. V. Lerner,Phys. Rev. 
Lett. {\bf 74}(1995), 2563.\\
 9. A. G. Aronov and A. D. Mirlin, Phys. Rev. {\bf 
B51}(1995),R6131.\\
 10. J. T. Chalker, I. V. Lerner and R. Smith, Phys. Rev. 
Lett.{\bf 77}(1996), 554; J. Math. Phys.{\bf 37}(1996),5061.\\
 11. J. T. Chalker, V. E. Kravtsov and I. V. Lerner, JETP 
Lett.{\bf 64}
(1996),386.



\newpage

\begin{center}
{\Large\bf Random Matrix Theories and the Problem of \\
 Anderson Localization II }


  {\bf Hiroshi Hasegawa}

{\it
 Research Center of Quantum Communications, Tamagawa University, Macida, 
 and \\
 Atomic Energy Research Institute, Nihon University, Kanda-Surugadai, 
Tokyo} 
\end{center}

\noindent
There exists a framework of computing {\it two-point correlation 
function} between
a pair of levels of a random eigenvalue sequence that is supposed to
arise from a perturbed hermitian $ H = H_0 +\lambda V$, where $H_0$ 
belongs to a
set of fully uncorrelated Hermitians (a Poisson ensemble) and $V$ to a 
set of Hermitians in the Wigner-Dyson class (one of the three standard 
RMT ensembles)[12-15]: it has been regarded as adapted for level 
statistics with Anderson transition[15].    This correlation function 
$R(r)$ has two remarkable characteristics. (1) its long-range 
behavior ($r > r_c$) reflects an attractiveness between the pair of 
levels ($R(r)>1$ [12,13]), and (2) $\int_{-\infty}^{\infty}(R(r)-1)dr=0$ [14]. 
We argue that (1) may be regarded as the characteristic of 
level statistics in a metallic phase[16] that could undergo {\it 
Anderson transition}, whereas (2) contradicts the existence of an 
intermediate compressibility[10,11].  Here, we present a new framework 
adapted for Anderson transition based on equilibrium statistical 
mechanics of level gas[18-22] (a Hamiltonian level dynamics for $ H=  
H_0 +\lambda V$) satisfying (1) and, furthermore, capable for the 
intermediate compressibility.

{\bf References}\\
12. F. Leyvraz and T. H. Seligman, J. Phys. A: Math. Gen.{\bf 
23}(1990),1555.\\
13. T. Guhr, Phys.Rev.Lett.{\bf 76}(1996),2258; 
Ann.Phys.(N.Y.){\bf 250}(1996),145.\\
14. H. Kunz and B. Shapiro, Phys. Rev. {\bf E58}(1998),400.\\
15. K. M. Frahm, T. Guhr and A. M\"{u}ller-Groeling, 
Ann.Phys.(N.Y.)
{\bf 270 }(1998),292.\\
16. R.A.Jalabert, J.-L.Pichard and C.W.J. Beenakker, Europhys. 
Lett. {\bf 24}(1993),1.\\
17. D. Weinmann and J.-L.Pichard, Phys.Rev.Lett.{\bf 
77}(1996),1556.\\
18. M. Gaudin, Nucl. Phys.{\bf 85}(1966),545; T. Yukawa, Phys. 
Rev. Lett.{\bf 54}(1985),1883; Phys. Lett.{\bf A116}(1986),227. \\
19. H. Hasegawa and J.-Z. Ma, J. Math. Phys.{\bf 39}(1998),2564.\\
20. H. Hasegawa {\it Information Theory and Statistical Mechanics 
of Random Matrices}, RIMS publication {\bf 1066}(1998),205; {\it Open 
Sys. and Information Dyn.}(1999),to be published.\\
21. J.-L. Pichard and B. Shapiro, J.Phys.I (France) {\bf 
4}(1994),623.\\
22. P.J. Forrester, Phys. Lett.{\bf A173}(1993),355.\\
23. D. Ruelle,{\it Statistical Mechanics}(W.A. Benjamin,Inc. New 
York, 1969).


\newpage

\begin{center}
\Large
{\bf  From small to large scale activity of 
spatiotemporal neural patterns} 
\normalsize

{\bf Viktor K. Jirsa}
  
{\em Center for Complex Systems and Brain Sciences,\\
Florida Atlantic University, Boca Raton, USA}
\end{center}                            

\normalsize
\noindent
The neuron is thought to be the central microscopic 
information-processing unit
in the brain. We review its morphology (see Braitenberg 
and Sch\"uz (1991)) 
and dynamics and discuss mathematical models of neuronal 
temporal activity
(Hodgkin-Huxley, integrate-and-fire and spike-response 
model; see e.g. Kistler
et al (1997) ).  Interconnectivity among neurons 
requires to
traverse to a description of neural networks. In many 
areas of the brain
neurons are organized in populations of units with 
similar properties which
allows to describe the mean activity of a neuronal 
population rather than the
temporal behavior of individual neurons. Spatiotemporal 
neural activity in 
terms of neural ensembles 
has been described by Wilson and Cowan (1972,1973) and 
Nunez (1974). 
Later approaches by Jirsa and Haken (1996) and others 
unified and generalized 
these models to a neural field theory on large 
spatiotemporal scales 
accessible to electroencephalography (EEG) and 
magnetoencephalography (MEG). 
The introduction of the notion of ``Functional units" by 
Jirsa and Haken (1996) 
provides a means to correlate behavioral patterns to EEG 
and MEG patterns (for
the specific realization of Bimanual Coordination see 
Jirsa et al (1998), for
reviews of large scale brain and behavioral dynamics see 
Kelso (1995), Nunez
(1995), Haken (1996)). 


\noindent{\bf References}
\\
Braitenberg V and Sch\"uz A (1991) {\em Anatomy of the 
cortex.
Statistics and geometry} (Springer, Berlin)
\\
Haken H (1996) {\em Principles of Brain Functioning} 
(Springer, Berlin)
\\
Jirsa V K and Haken H  (1996) {\em Phys. Rev. Let.} {\bf 
77} 960
\\
Jirsa V K, Fuchs A and Kelso J A S (1998) {\em Neur. 
Comp.} {\bf 10} 2019 
\\
Kelso J A S (1995) {\em Dynamic Patterns. The 
Self-Organization of Brain and
Behavior} (The MIT Press, Cambridge, Massachusetts)
\\
Kistler W M, Gerstner W and van Hemmen J L (1997) {\em 
Neur. Comp.} {\bf 9}
1015
\\
Nunez P L (1974) {\em Math. Biosci.} {\bf 21} 279
\\
Nunez P L (1995) {\em Neocortical Dynamics and Human EEG 
Rhythms} (Oxford
University Press, Oxford)
\\
Wilson H R and Cowan J D (1972) {\em Biophys. J.} {\bf 
12} 1
\\
Wilson H R and Cowan J D (1973) {\em Kybernetik} {\bf 
13} 55




\newpage



\begin{center}
\Large
{\bf  Spatiotemporal pattern formation in systems with 
heterogeneous connection
topology}                             
\normalsize

{\bf Viktor K. Jirsa}
  
{\em Center for Complex Systems and Brain Sciences,\\
Florida Atlantic University, Boca Raton, USA}
\end{center}

\noindent
We briefly review spatiotemporal pattern formation via 
self-organization in 
open non-equilibrium systems of different nature, e.g. 
in physical systems as 
in hydrodynamics or the laser, in chemical systems as in 
the Belousov 
Zhabotinsky reaction (see Haken 1975, Cross and 
Hohenberg 1993 for reviews). 
In these systems the underlying connection topology is 
typically homogeneous, 
i.e. translationally invariant, which provides the basis 
for modelling their
spatiotemporal dynamics by nonlinear partial 
differential equations as in the
Ginzburg-Landau theory (see Haken (1983)). This approach 
has been applied 
successfully to a class of biological systems for which 
the connection 
topology may be assumed to be regular such as coat 
markings of mammals or fish 
(see Murray (1993) for review). However, many biological 
systems such as the 
human cortex show homogeneous connectivity with 
additional strongly 
heterogeneous projections from one area to another (see 
Braitenberg and 
Sch\"uz 1991, Felleman and Van Essen 1991), but still 
perform coherent 
spatiotemporal pattern formation (see Kelso (1995)). 
Such heterogeneous
connectivity may undergo temporal changes via mechanisms 
of asymmetric
Hebbian learning or synaptic plasticity (see Markram et 
al (1997)). Here we
show mathematically and numerically how the global 
spatiotemporal dynamics of a
heterogeneously connected system can be controlled via 
local changes in the
connection topology and thus be guided through a series 
of non-equilibrium
phase transitions.  
We provide a mathematical description of the system in 
terms of a nonlinear 
integral equation which is retarded by delays via 
propagation of signals along 
connecting pathways (see Jirsa and Haken (1996)). Our 
formulation allows to 
use the connectivity as a topological control parameter 
which could not be 
achieved by means of partial differential equations in a 
straight forward way. 


\noindent{\bf References}
\\
Braitenberg V and Sch\"uz A (1991) {\em Anatomy of the 
cortex.
Statistics and geometry} (Springer, Berlin)
\\
Cross M C and Hohenberg P C (1993) {\em Rev. Mod. Phys.} 
{\bf 65} 851 
\\
Felleman D J and Van Essen D C (1991) {\em Cerebral 
Cortex} {\bf 1} 1
\\
Haken H (1975) {\em Rev. Mod. Phys.} {\bf 47} 67 
\\
Haken H (1983) {\em Synergetics. An Introduction} 
(Springer, Berlin)
\\
Jirsa V K and Haken H  (1996) {\em Phys. Rev. Let.} {\bf 
77} 960
\\
Kelso J A S (1995) {\em Dynamic Patterns. The 
Self-Organization of Brain and
Behavior} (The MIT Press, Cambridge, Massachusetts)
\\
Markram H, L\"ubke J, Frotscher M and Sakmann B (1997) 
{\em Science} {\bf 275} 
213 
\\
Murray J D  (1993) {\em Mathematical Biology} (2nd ed., 
Springer, Berlin 
Heidelberg New York)

\newpage

\begin{center}
\Large
{\bf Control of chaos in the heart: New approaches, problems}
\normalsize

{\bf  Alain Pumir and Valentin Krinsky}

 {\em Institut Non Lin\'eaire de Nice}\\
{\em U.M.R. 6618, C.N.R.S. Universit\'e de Nice Sophia - 
Antipolis,  Valbonne, France.}

 \end{center}                           

\normalsize
\noindent

We review the main ideas and results in the mathematical theory of 
chaotic wave propagation in cardiac muscle.  Mechanisms of chaos 
and approaches to control the chaos  are understood    for 
ordinary differential equations (d.e.), but   for partial d.e, 
only isolated results are published.
In cardiac muscle, the   chaotic regime  known as fibrillation  is 
 the main reason of mortality in industrial world. A very strong   
electrical discharge (defibrillation) eliminates the chaos. 
Mechanisms of defibrillation are poorly understood. Important is 
to study alternative approaches to eliminate the chaos.

For cardiac muscle (and excitable media in general), the initial 
stage in transition to  chaos is formation of rotating vortices.  
Elimination of vortices at this stage effectively prevents chaos 
development.    We study two different kind of vortices, free 
vortices, and vortices pinned to a  defect.  We found that a weak 
shock, with an amplitude an
order of magnitude less than the defibrillating shock, may
unpin the vortices rotating around the defects (obstacles). The
unpinning results from a depolarization of the tissue near the 
obstacle,
induced by an external electric field within a distance of order 
electrical space constant in cardiac muscle $\sim 1mm $ (Weidmann, 
1970).
Unpinning was observed both in the Fitz Hugh model of 
excitable tissue,
and in a specific Beeler-Reuter model of cardiac tissue. This 
theoretical
observation suggests that anatomical reentries can be transformed
into functional reentries, an effect that can be tested
in experiments with cardiac muscle.

\noindent{\bf References}\\
J. Keener and J. Sneyd.
{\it Mathematical Physiology,}
Springer- Verlag, New York, 1998.
\\
Pumir, A., Plaza, F., Krinsky , V.
{\it Control of rotating waves in cardiac muscle:
analysis of the effect of the electric field.
Proc Roy Soc B,} {\bf 257}, 129-134, 1994.  0D
\\
A. Pumir, V. Krinsky.
{\it How does an electric field defibrillate  cardiac muscle?
Physica D,} {\bf 91}, 205 - 219, 1996.
\\
 V. Krinsky and  A.Pumir
{\it Models of defibrillation of cardiac tissue 
Chaos,} {\bf 8}, n.1, p.188-203,  1998
\\
 V. Krinsky
{\it  Qualitative theory of reentry   
  In  Cardiac  Electrophysiology, 46rom Cell to Bedside,} D.P. 
Zipes and J.Jalife, editors,   chapter 38,   Saunders, 
Philadelphia, 1998.

\newpage


\begin{center}
\Large
{\bf Phase dynamics --- Its conceptual universality\\ and different 
specific forms --- }
\normalsize

{\bf Yoshiki Kuramoto}
  
{\em  
Graduate School of Sciences, Kyoto University, Japan
}
\end{center}                            

\normalsize
\noindent
Our understanding of the dynamics of many nonlinear 
dissipative systems relies crucially upon the possibility that  under 
various conditions
complicated nonlinear evolution equations could be reduced to far simpler
forms. This lecture is devoted to clarifying the theoretical grounds for
such dynamical reduction with particular emphasis on the phase dynamics.
\par 
We first argue that, explicit or implicit,
virtually all theories of reduction involve commonly one universal feature, 
namely, a perturbative construction of an attracting invariant manifold 
together with a definition of an appropriate coordinate system on
this manifold in such a way that the evolution law there may look as simple as
possible. This assertion is then demonstrated for the two of the most 
fundamental
reduction methods, i.e., the center-manifold reduction and the phase
reduction. Here, the trivial invariant manifold to start with 
is given by the critical eigenspace in the first case, and the neutral 
space spanned by the Goldstone mode in the second case. We also discuss
how the reduction idea developed for ODEs, where such perturbative methods
presuppose low-dimensionality
of the invariant manifold, can immediately be carried over to PDEs with 
large spatial extension for which the dimension of the invariant manifold
should practically be infinite. From such a viewpoint, the Chapman-Enskog
reduction, a monumental reduction theory of nonequilibrium statistical 
mechanics, turns out almost identical in structure with the phase
dynamics. 
\par 
As a separate topic of reduction, a different form of phase reduction
particularly adapted to coupled oscillators systems will be presented, 
and we demonstrate its usefulness through the derivation of phase 
equations in various situations of practical interest.  

\noindent{\bf References}
\\
Mori H and Kuramoto Y 1998 {\em Dissipative Structures and Chaos}
(Berlin Heidelberg: Springer-Verlag) Ch.\ 5\\
Chapman S and Cowling T.\ G.\ 1970 {\em The Mathematical Theory of
Nonuniform Gases} 3rd. Ed. (Cambridge: Cambridge Univ.\ Press)\\
Kuramoto Y 1984 {\em Chemical Oscillations, Waves, and Turbulence}
(Berlin: Springer-Verlag) \\

\newpage

\begin{center}
\Large
{\bf Collective dynamics of coupled oscillators 
}
\normalsize

{\bf Yoshiki Kuramoto}
  
{\em Graduate School of Sciences, Kyoto University, Japan}
\end{center}                            

\normalsize
\noindent
Numerous examples of interacting rhythmic processes are found in various
fields of science including physics, chemistry and biology, and their
proper mathematical description can be provided by dynamical systems of
coupled limit-cycle oscillators. In this lecture, we will discuss 
the dynamics of large populations of oscillators with mean-field coupling,
and present an overview of various theoretical approaches developed up to 
this day since the pioneering work of Winfree (1967). We describe
in some detail a prototypal phase oscillator model with distributed
native frequencies (Kuramoto 1975) which admits
an exact solution for the phase transition due to mutual synchronization.
This model and its generalizations have been analyzed more deeply, and also
found some important applications. Some of the important works among them 
will be surveyed. They include Strogatz et al's theory on the stability of
the collective states, Crawford's center-manifold reduction of the model
with external noise, Daido's generalization of the coupling function
in terms of the order function, and Wiesenfeld et al's proof of 
the equivalence of the original model with circuit equations for
certain series arrays of Josephson 
junctions.  
We also describe the possibility of nontrivial forms of collective dynamics,
e.g., clustering, bursting and collective chaos, some of which may have 
implication in biological information processing.
\par
Aside from collective dynamics, there are many applications of the 
phase reduction method itself, e.g, to the construction 
of the phase-coupling function for various realistic models, 
which will also be touched upon.

\noindent{\bf References}
\\
Winfree A T 1967 {\em J.\ Theor.\ Biol.} {\bf 16} 15\\
Kuramoto Y 1975 in {\em International Symposium on Mathematical Problems
in Theoretical Physics}, Lecture Notes in Physics Vol.\ 39, ed. by Araki H
(Berlin: Springer-Verlag) 420\\
Strogatz S H et al. 1992 {\em Phys.\ Rev.\ Lett.} {\bf 68} 2730\\
Crawford J D 1994 {\em J.\ Stat.\ Phys.} {\bf 74} 1047\\
Daido H 1996 {\em Physica D} {\bf 91} 24\\
Wiesenfeld et al 1998 {\em Phys.\ Rev.\ Lett.} {\bf 57} 1563\\
Golomb D et al. 1992 {\em Phys.\ Rev.\ A} {\bf 45} 3516\\
Hansel D et al. 1993 {\em Phys.\ Rev.\ E} {\bf 48} 3470\\
Hakim V and Rappel W J 1992 {\em Phys.\ Rev.\ A} {\bf 46} R7347\\
Nakagawa N and Kuramoto Y 1993 {\em Prog.\ Theor.\ Phys.} {\bf 89} 313\\
Han S K et al. 1995 {\em Phys.\ Rev.\ Lett.} {\bf 75} 3190

\newpage

\begin{center}
\Large
{\bf Turbulence with multiscaling in large assemblies
of oscillators}
\normalsize

{\bf Yoshiki Kuramoto}
  
{\em 
Graduate School of Sciences, Kyoto University, Japan
}
\end{center}                            

\normalsize
\noindent
We discuss a universal class of turbulent states discovered recently in 
a wide variety of nonequilibrium systems including nonlocally coupled
chaotic and non-chaotic oscillators, arrays of uncoupled oscillators 
with long-wave random forcing, and certain types of 3-component 
reaction-diffusion systems. The turbulence here is characterized by 
multiscaling properties of the structure functions, which is reminiscent 
of fully developed Navier-Stokes turbulence. Specifically, various moments
of the amplitude increments between two points exhibit a power-law 
dependence on the mutual distance. As a consequence, the amplitude profile
is fractal with multi-affine properties. Remarkably, the fractal dimension
changes continuously with the system parameter, and there is a critical 
condition beyond which the amplitude profile completely loses the
spatial continuity, disintegrating into infinitesimal pieces. 
We develop a theory based on a multiplicative stochastic process obeyed by
the amplitude increments, and discuss its approximate stationary solutions
from which all multiscaling properties as well as the 
continuity-discontinuity transition can be explained. 
\par
It is also argued that the field defined by a coarse-grained spatial derivative
of the amplitude is characterized by multifractals, which is again
reminiscent of the multifractal nature of the energy dissipation field
in fully developed turbulence. Such properties are, however, by no means 
direct consequences
of the multiscaling of the original field, and some attempts will be made
toward their understanding.

\noindent{\bf References}
\\
Kuramoto Y 1995 {\em Prog.\ Theor.\ Phys.} {\bf 94} 321\\
Kuramoto Y and Nakao H 1996 {\em Phys.\ Rev.\ Lett.} {\bf 76} 4352\\
Kuramoto Y and Nakao H 1997 {\em Phys.\ Rev.\ Lett.} {\bf 78} 4039\\
Kuramoto Y, Battogtokh D and Nakao H 1998 {\em Phys.\ Rev.\ Lett.} {\bf 81} 3543\\
Nakao H 1998 {\em Phys.\ Rev.\ E} {\bf 58} 1591\\
Nakao H and Kuramoto Y 1999 {\em Eur.\ Phys.\ J.\ B} (to appear)\\
Bohr T et al. 1998 {\em Dynamical Systems Approach to Turbulence}
(Cambridge) \\
Frisch U 1995 {\em Turbulence --- The Legacy of A.\ N.\ Kolmogorov}
(Cambridge)   

\newpage

\begin{center}
\Large
{\bf  Chaos and order in high energy physics and Yang-Mills theories}
\normalsize

{\bf Viatcheslav I. Kuvshinov}
  
{\em Institute of Physics, Minsk, Belarus}
\end{center}                            

\normalsize
\noindent

Chaos and fractal signatures in high-energy physics processes are 
discussed. In particular intermittency phenomenon, fractal jets and 
clusters are analysed. 

All four fundamental physical interactions have Yang-Mills gauge 
nature and all of them have chaotic classical solutions in specific cases. 
The talk reviews some situations when YM field theories have chaos--order 
transition (at corresponding controlling parameters): non-Abelian nature, 
monopoles, sphalerons, other topological solutions, Higgs fields, etc are 
discussed as sources of chaos and order. For example, it is known that 
Higgs fields---a very important element of gauge theories giving masses 
to the particles---in its vacuum state can suppress chaos. Here it is shown 
that non-vacuum states of Higgs fields can both suppress and induce 
chaos. 

Numerical and analytical methods of study of chaos properties of YM 
theories are reviewed. 

The problem of the description of quantum chaos in second quantized field HEP 
theories is mentioned and its physical role questioned.

Some unsolved problems of modern HEP models such as  
confinement of color quarks and gluons  and their possible connections with 
presence of chaos as immanent property of YM-equations are discussed. 

\noindent{\bf References}
\\
Babichev L.F., Klenitsky D.V.,  Kuvshinov V.I. (1995){\em Phys.Lett.} 
{\bf B345} 
269
\\
De Wolf E.A.,  Dremin I.M.,  Kittel W. (1996){\em Phys.Rep.} {\bf 270} 1
\\
Hwa R. {\em Correlations and Fluctuations} W.Sci. (1996)pp.303-313
\\
Kawabe T.,  Ohta S. (1991){\em Phys.Rev.}{\bf D44} 1274
\\
Kawabe T. (1993){\em J.Phys.A.}:Math.Gen.{\bf 26} L1131
\\
Kawabe T., Ohta S. (1994){\em Phys.Lett.}{\bf B334} 127
\\
Kuvshinov V.I.(1995) {\em Chaos and Complexity Editions Frontieres} pp.197-200
\\
Kuvshinov V.I. et al (1998){\em NPCS1} 114
\\
Kuvshinov V.I.,  Kuz'min (1999) {\em Nonlinear Phenomena in Complex Systems} Minsk
\\
Mandelbaum G. (1995) {\em Chaos and Complexity Editions Frontieres} pp.193-196
\\
Matinyan S.G.,  Prokhorenko E.B.,  Savidy G.K.(1986){\em JETF Lett} {\bf 44} 138
\\
Rugh S.E.(1994) {\em Aspects of Chaos of Fundamental Interactions. Part 1 Non-
Abelian Gauge Fields Licentiate Thesis.} The Niels Bohr Institute
\\
Salasnich L. (1995) {\em Mod.Rev.Lett.} {\bf A10} 3119
\\
Salasnich L. (1997) Symmetry in Physics,  JINR Dubna
\\
Savidy G.K.(1984) {\em Nucl.Phys} {\bf B246} 302
\\
Veneziano G. (1997) {\em Int. Conf. on HEP},  Springer pp.324-336     

\newpage

\begin{center}
\Large
{\bf  Pattern formation in bacterial colonies}
\normalsize

{\bf Mitsugu Matsushita}
  
{\em Department of Physics, Chuo University,
 Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
We have studied the growth mechanism and morphological change in colony
 formation of bacteria from the viewpoint of physics of pattern formation.  
Even very small number of bacterial cells, once they are inoculated on the
 surface of an appropriate medium such as semi-solid and nutrient-rich agar
 plate and incubated for a while, repeat the growth and cell division
 many times.  
Eventually the cell number of the progeny bacteria becomes huge, and they
 swarm on the medium to form a visible colony.  
The colony differs in size, form, and color according to bacterial species.  
It also changes its form sensitively with the variation of environmental
 conditions.  
This implies that although usual bacteria such as {\em Escherichia coli} are
 regarded as single cell organisms, they never make their colony independently
 and randomly but somehow collaborate multicellularly.  
We have thus tried to extract some simple and universal behaviors in growth
 from such complex bacterial systems.  

Here we varied only two parameters to investigate the colony growth;
 concentrations of nutrient $C_n$ and agar $C_a$ in a thin agar
 plate as the incubation medium.  
Other parameters specifying experimental conditions such as temperature were
 kept constant.  
We mainly used a typical bacterial species {\em Bacillus subtilis}.  
Otherwise the experimental procedures are standard.  
It was found that colonies show characteristic patterns in the specific
 regions of values of $C_n$ and $C_a$ in the morphological diagram and
 the patterns change drastically from one region to another.  
They were classified into five types; fractal DLA-like, compact Eden-like,
 concentric ring-like, simple disk-like and densely branched DBM-like.  
We have experimentally elaborated characteristic properties for each of these
 colony patterns.  

We have also tried to construct a phenomenological but unified model which
 produces characteristic colony patterns observed in our experiments.  
It is a reaction-diffusion type model for the population density of bacterial
 cells and the concentration of nutrient.  
The essential assumption is that there exist two types of bacterial cells;
 {\em active} cells that move actively, grow and perform cell division, and
 {\em inactive} ones that do nothing at all.  
We have varied values of the initial nutrient concentration and the ratio of
 diffusion rate of bacterial cells to that of nutrients.  
They correspond to the specified nutrient concentration in agar plates and
 the substrate softness in our experiments, respectively.  
Our model is found to be able to globally reproduce all the colony patterns
 described above, and is phenomenologically quite satisfactory.

\noindent{\bf References}
\\
Matsushita M 1997 in {\em Bacteria as Multicellular Organisms}
 eds. J A Shapiro and M Dworkin (New York: Oxford UP) pp366-393
\\
Matsushita M, Wakita J and Matsuyama T 1995 in {\em Spatio-Temporal Patterns in
 Nonequilibrium Complex Systems}, Proceedings Volume XXI of Santa Fe Institute
 Studies in the Sciences of Complexity, eds. P E Cladis and P Palffy-Muhoray
 (Reading: Addison-Wesley) pp609-618
\\
Matsuyama T and Matsushita M 1993 {\em CRC Critical Reviews in Microbiology}
 {\bf 19} 117
\\
Wakita J, R\`afols I, Itoh H, Matsuyama T and Matsushita M 1998 {\em J. Phys.
 Soc. Jpn.} {\bf 67} 3630
\\
Wakita J, Itoh H, Matsuyama T and Matsushita M 1997 {\em J. Phys. Soc. Jpn.}
 {\bf 66} 67
\\
Matsushita M, Wakita J, R\`afols I, Itoh H, Matsuyama T, Sakaguchi H and
 Mimura M 1998 {\em Physica A} {\bf 249} 517
\\
\newpage
\begin{center}
\Large
{\bf  Spatiotemporal patterns produced by bacteria}
\normalsize

{\bf Mitsugu Matsushita}
  
{\em Department of Physics, Chuo University,
 Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
The growth of bacterial colonies is a ``Treasure Island'' of pattern
 formation.  
Apparently, the pattern-forming members (bacterial cells) are strongly
 interacting, self-reproducible and self-driven ``particles'', very different
 from conventional physical systems.  
Some colony patterns grown on the surface of agar plates can be understood in
 terms of well-defined models of pattern formation such as DLA, but not in all
 cases.  

Here we used a bacterial species {\em Proteus mirabilis}.  
This species has been well-known for more than a century to form
 macroscopically almost perfect concentric-ring like colonies with
 approximately equal spacings on the surface of semi-solid substrate such as
 an agar plate.  
However, when this {\em Proteus mirabilis} was inoculated and incubated on
 the surface of a very soft agar medium with rich nutrient, we observed that
 entirely different, spatiotemporal patterns emerge inside a colony after
 the colony spreads over the whole surface of the agar medium.  
These spatiotemporal patterns include target and rotating spiral patterns.  
In this sense the emerging patterns are very similar to those seen in
 dissipative systems such as chemical oscillators and liquid-crystalline
 electrohydrodynamic convective systems.  
They also seem to exhibit characteristic features of spatiotemporal chaos.  
Microscopic observations revealed that the two-dimensional collective motion
 of bacterial cells seems to be responsible for the formation of the observed
 spatiotemporal patterns.  

Emphasis is also put on the construction of morphological diagram for
 obtaining some clue to elucidate the growth mechanism and sometimes finding
 new morphologies as well.  
Examples are shown for bacterial colony formation and crystal growth in
 gelatin gels.  

\noindent{\bf References}
\\
Matsushita M 1997 in {\em Bacteria as Multicellular Organisms}
 eds. J A Shapiro and M Dworkin (New York: Oxford UP) pp366-393
\\
Rauprich O, Matsushita M, Weijer C J, Siegert F, Esipov S E and Shapiro J A
 1996 {\em J. Bacteriology} {\bf 178} 6525
\\
Shimada Y, Nakahara A, Matsushita M and Matsuyama T 1995
 {\em J. Phys. Soc. Jpn.} {\bf 64} 1896
\\
Moriyama O and Matsushita M 1995
 {\em J. Phys. Soc. Jpn.} {\bf 64} 1081
\\
Nakahara A, Shimada Y, Wakita J, Matsushita M and Matsuyama T 1996
 {\em J. Phys. Soc. Jpn.} {\bf 65} 2700
\\
Suda J, Nakayama T, Nakahara A and Matsushita M 1996
 {\em J. Phys. Soc. Jpn.} {\bf 65} 771
\\
Suda J, Nakayama T and Matsushita M 1996
 {\em J. Phys. Soc. Jpn.} {\bf 67} 2981

\newpage

\begin{center}
\Large
{\bf  Formation of colony patterns by a bacterial cell population}
\normalsize

{\bf Mitsugu Matsushita}
  
{\em Department of Physics, Chuo University,
 Tokyo, Japan}
\end{center}                            

\normalsize
\noindent
Bacterial species {\it Bacillus subtilis} is known to exhibit
 various colony patterns, such as diffusion-limited aggregation (DLA)-like,
 compact Eden-like, dense-branching-morphology (DBM)-like,
 concentric-ring-like and disk-like, depending on the substrate softness and
 nutrient concentration.  
We have established the morphological diagram of colony patterns, and examined
 and characterized both macroscopically and microscopically
 how they grow.  
For instance, we have found that there seem to be two kinds of bacterial
 cells; active and inactive cells, the former of which drive colony
 interfaces outward.  
The active cells are particularly distinguished from the inactive ones at
 the tips of growing branches of a DBM-like colony as the characteristic
 fingernail structure.  
We have also found that the concentric-ring-like colony is formed as a
 consequence of alternate repetition of advancing and resting of the growing
 interface which consists of active cells.  

Based on our observations, we have tried to construct a phenomenological but
 unified model which produces characteristic colony patterns.  
It is a reaction-diffusion type model for the population density of bacterial
 cells and the concentration of nutrient.  
The essential assumption is that there exist two types of bacterial cells;
 {\em active} cells that move actively, grow and perform cell division, and
 {\em inactive} ones that do nothing at all.  
We have varied values of the initial nutrient concentration and the ratio of
 diffusion rate of bacterial cells to that of nutrients.  
They correspond to the specified nutrient concentration in agar plates and
 the substrate softness in our experiments, respectively.  
Our model is found to be able to globally reproduce all the colony patterns
 seen in the experimentally obtained morphological diagram, and is
 phenomenologically quite satisfactory.

\noindent{\bf References}
\\
Matsushita M 1997 in {\em Bacteria as Multicellular Organisms}
 eds. J A Shapiro and M Dworkin (New York: Oxford UP) pp366-393
\\
Matsushita M, Wakita J and Matsuyama T 1995 in {\em Spatio-Temporal Patterns in
 Nonequilibrium Complex Systems}, Proceedings Volume XXI of Santa Fe Institute
 Studies in the Sciences of Complexity, eds. P E Cladis and P Palffy-Muhoray
 (Reading: Addison-Wesley) pp609-618
\\
Matsuyama T and Matsushita M 1993 {\em CRC Critical Reviews in Microbiology}
 {\bf 19} 117
\\
Wakita J, R\`afols I, Itoh H, Matsuyama T and Matsushita M 1998 {\em J. Phys.
 Soc. Jpn.} {\bf 67} 3630
\\
Wakita J, Itoh H, Matsuyama T and Matsushita M 1997 {\em J. Phys. Soc. Jpn.}
 {\bf 66} 67
\\
Matsushita M, Wakita J, R\`afols I, Itoh H, Matsuyama T, Sakaguchi H and
 Mimura M 1998 {\em Physica A} {\bf 249} 517

\newpage

\begin{center}
\Large
{\bf Stochastic resonance}
\normalsize

{\bf Peter V.E.\ McClintock}
  
{\em Department of Physics, Lancaster University, UK}
\end{center}

\normalsize
\noindent

Stochastic resonance (SR) is commonly said to occur when a weak
periodic signal in a nonlinear system is enhanced by an increase of the
ambient noise intensity; a stronger definition requires that the
signal/noise {\it ratio} (SNR) should also increase. SR has been much
in the news recently, partly on account of its wide occurrence in many
areas of science. After being introduced as a possible explanation of
the earth's ice-age cycle (Benzi et al 1981, Nicolis 1982), SR has
subsequently been observed or invoked in contexts that include lasers
(e.g.\ McNamara et al 1988), passive optical systems, tunnel diodes, a
Brownian particle in an optical trap, a magnetoelastic ribbon, crayfish
and rat mechanoreceptors, a bistable SQUID (superconducting quantum
interference device), arrays of SR elements, ion channels, magnetic
systems, the El Nino phenomenon, social ills, various types of bistable
electronic model including coexisting periodic attractors, subcritical
bifurcations, systems with thresholds, transient dynamics, a quantum
2-level system, an array of coupled bistable systems, a system driven
by quasimonochromatic (harmonic) noise, excitable neurons, and chemical
systems. There have been several general scientific articles (e.g.\
Moss and Wiesenfeld 1995, Bulsara and Gammaitoni 1996) and two largely
complementary topical reviews (Dykman et al 1995, Gammaitoni et al
1998).

We present a succinct introductory review of SR discussing its
physical nature, the insights that can be obtained by treating it as a
linear response phenomenon (Dykman et al, 1990), and the way in which
electronic experiments (Luchinsky et al 1998) have tested the
approximate analytic theory and extended the known range of occurrence
of SR far beyond the systems characterised by static bistable
potentials in which it was originally discovered.

\noindent{\bf References}
\\
Benzi R, Sutera S, and Vulpiani A 1981 {\it J. Phys. A} {\bf 14}
L453
\\
Bulsara A R and Gammaitoni L 1996 {\it Physics Today} March 39
\\
Dykman M I, Mannella R, McClintock P V E and Stocks N G 1990
{\it Phys. Rev. Lett.} {\bf 65} 2606
\\
Dykman M I, McClintock P V E, Stein N D and Stocks N G 1992 {\it
Phys. Rev. Lett.} {\bf 68} 2718
\\
Dykman M I, Mannella R, McClintock P V E and Stocks N G 1993
{\it Phys. Rev. Lett.} {\bf 70} 874
\\
Dykman M I and McClintock P V E 1998 {\it Nature} {\bf 391} 344
\\
Dykman M I, Haken H, Gang Hu, Luchinsky D G, Mannella R, 
McClintock P V E, Ning C Z, Stein N D and Stocks N G 1993 {\it Phys.
Lett. A} {\bf 180} 332
\\
Dykman M I, Luchinsky D G, Mannella R, McClintock P V E, Stein N
D and Stocks N G 1995 {\it Nuovo Cimento D} {\bf 17} 661
\\
Gammaitoni L, H\"anggi, Jung P and Marchesoni F 1998 {\it
Rev. Mod. Phys.} {\bf 70} 223
\\
Kaufman I Kh, Luchinsky D G, McClintock P V E, Soskin S M and
Stein N D 1998 {\it Phys. Rev. E} {\bf 57} 78
\\
Luchinsky D G, McClintock P V E and Dykman M I 1998 {\it Rep.\ Prog.\ Phys.}
{\bf 61} 889
\\
McNamara B, Wiesenfeld K and Roy R 1988 {\it Phys. Rev. Lett.}
{\bf 60} 2626
\\
Nicolis C 1982 {\it Tellus} {\bf 34} 1
\\
Stocks N G, Stein N D, Soskin S M and McClintock P V E 1992
{\it J. Phys. A: Math. Gen.} {\bf 25} L1119
\\
Stocks N G, Stein N D and McClintock P V E 1993 {\it J. Phys. A:
Math. Gen.} {\bf 26} L385

\newpage

\begin{center}
\Large
{\bf Large fluctuations and optimal paths}
\end{center}
\normalsize

\noindent Most of the interesting and important events that occur in
fluctuating nonlinear dynamical systems, e.g.\ the transitions between
attractors giving rise to stochastic resonance or transport in Brownian
ratchets, are attributable to {\it large rare fluctuations}. Their
analysis requires both evaluation of the probability of the system
occupying a state remote from the attractor and identification of the
fluctuational path through which it reaches that point. Starting from
Boltzmann (1904), a huge body of theory has been developed during this
century; the modern understanding dates from Onsager and Machlup
(1953). But it was not until recently that a method was established
(Dykman et al 1992) by which {\it optimal paths}, i.e.\ the most
probable fluctuational paths, can be measured experimentally.

Recent progress in the area, with particular reference to experiments
on optimal paths in far-from-equilibrium systems, will be discussed.
Like rays in geometrical optics or WKB trajectories in quantum
mechanics, the pattern of optimal paths displays singularities. The
reasons why some of these are physically observable in electronic
experiments, and others are not, will be accounted for though the
topological insights introduced by Dykman et al (1994). Measurements of
an exit location distribution for a noisy nonconservative system
(Luchinsky et al 1999), leading to the observation of {\it saddle-point
avoidance}, will be discussed.



\noindent{\bf References}
\\
Array\'as M, Casado J M, G\'omez Ord\'o\~nez J, McClintock P V E,
Morillo M, Stein N D 1998  {\it Phys.\ Rev.\ Lett.} {\bf 80}
2273
\\
Berry M V 1976 {\it Adv. Phys.} {\bf 25} 1
\\
Boltzmann L, 1904 ``On Statistical Mechanics'', address given to
the Scientific Congress in St. Louis, 1904 reprinted in {\it
Theoretical Physics and Philosophical Problems} ed B McGuinness
(Dordrecht: Reidel) pp 159--172
\\
Bray A J and McKane A J 1989 {\it Phys.\ Rev.\ Lett.} {\bf 62} 493
\\
Chinarov V A, Dykman M I and Smelyanskiy V N  1993 
{\it Phys. Rev. E} {\bf 47} 2448
\\
Dykman M I, Krivoglaz M A and Soskin S M 1989 in {\it Noise in
Nonlinear Dynamical Systems} ed F Moss and P V E McClintock (Cambridge,
Cambridge University Press) vol. 2 pp 347--380
\\
Dykman M I, McClintock P V E, Stein N D and Stocks N G 1992 {\it
Phys. Rev. Lett.} {\bf 68} 2718
\\
Dykman M I, Millonas M M and Smelyanskiy V N 1994 {\it Phys.
Lett. A} {\bf 195} 53
\\
Dykman M I, Luchinsky D G, McClintock P V E and Smelyanskiy V N 1996
{\it Phys. Rev. Lett.} {\bf 77} 5229
\\
Freidlin M I and Wentzell A D 1984 {\it Random Perturbations in
Dynamical Systems} (New York: Springer-Verlag).
\\
Graham R 1989 in {\it Noise in Nonlinear Dynamical Systems}
edited by F. Moss and P. V. E. McClintock (Cambridge: Cambridge
University Press) vol 1, pp 225--278
\\
Haken H 1975  {\it Rev. Mod. Phys.} {\bf 47} 67
\\
Jauslin H R 1987 {\it J. Stat. Phys.} {\bf 42} 573
\\
Luchinsky D G 1997 {\it J. Phys. A} {\bf 30} L577
\\
Luchinsky D G and McClintock P V E 1997 {\it Nature} {\bf 389} 463
\\
Luchinsky D G, Maier R S, Mannella R, McClintock P V E and
Stein D L 1997 {\it Phys.\ Rev.\ Lett.} {\bf 79} 3109
\\
Luchinsky D G, Maier R S, Mannella R, McClintock P V E and
Stein D L 1999,  {\it Phys.\ Rev.\ Lett.} {\bf 82} 1806
\\
Maier R S and Stein D L 1996 {\it J. Stat. Phys.} {\bf
83} 291
\\
Onsager L and Machlup S 1953 {\it Phys. Rev.} {\bf 91} 1505--1512

\newpage

\begin{center}
\Large
{\bf Chaos in economic systems}
\normalsize

{\bf Erik Mosekilde}
  
{\em 
Center for Chaos and Turbulence Studies,
Department of Physics,\\
The Technical University of Denmark, Denmark} 
\end{center}                            
                                              
The basic assumption in most economic theory is that agents act in
accordance with rational expectations about the future. This implies
that all relevant information supplied to a market is immediately
taken into account, and the noise-like behavior observed in many
economic time series is considered to represent this fast and accurate
adjustment process. Nonetheless, the question remains whether the
concept of efficient markets is consistent with actual economic
conditions. The systematic phase shifts and lags observed between
various economic variables bear evidence of dynamic phenomena with
relatively long adjustment times, and asymmetries in the temporal
variations indicate the presence of nonlinear interactions.

In order to explore the nature of human decision making behavior in
complex, nonlinear systems, we have performed a series of role playing
experiments with a simulated microeconomic system. MIT students of
management and experienced managers from major US companies were asked
to operate a four-stage production distribution chain to ensure a
stable supply of beer. The observed performance was systematically
sub-optimal, leading in many cases to operational costs of more than
ten times the optimal. By virtue of the built-in delays and nonlinear
constraints, most players were unable to ensure a stable operation,
and large-scale fluctuations with characteristic amplitudes and phase
lags developed.

We have formulated a four-parameter anchoring and adjustment
heuristics for the ordering decisions in the game. With
econometrically estimated parameters for each individual participant,
this heuristics is found to reproduce the behavior of the system quite
well. At the same time it explains the sources of poor performance in
the game in term of insufficient levels of inventories, too aggressive
stock adjustments, or inadequate account of supply line variations.

Computer simulations with the estimated ordering heuristics produce
extremely complicated modes of behavior. For many parameter
combinations, including combinations within the range of realistic
policies, the system shows chaotic, hyperchaotic and higher-order
hyperchaotic solutions. It is also possible to observe various forms
of chaos-chaos intermittency.

{\bf References}
\\
J.S. Thomsen, E. Mosekilde, and J.D. Sterman, \textit{Hyperchaotic Phenomena 
in Dynamic Decision Making}, J. Syst. Analysis and Mod. Simulation {\bf 9}, 
137 (1992)
\\
O. Sosnovtseva and E. Mosekilde, \textit{Torus Destruction and Chaos-Chaos 
Intermittency in the Beer Model}, Int. J. Bifurcation and Chaos {\bf 7},
 1225 (1997)
\\
E. Mosekilde, \textit{Topics in Nonlinear Dynamics: Applications to Physics, 
Biology and Economic Systems} (World Scientific, Singapore, 1996).

\newpage                                   
\begin{center}
\Large
{\bf Synchronization of chaotic oscillators}
\normalsize

{\bf Erik Mosekilde}
  
{\em 
Center for Chaos and Turbulence Studies,
Department of Physics,\\
The Technical University of Denmark, Denmark} 
\end{center}                               
                                           

The ability of interacting, nearly identical chaotic oscillators to
synchronize their motions has been demonstrated for a variety of
different physical, chemical, and biological systems, and applications
for chaos suppression, for monitoring of dynamical systems, and for
different communication purposes are actively being pursued by many
investigators. An interesting question concerns the starting
conditions for which synchronization can be achieved. Other questions
relate to the sensitivity of the synchronized state to a small
parameter mismatch between the interacting oscillators and to the
behavior of the system, once synchronization breaks down.

Riddling of the basin of attraction for the synchronized chaotic state
arises when particular orbits embedded in this state become unstable
to asynchronous perturbations while the state remains attractive on
the average. Under these conditions the neighborhood of any point from
which the trajectories are attracted to the synchronized state will
contain initial conditions that are repelled from it. On-off
intermittency is an extreme form of intermittent bursting that can
sometimes be observed on the other side of the blowout bifurcation
where the synchronized state loses its average attraction.

Considering as a simple example two coupled logistic maps, the lecture
presents three different scenarios for the onset of riddling and for
the subsequent transformations of the basins of attraction. It is
demonstrated how the concepts of absorbing and mixed absorbing areas
play an essential role in explaining the different types of behavior:
The transition from local to global riddling, the distinction between
different types of noise response, and the occurrence of on-off
intermittency. Considering hereafter a system of two coupled
R\"{o}ssler systems it is shown how similar phenomena arise in time
continuous systems.

{\bf References}
\\
Yu. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde, 
\textit{Transverse Instability and Riddled Basins in a System of Two 
Coupled Logistic Maps}, Phys. Rev. E {\bf 57}, 2713 (1998)
\\
Yu. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde, 
\textit{Role of the Absorbing Area in Chaotic Synchronization}, Phys. 
Rev. Lett. {\bf 80}, 1638 (1998)
\\
E. Mosekilde, \textit{Topics in Nonlinear Dynamics: Applications 
to Physics, Biology and Economic Systems} (World Scientific, Singapore, 1996).

\newpage                                   

\begin{center}
\Large
{\bf Chaos in living systems}
\normalsize

{\bf Erik Mosekilde}
  
{\em 
Center for Chaos and Turbulence Studies,
Department of Physics,\\
The Technical University of Denmark, Denmark} 
\end{center}                

Physiological control systems offer one of the most interesting areas
of application for nonlinear dynamics. Complementary to the concept of
homeostasis that has dominated physiological thinking for such a long
time, unstable phenomena are increasingly being recognized as
significant for the regulation and function of normal physiological
systems. Many hormonal systems, for instance, operate in a pulsatile
mode with the release of insulin, growth hormone and luteinizing
hormone typically occurring with one-three hour intervals.

By regulating the excretion of salts and water the kidneys play an
important role in regulating the arterial blood pressure. To protect
their own function and secure a relatively constant supply of blood,
the kidneys also dispose of mechanisms that can compensate for
variations of the arterial pressure. This ability rests at least
partly with controls in the individual nephron, primarily the
so-called tubuloglomerular feedback (TGF). This is a negative feedback
that adjusts the diameter of the incoming arteriole in response to
variations in the chloride concentration of the fluid that leaves the
nephron.

Experiments on rat kidneys performed at The Department of Medical
Physiology, University of Copenhagen have demonstrated that the time
delays associated with the TGF mechanism tend to produce oscillations
in the nephron pressure and flow regulation with a typical period of
20-30 sec. For rats with elevated blood pressure, the oscillations are
often chaotic. Moreover, both regular and chaotic oscillations can be
observed to synchronize between neighboring nephrons.

The lecture presents a detailed physiological model of the feedback
controls of the individual nephron. A bifurcation analysis of this
model reveals the existence of so-called crossroad structures of
overlapping saddle-node and period-doubling bifurcations that can
explain the experimental observations. The model is subsequently
extended to consider the coupling between two neighboring nephrons
that share the same afferent arteriole. The coupled nephron model
reveals a variety of different synchronization phenomena including
in-phase and anti-phase synchronization of regularly oscillating
nephrons and complete and partial synchronization of chaotic nephrons.

{\bf References}
\\
M. Barfred, E. Mosekilde, and N.-H. Holstein-Rathlou, \textit{Bifurcation 
Analysis of Nephron Pressure and Flow Regulation}, Chaos {\bf 6}, 280 (1996)
\\
J. Sturis et al., \textit{Phase-Locking Regions in a Forced Model of Slow 
Insulin and Glucose Oscillations}, Chaos {\bf 5}, 193 (1995)
\\
E. Mosekilde, \textit{Topics in Nonlinear Dynamics: Applications to Physics, 
Biology and Economic Systems} (World Scientific, Singapore, 1996).


\newpage


\begin{center}
\Large
{\bf  Asymptotic expansions beyond all orders: asymptotic description 
of chaotic dynamics
}
\normalsize


{\bf Katsuhiro Nakamura$^\dagger$ and Hiroyuki Kushibe$^\ddagger$}
  
{\em $^\dagger$Department 
of Applied Physics, Osaka City University, Osaka, Japan\\
$^\ddagger$Department 
of Applied Biochemistry, Osaka City University, Osaka, Japan
}
\end{center}                            

\normalsize
\noindent
Heteroclinic or homoclinic structures are  the well-known mechanism for a 
genesis of chaos in conservative dynamical systems. While  numerical
iterations of low-dimensional maps easily provide these complicated
structures,  it is very difficult to derive them analytically, as addressed
by Poincar\'e more than a century ago.

       However, the difficulty  can now be overcome by using a scheme of 
the  asymptotic expansions beyond all orders. The  scheme  was recently
sharpened with a help of  the theoretical tool of Borel summability and
Stokes phenomena. The pioneering works by Hakim and Mallick(1993) 
and Gelfreich et al(1994) exploited the asymptotics beyond all orders 
to elucidate the angle of separatrices splitting of  the standard map.      

       It is highly desirable to develop the scheme further to describe the
chaotic dynamics, i.e., the stretching and folding of the unstable manifold
impinging upon the hyperbolic fixed point.   Tovbis et al (1998),
Nakamura and Hamada(1996) and Nakamura (1997) attempted to describe 
analytically the chaotic dynamics for the Henon and cubic maps. However, 
they failed to derive the Stokes multiplier precisely. In order to overcome 
this problem, some new systematic approach should be invented.

      In this talk, dealing with the standard and cubic maps., i.e., 
time-discrete dynamical systems with cosine and double-well  potentials, 
respectively, we shall provide an asymptotic analytical description of  the  
complicated heteroclinic and homoclinic structures on extremely  fine
scales and compare the issue with that of  an exact numerical iteration of
the maps. Technical problems around the Borel summability and Stokes 
phenomena are also discussed in detail. The talk is based on the
latest work by Nakamura and Kushibe (1999).


\noindent{\bf References}
\\
Gelfreich V G, Lazutkin V F and  Svanidze N V  1994  
{\em Physica D} {\bf71}  82
\\
Hakim V and Mallick K 1993  {\em Nonlinearity}  {\bf 6}  57.
\\
Nakamura K  1997 {\em Quantum versus Chaos: Questions emerging from    
     Mesoscopic Cosmos} (Dordrecht: Kluwer Academic).
\\
Nakamura K and Hamada M 1996 {\em J. Phys.A} {\bf 29}  7315; 
\\
Nakamura K and Kushibe H 1999, {\em preprint}.
\\
Tovbis A , Tsuchiya  M and Jaffe C 1998 {\em Chaos} {\bf 8} 665.

\newpage

\begin{center}
\Large
{\bf  Semiclassical theory of \\
mesoscopic 3-dimensional billiards:\\ 
orbital magnetism and persistent current}

\normalsize
{\bf Jun Ma and Katsuhiro Nakamura}
  
{\em 
Department of Applied Physics, Osaka City University, Osaka, Japan}
\end{center}                            

\normalsize
\noindent
In the mesoscopic billiards, chaos is known to play  an important role in
the orbital diamagnetism as pointed out by Nakamura(1993). While subsequent
papers by Oppen(1994) and Richter et al(1996) dealt with mesoscopic 2-d
billiards from the semiclassical viewpoint, quite few works have appeared
about the rich phenomena in orbital diamagnetism of 3-d billiards.

       In this talk, the magnetic susceptibility of ballistic 3-d billiards is 
investigated on the basis of the independent-electron model, for both 
single-billiard and ensemble-averaged cases. Using a semiclassical method, 
three kind of 3-d billiards subjected to weak uniform magnetic field are 
analyzed: spherical shell billiards; chaotic 3-d billiards but having SO(2) 
rotational symmetry; completely chaotic 3-d billiards without any 
continuous symmetry. We particularly elucidate the role of geometric 
symmetry in the susceptibility. According to different symmetries, 
different trace formulas (e.g., the Gutzwiller trace formula extended 
so as to incorporate continuously-degenerate periodic orbits) are exploited.

       We first calculate the magnetic susceptibility for a spherical shell
at finite temperature. Periodic orbits lying on great circular planes that
share the center with both of the inner and outer spherical shells give 
rise to the susceptibility.  The obtained result is compared with the 
corresponding results for a SO(2) rotational-symmetric chaotic billiard 
and for a completely chaotic 3-d billiard. Furthermore, the asymmetry 
in the magnetic response is found for 3-d billiard with SO(2) symmetry. 

       The analysis is also made on the persistent current in the 
corresponding three kinds of 3-d ``shell" billiards. The talk is based on 
the latest works by Ma and Nakamura (1999).


\noindent{\bf References}
\\
Ma J and Nakamura K, {\em preprints 1\&2}.
\\
Nakamura K 1993 {\em Quantum Chaos: a New Paradigm of Nonlinear Dynamics}
     (Cambridge:  Cambridge University Press).
\\
Richter K et al 1996 {\em Phys. Rep.} {\bf 276}  1. 
\\
von Oppen F 1994 {\em Phys. Rev. B} {\bf 50} 17151.

\newpage

\begin{center}
\Large
{\bf Quantum mesoscopic systems: \\
I. Random matrix theory of the conductance of a chaotic cavity}
\normalsize

{\bf Jean-Louis Pichard}
  
{\em CEA Saclay, Service de Physique de l'Etat Condens\'e,\\
 91191 Gif sur Yvette cedex, France}
\end{center}                            

\normalsize
\noindent
In this first lecture, the conductance of a ballistic quantum dot 
(having classical chaotic dynamics and being coupled by ballistic 
point contacts to two electron reservoirs) will be described on the 
single assumption that its scattering matrix is a member of Dyson's 
circular ensemble. The weak-localization corrections to the average 
conductance and the universal amplitude of the conductance fluctuations 
will be calculated. This example will show how the methods of random 
matrix theory can be implemented for describing quantum transport  
when the electron-electron interaction remains negligible (non interacting 
quasi-particles). The results discussed in this lecture have been published 
in Jalabert, Pichard and Beenakker (1994) and in Baranger and 
Mello (1994). Reviews for the random matrix theory of quantum transport 
can be found in Pichard (1991), Stone, Mello, Muttalib and Pichard (1991), 
Beenakker (1997) and Guhr, M\"uller-Groeling, and Weidenm\"uller (1998).

\noindent{\bf References}
\\
R.A. Jalabert, J.-L. Pichard and C.W.J. Beenakker, 1994,  
{\em Europhys. Lett.} {\bf 27}, 255.
\\
H. Baranger and P.A. Mello, 1994, {\em Phys. Rev. Lett.}
{\bf 73}, 142.
\\
J.-L. Pichard, 1991, in {\em Quantum coherence in Mesoscopic Systems}, 
B. Kramer ed, NATO-ASI Series B, Physics Vol 254, 369-400.
\\
A.D. Stone, P.A. Mello, K.A. Muttalib and J.-L. Pichard, 1991, in {\em 
Mesoscopic Phenomena in Solids} eds. B.L. Altshuler, P.A. Lee and 
R.A. Webb (Amsterdam: North-Holland), 369-449.
\\
C.W.J. Beenakker, 1997, {\em Rev. of Modern Phys.} {\bf69},731.
\\
T. Guhr, A.  M\"uller-Groeling and H. Weidenm\"uller, 1998,
{\em Phys.Rep.} {\bf 299}, 189.
\\

\newpage

\begin{center}
\Large
{\bf Quantum mesoscopic systems: \\
II. Chaotic mixing of the one body states by 
electron-electron interactions and delocalization in one dimension.}
\normalsize

{\bf Jean-Louis Pichard}
  
{\em CEA Saclay, Service de Physique de l'Etat Condens\'e, 
Gif sur Yvette, France}
\end{center}                            

\normalsize
\noindent
 
  The effect of electron-electron interactions when the one body states 
are localized will be introduced in a simple limit: two interacting 
particles (TIP) in a disordered chain. This problem was introduced 
by Shepelyansky (1994) and has been studied by Pichard et al (1996-1998). 
The TIP localization length will be estimated and the TIP spectral 
statistics will be discussed. The strength of the interaction leading 
to a maximum mixing of the one body states and to a maximum enhancement 
of the TIP localization length will be derived. Then, we will consider 
the ground state properties of a finite density of spinless fermions with 
short range interactions in a disordered chain. The enhancement of the 
persistent current driven by an Aharonov-Bohm flux when the system is 
between the Anderson (no interaction) and the Mott limits (correlated array 
of charges for strong interaction) will be discussed following Schmitteckert 
et al (1998). 

\noindent{\bf References}
\\
D. L. Shepelyansky, 1994, {\em Phys. Rev. Lett. } {\bf 73}, 2607.
\\
D. Weinmann and J.-L. Pichard, 1996, {\em Phys. Rev. Lett.}
{\bf 77}, 1556.
\\
X. Waintal and J.-L. Pichard, 1999, {\em Eur. Phys. J.} {\bf B6}, 117.
\\
X. Waintal, D. Weinmann and J.-L. Pichard, 1999, {\em Eur. Phys. J } 
{\bf B7}, 451.
\\
X. Waintal, S. de Toro-Arias and J.-L. Pichard, 1999, {\em Eur. Phys. J } 
{\bf B}, in press.
\\
P. Schmitteckert, R. Jalabert, D. Weinmann and J.-L. Pichard, 1998,
{\em Phys. Rev. Lett.} {\bf 81}, 2308.
\\

\newpage

\begin{center}
\Large
{\bf Quantum mesoscopic systems: \\
III. Two dimensional gases of charges with Coulomb repulsions: 
A new metal between the Fermi glass and the Wigner crystal.}
\normalsize

{\bf Jean-Louis Pichard}
  
{\em CEA Saclay, Service de Physique de l'Etat Condens\'e,
 Gif sur Yvette, France}
\end{center}                            

\normalsize
\noindent
 
 The ground state and the low energy excitations of electron (or hole) 
gases confined in a two dimensional random potential are the subject 
of this third lecture. Without interaction, one cannot have 
a two dimensional metal according to the scaling theory of localization. 
Experiments pioneered by Kravchenko et al (1994) (and nowadays confirmed 
by many others experimental groups) have given evidences for unexpected 
insulator-metal transitions. We will review the results obtained by numerical 
studies of small clusters: When the Coulomb energy to Fermi energy ratio 
$r_s$ is small, the system is an Anderson insulator while one has a pinned 
Wigner crystal in the large $r_s$ limit. For intermediary $r_s$, there are 
growing numerical (Benenti et al 1999) and experimental evidences supporting 
the existence of a new metal with an ordered flow of enhanced persistent 
currents. 

\noindent{\bf References}
\\
S.V. Kravchenko, G.V. Kravchenko, J.E. Fourneaux, J.E. Pudalov 
and M. D'Iorio, 1994 {\em Phys. Rev. } {\bf B 50}, 8038.
\\
G. Benenti, X. Waintal and J.-L. Pichard, 1999, {\em cond-mat/} 9904096.
\\
G. Benenti, X. Waintal, J.-L. Pichard and D.L. Shepelyansky, 1999, 
{\em cond-mat/} 9903339.
\\

\newpage

\begin{center}
\Large
{\bf Integrability and transport in quantum many body systems}
\normalsize

{\bf Peter Prelov\v sek}

{\em Faculty of Mathematics and Physics, and J. Stefan Institute,
Ljubljana, Slovenia}
\end{center}

\normalsize
\noindent
The particle transport in macroscopic many-fermion quantum systems at
finite temperatures will be discussed. It will be argued that there is
a fundamental difference between the integrable and nonintegrable
systems [1]. Even when the current is not a conserved quantity the
integrable models show a dissipationless transport, they can in fact
behave as ideal conductors in the metallic regime and as ideal
insulators in the insulating regime, while nonintegrable systems are
expected to behave as generic conductors or resistors [2]. In this
connection the role of conservation laws will be pointed out [3] and a
relation to the level statistics will be examined.

\noindent{\bf References}\\
{[1]} H. Castella, X. Zotos, and P. Prelov\v sek, Phys. Rev. Lett. {\bf
74}, 972 (1995).
\\
{[2]} X. Zotos and P. Prelov\v sek, Phys. Rev. B {\bf 53}, 983
(1996).
\\
{[3]} X. Zotos, F. Naef, and P. Prelov\v sek, Phys. Rev. B {\bf 55}, 11029
(1997).

\newpage

\begin{center}
\Large
{\bf Quantum Poincar\' e mapping}
\normalsize

{\bf Toma\v z Prosen}

{\em Physics Department, Faculty of Mathematics and Physics, \\
University of Ljubljana, Slovenia}
\end{center}

\normalsize
\noindent
The lecture will give a review of the recent theoretical work 
on the quantization of Poincar\' e mapping on the {\em surface of section} 
(SOS) of general bounded and autonomous Hamiltonian systems.
  
We will begin, historically, to describe the {\em semi-classical} 
construction by Bogomolny [1] of the approximate {\em quantum
Poincar\' e mapping} (QPM). Then we will 
formulate the scattering approach to quantization, which has been 
introduced for billiards by Smilansky et al [2], and which can be 
reinterpreted as an approach to exact quantization on SOS. We will 
show [3] that one can generally construct the {\em exact} QPM 
$T(E) = T_\uparrow(E) T_\downarrow(E)$, as the product of two 
scattering operators $T_{\uparrow,\downarrow}$ of the associated scattering 
problems which are obtained by amputing upper/lower half of configuration 
space along the configurational SOS and attaching a semi-infinite flat 
`wave-guide' instead. Although this form of QPM enables one to write an exact 
{\em quantization condition} (QC) as $\det(1-T(E))=0$ and has the correct 
semi-classical limit of Bogomolny (as $\hbar\rightarrow 0$), it has
a disadvantage of not being strictly unitary operator, so it does not 
conserve probability, 
and hence, strictly speaking, it cannot be interpreted as
an evolution operator on SOS. 

However, we propose a simple and almost unique technique of
{\em unitarization} of QPM [4] $T(E) \longrightarrow {\cal T}(E)$,
which yields the strictly unitary QPM ${\cal T}(E)$ on the Hilbert
space of $L^2$ functions over the configurational SOS
(which has one degree of freedom less than the full space problem),
and which at the same time preserves (i) the (semi)classical limit, 
as well as (ii) the exact QC, namely, 
$E$ is an eigenenergy iff $\det(1-{\cal T}(E))=0$,
or alternatively, iff one of the eigenphases $\varphi_n(E)$ of the
unitary QPM ${\cal T}(E)$ crosses zero, $e^{-i\varphi_n(E)}=1$.

The theoretical ideas will be worked out explicitly on an example of
a {\em semi-separable} system (Prosen 1996), a simple two-degree-of-freedom 
dynamical system which is separable above and below SOS, but is
discontinuous on SOS, hence in full space it is non-separable, non-integrable
and possibly chaotic (depending on parameters).


\noindent{\bf References}
\\
$[1]$ Bogomolny E B 1992 {\em Nonlinearity} {\bf 5} 805; 
1990 {\em Comments At.Mol.Phys.} {\bf 25} 67
\\
$[2]$ Doron E and Smmlansky U 1992 {\em Nonlinearity} {\bf 5} 1055,
Dietz B and Smilansky U 1993 {\em Chaos} {\bf 3} 581,
Schanz H and Smilansky U 1995 {\em Chaos, Solitons \& Fractals} {\bf 5} 1289
\\
$[3]$ Prosen T 1995 {\em J. Phys. A: Math. Gen.} {\bf 28} 4133;
1994 {\em J. Phys. A: Math. Gen.} {\bf 27} L709;
1997 {\em Open Sys.\& Information Dyn.} {\bf 4} 339
\\
$[4]$ Prosen T 1996 {\em Physica} {\bf D 91} 244;
1995 {\em J. Phys. A: Math. Gen.} {\bf 28} L349


\newpage


\begin{center}
\Large
{\bf Quantum chaos, ergodicity and mixing in 
generic many-body systems in thermodynamic limit}
\normalsize

{\bf Toma\v z Prosen}

{\em Physics Department, Faculty of Mathematics and Physics, \\
University of Ljubljana, Slovenia}
\end{center}

\normalsize
\noindent
We address an old unsolved problem of (quantum) statistical
mechanics, namely the justification of the ergodic hypothesis, 
and consequently, the `derivation' or dynamical understanding of 
macroscopic statistical laws, e.g. the transport laws such as Ohm's, Fourier's 
or Fick's law, from the (non-integrable) microscopic equations of motion.
To this end we propose to study (i) the dynamics (direct time-evolution), 
and (ii) the existence of dynamical conservation laws (integrals of motion),
of simple quantum-many body dynamical system for which thermodynamic
limit can be approached numerically. This we do [1] 
by means of efficient (and presumably novel) numerical methods, 
namely, by (i) multi-dimensional (anti)symmetrized Fast Fourier 
Transformation, and (ii) solving Galerkin-like variational problems in
truncated operator Hilbert spaces of quantum observables of an
infinite many-body system, respectively.

We define $[1]$ a generic but simple non-integrable quantum 
{\em many-body} system of {\em locally} interacting particles, namely a 
kicked $t-V$ model of spinless fermions on 1-dim lattice (equivalent to a 
kicked Heisenberg XX-Z spin$-1/2$ chain). Statistical properties of 
dynamics (quantum ergodicity, quantum mixing $[2]$) and the nature of 
quantum transport in {\em thermodynamic limit} are considered as the kick 
parameters (which control the degree of non-integrability) are varied.

We find and demonstrate {\em ballistic} transport 
and non-ergodic, non-mixing dynamics (implying infinite conductivity at 
all temperatures) in the {\em integrable} regime of zero or very small kick 
parameters (as conjectured already in $[3]$), and more generally and 
important, also in {\em non-integrable} regime of {\em intermediate} values 
of kicked parameters, whereas only for sufficiently large kick parameters we 
recover quantum ergodicity and mixing implying normal (diffusive) transport. 
Based on numerical results we propose a (dynamical) phase transition
with an order parameter $D$ (charge stiffness, which is the time-averaged
current auto-correlation function) from {\em non-ergodic} and
{\em non-mixing} quantum dynamics (ordered phase, $D > 0$)
to ergodic and mixing dynamics (disordered phase, $D = 0$) in the
thermodynamic limit.

Similarity with order-to-chaos transition of generic non-integrable
one-particle quantum systems in semi-classical limit is striking, 
and we mention a result [4] on close formal 
relation between dynamics of a particular class of non-linear quantum 
many body systems (in thermodynamic limit) and an associated class of 
{\em one-body} quantum systems (in classical limit).
\vfil

\noindent{\bf References}
\\
$[1]$ Prosen T 1998 {\em Phys. Rev. Lett.} {\bf 80} 1808;
1998 {\em J. Phys. A: Math. Gen.} {\bf 31} L645;
1998 `Ergodic properties of a generic non-integrable quantum many-body
system in thermodynamic limit', preprint cond-mat/9808150
\\
$[2]$ Jona-Lasinio G and Presilla C 1996 {\em Phys. Rev. Lett.} 
{\bf 77} 4322.
\\
$[3]$ Castella H, Zotos X, and Prelov\v sek P 1995 {\em Phys. Rev. Lett.} 
{\bf 74} 972; 
Zotos X and Prelov\v sek P, 1996 {\em Phys. Rev. B} {\bf 53} 983;
Zotos X, Naef F, and Prelov\v sek P 1997 {\em Phys. Rev. B} {\bf 55} 11029
\\
$[4]$ Prosen T 1998 `A Map from 1d Quantum Field Theory to Quantum Chaos on a
2d Torus', preprint cond-mat/9809211

\newpage

\begin{center}
\Large
{\bf
Chaotic dynamics and statistical mechanics  in  Hamiltonian 
systems with many degrees of freedom
}
\normalsize

{\bf Andrea Rapisarda}
  
{\em

Dipartimento di Fisica,  Universit\'a di Catania, \\
and 
Istituto Nazionale 
di Fisica Nucleare, Sezione di Catania, Italy\\
}
\end{center}                            

\normalsize
\noindent
We discuss the connection between chaotic dynamics 
and equilibrium statistical mechanics in systems 
with many degrees of freedom.
The talk will focus in particular on  the links  between 
microscopic chaos and phase transitions 
in the Hamiltonian Mean Field model (HMF), Antoni and Ruffo (1995). 
The latter is  a simple toy model of 
$N$ fully-coupled rotators which shows a second order phase transition. 
Recently  
the equilibrium ~ properties and the dynamical ~ behavior of ~ HMF~  
have been  ~ investigated numerically and analytically in several papers 
by Latora, Rapisarda and Ruffo (1998),(1999) . Our main result is that  
both the largest Lyapunov exponent and Kolmogorov-Sinai entropy 
show a peak at the critical energy $U_c$. 
This result is checked to be valid in the limit $N \to \infty$. 
Scale law behaviors for the Lyapunov exponents are derived, and an 
important link between chaoticity and the 
thermodynamical fluctuations, expressed by kinetic energy 
fluctuations or specific heat, is found.   
The canonical thermodynamical Vlasov solution has been  successfully checked
in numerical simulations. In general, chaotic dynamics provides the mixing 
property in phase space necessary for obtaining equilibration. However
relaxation to equilibrium can be very slow  and grows with $N$ 
close to the critical point where metastable states are found.  
Possible connection to anomalous 
transport properties  and Levy walks will be   discussed in connection 
to the out-of-equilibrium regime.  
Our results seem to be  very general 
and  confirm a previous pioneering work by Bonasera, Latora and Rapisarda
 1995. Lately similar features have been 
found also in other models by Caiani et al (1998),
 Anteneodo and Tsallis (1998),
Torcini and Antoni (1999).  
The results discussed for this toy model  can be very useful 
in order to understand the  multifragmentation phase transition
in excited nuclei, Atalmi et al (1998), and clusters , 
Farizon et al (1998),  and could be important 
also in astrophysics for 
autogravitating systems,  Yawn and Miller
(1997).  
In general this new scenario  provides an interesting bridge 
between Hamiltonian chaos in systems with many degrees of freedom 
and  statistical mechanics. 



\noindent{\bf References}
\\
Anteneodo C and Tsallis C 1998 
{\em Phys. Rev. Lett } {\bf 80} 5313
\\
Antoni M and Ruffo S 1995 {\em Phys. Rev.} {\bf E 52}
2361
\\
Atalmi A, Baldo M, Burgio G F and Rapisarda A 1998 {\em Phys. Rev.  } 
{\bf C 58 } 2238
\\
Bonasera A, Latora V and Rapisarda A 1995 
{\em Phys. Rev. Lett.} {\bf 75}
3434
\\
Caiani L, Casetti L, Clementi C and  Pettini G, Pettini M and Gatto R 
1998 {\em Phys. Rev. } {\bf E 57} 3886
\\
Farizon B et al  1998 
{\em Phys. Rev. Lett.} {\bf 81}
4108
\\
Latora V, Rapisarda A and Ruffo S 1998 {\em Phys. Rev. Lett. } {\bf 80}
698
\\
Latora V, Rapisarda A and Ruffo S 1999 {\em Physica  } {\bf D}
in press, chao-dyn/9803019
\\
Torcini A and Antoni M 1999 
{\em Phys. Rev. } {\bf E} in press, cond-mat/9808068
\\
Yawn K R and Miller B N 1997 
{\em Phys. Rev. Lett.} {\bf 79}
3561
\\

\newpage


\begin{center}
\Large
{\bf Chaos in quantum transport:\\
From electron billiards to Coulomb blockade }
\normalsize

{\bf Klaus Richter}
  
{\em Max-Planck-Institut f\"ur Physik komplexer Systeme,\\
Dresden, Germany}
\end{center}                            

\normalsize
\noindent
We review aspects of classical and quantum transport through 
electronic mesoscopic systems. In the first part of the talk
we discuss properties of open conductors which represent
realizations of non-interacting chaotic quantum billiards. We 
introduce semiclassical concepts  
which allow us to relate quantum coherence effects in the 
conductance to properties of classical irregular scattering
in these billiards (Baranger et al.\ 1993, Stone 1995, Richter 1999).
We highlight the achievements of these
approaches for the understanding of related recent experiments
on transport through semiconductor nanostructures.
We furthermore discuss problems of the standard semiclassical 
transport theory to account adequately for weak localization,
a quantum correction to the average conductance,
in ballistic systems. We outline new theoretical advances 
which may cope with these problems and introduce the concept
of the Ehrenfest time in the context of
chaotic transport (Aleiner and Larkin 1996). 
These approaches enable us
to extract from the {\em quantum} weak localization
correction the mean Liapunov exponent of the 
{\em classical} counterpart of a ballistic mesoscopic
system (Yevtushenko et al.\ 1999).

Upon reducing the coupling of the quantum dot (billiard) to the
leads, interaction effects become important  (Kastner 1992). In the second part
of the talk we thus describe novel interaction phenomena 
in this {\em Coulomb blockade} regime; namely strong fluctuations
in the conductance-peak heights and spacings. (Experimentally)
observed fluctuations in the related ground state energies do not follow
Wigner-Dyson (GOE- or GUE-) statistics
(Sivan et al.\ 1996, Simmel et al.\ 1999)
 and call for refined random
matrix approaches and a more careful consideration of level
statistics of interacting electrons in chaotic quantum dots.
We review recent theoretical approaches and discuss in particular
the interplay between electron-electron interactions and the
confinement geometry (integrable/chaotic) of ballistic quantum dots.
We do this 
on the basis of both exact diagonalization for few particles and 
self-consistent Hartree-Fock calculations for a larger number of electrons.

\noindent{\bf References}
\\
Baranger, H.U., R.A.~Jalabert, and A.D.~Stone, Chaos {\bf 3}, 665 (1993).
\\
Stone, A.D.\ in  {\it Mesoscopic Quantum Physics,} 
        edited by E. Akkermans, G. Montambaux, J.-L.\ Pichard, and J.\
       Zinn-Justin (Elsevier, New York, 1995).
\\
Richter, K.,  {\em Semiclassical Theory of Mesoscopic Quantum Systems},
                 Springer Tracts in Modern Physics, in print (1999).
\\
Aleiner I.L.\ and A.I.\ Larkin, Chaos, Solitons \& Fractals {\bf 8}, 1179 
(1997).
\\
 Oleg Yevtushenko, Gerd L\"utjering, Dieter Weiss, and
         Klaus Richter, preprint MPIPKS-9902004 (1999).
\\
Kastner, M.,  Rev.~Mod.~Phys.\ {\bf 64} 849 (1992).
\\
Sivan, U.,
R. Berkovits, Y. Aloni, O. Prus, A. Auerbach and G. Ben-Yosef,
Phys.\ Rev.\ Lett.\ {\bf 77}, 1123 (1996).
\\
Simmel, F.,  D.\ Abusch-Magder, D.A.\ Wharam, M.A.\ Kast\-ner,
 and J.P.\ Kott\-haus (cond-mat 9901274).

\newpage

\begin{center}
\Large
{\bf  Topics in quantum chaos of generic systems I \& II}
\normalsize

{\bf Marko Robnik}
  
{\em Center for Applied Mathematics and Theoretical Physics,\\
University of Maribor, Maribor, Slovenia}
\end{center}                            

\normalsize
\noindent
We review the main ideas and results in the
stationary problems of quantum chaos in generic (mixed) systems,
whose classical dynamics has regular (invariant tori) and
chaotic regions coexisting in the phase space. First we
discuss the universality classes of spectral fluctuations
(GOE/GUE for ergodic systems, and Poissonian for integrable
systems). We explain the problems in the
calculation of the invariant (Liouville) measure of
classically chaotic components, which has recently been
studied by Robnik et al (1997) and by Prosen and Robnik (1998).
Then we describe the Berry-Robnik (1984) picture,
which is claimed to become exact in the strict semiclassical
limit $\hbar\rightarrow 0$.
However, at not sufficiently small values of $\hbar$ we see
a crossover regime due to the localization properties of
stationary quantum states where Brody-like behaviour with
the fractional power law level repulsion is observed in
the corresponding quantal energy spectra.

\noindent{\bf References}
\\
Aurich R, B\"acker A and Steiner F 1997 {\em Int. J. Mod. Phys.} {\bf 11}
805
\\
Berry M V 1983 in {\em Chaotic Behaviour of Deterministic Systems}
eds. G Iooss, R H G Helleman and R Stora  (Amsterdam: North-Holland)
pp171-271
\\
Berry M V 1991 in {\em Chaos and Quantum Physics} eds.
M-J Giannoni, A Voros and J Zinn-Justin (Amsterdam: North-Holland)
pp251-303
\\
Berry M V and Robnik M 1984 {\em J. Phys. A: Math. Gen.} {\bf 17} 2413
\\
Bohigas O 1991 in {\em Chaos and Quantum Physics} eds.
M-J Giannoni, A Voros and J Zinn-Justin (Amsterdam: North-Holland) pp87-199
\\
Bohigas O, Giannoni M.-J. and Schmit C 1984 {\em Phys. Rev. Lett.} {\bf
25} 1
\\
Casati G and Chirikov B V 1994 in {\em Quantum Chaos: Between Order and
Disorder} eds. G. Casati and B.V. Chirikov (Cambridge: Cambridge University
Press)
\\
Guhr T, M\"uller-Groeling A and Weidenm\"uller H A 1998,
{\em Phys.Rep.} {\bf 299} 189
\\
Li Baowen and Robnik M 1994 {\em J. Phys. A: Math. Gen.} {\bf 27} 5509
\\
Li Baowen and Robnik M 1995a {\em J. Phys. A: Math. gen.} {\bf 28} 2799
\\
Li Baowen and Robnik M 1995b {\em J. Phys. A: Math. gen.} {\bf 28} 4843
\\
Prosen T and Robnik M 1993a {\em J. Phys. A: Math. Gen.} {\bf 26} L319
\\
Prosen T and Robnik M 1993b {\em J. Phys. A: Math. Gen.} {\bf 26} 1105
\\
Prosen T and Robnik M 1993c {\em J. Phys. A: Math. Gen.} {\bf 26} 2371
\\
Prosen T and Robnik M 1993d {\em J. Phys. A: Math. Gen.} {\bf 26} L37
\\
Prosen T and Robnik M 1994a {\em J. Phys. A: Math. Gen.} {\bf 27} L459
\\
Prosen T and Robnik M 1994b {\em J. Phys. A: Math. Gen.} {\bf 27} 8059
\\
Robnik M and Prosen T 1997 {\em J. Phys. A: Math. Gen.} {\bf 30} 8787
\\
Robnik M 1984 {\em J. Phys. A: Math. Gen.} {\bf 17} 1049
\\
Robnik M 1988 in {\em "Atomic Spectra and Collisions in External Fields"}, eds.
K T Taylor, M H Nayfeh and C W Clark, (New York: Plenum) pp265-274
\\
Robnik M 1998 {\em Nonlinear Phenomena in Complex Systems} {\bf 1} 1

\newpage


\begin{center}
\Large
{\bf Chaotic Jung scattering maps versus topological chaos of the 
Hamiltonian  flow}
\normalsize

{\bf Thomas H. Seligman}

{\em Centro Internacional de Ciencias\\
and Centro de Ciencias F\'isicas, University of Mexico, UNAM,\\
Cuernavaca, M\'exico}
\end{center}

\normalsize
\noindent
 The classical scattering map proposed by Jung is the classical 
 analog of the quantum S-matrix. It therefore contains the 
 information we can obtain by asymptotic measurement for a classical 
 scattering system. A detailed analysis of the relation of 
 integrability and chaos for the Jung map and the Hamiltonian flow 
 shows that the connection is complicated and dependent of the choice 
 of the ``unperturbed Hamiltonian'' $ H _ 0 $. For the typical choice 
 of $ H _ 0 $, namely the free particle, we can readily construct 
 examples where the flow is integrable, but the scattering map is 
 chaotic. This is due to the incompatibility of the integrals of 
 motion of the Hamiltonian $ H $ with those of $ H _{0}$. To transfer 
 the integrability of $ H $ to the integrability of the 
 Jung map it is necessary and sufficient that $ H_{0 }$ conserves at 
 least asymptotically the integrals of motion of $ H $.
 
 This implies that it is not particularly simple to distinguish chaos 
 or integrability from scattering functions. Some simple examples 
 illustrate that selfsimilar and even fractal structures in the 
 scattering functions may result from integrable Hamiltonians.
 
 We may pass from classical to quantum mechanics if we realize that the Jung 
 map is a canonical map of channel space (considered as a symplectic 
 manifold or phase space) onto itself and that the 
 S-matrix is the unitary representation of this map. This implies 
 that the integrability properties of this map determine those of the 
 S-matrix. The criteria of integrability carry over easily to the 
 quantum case. 
 
 There is a further interesting aspect that results from the 
 interpretation of the S-matrix as a representation of the Jung map.
 The statistical properties of the spectrum of the S-matrix
  not determined by integrability or chaos in the 
 Hamiltonian flow, but rather by the corresponding properties of the 
 Jung map. This can be understood in terms of  a series of arguments 
 centered around the concept of { \it structural invariance}, which 
 establish the connection between chaos and random matrix 
 theory; This connection is based on a probabilistic argument which
 allows for all the exceptions and special cases that exist in this context. 
 It is short of being a proof because the existence of an invariant 
 measure on the group of invertible canonical transformations has to be 
 established.
 
 
\noindent{\bf References}
\\
C. Jung, J. Phys. {\bf 19} (1986) 1345
\\
C. Jung, T.H. Seligman  Phys Rep  {\em Int. J. Mod. Phys.} {\bf 11} 
(1997) 805
\\
C. Jung, C. Mej\'\i a and T.H. Seligman,  Phys. Lett. A {\bf 198} (1995) 306
\\
T.H. Seligman, Proceedings Wigner Symposium; Goslar 1991 
(World Scientific, Singapur,) (1992); T.H.  Seligman, {\it Quantum Chaos} 
Eds. G.Casati and B. Chirikov
Cambridge University Press 1995 p. 577; F. Leyvraz and T.H. Seligman, 
Phys Lett. {\bf A 168}, (1992), 348-352
 \\

\newpage

\begin{center}
\Large
{\bf Unitary quantum Poincar\'e maps and Rydberg molecules}
\normalsize

{\bf Thomas H. Seligman}

{\em Centro Internacional de Ciencias\\
and Centro de Ciencias F\'\i sicas, University of Mexico, UNAM,\\
Cuernavaca, M\'exico}
\end{center}

\normalsize
\noindent
Multichannel quantum defect theory (MQDT) has proven very successful for the 
calculation of Rydberg molecules. This theory uses results from scattering
theory to obtain information about states near, but below threshold.
In a particular approximation for the classical system, Lombardi showed
that MQDT provides the exact quantum solution. 
As this approximate system may be characterized by a symplectic or 
canonical map, MQDT can find a much more transparent interpretation
as a unitary representation of this map. A more careful analysis 
shows that indeed MQDT is a true quantum Poincar\'e map (QPM). Remember that 
Bogomolny's definition of a QPM is semi-classical. By consequence the map 
becomes
unitary only in the limit $ \hbar = 0 $. 
Prosen has proposed a very elegant unitarization of the QPM at the 
price of obtaining an infinite dimensional matrix.
In our case the QPM is constructed directly from unitary transformations 
between body fixed and space fixed frames as well as the phase shifts 
typical of MQDT, thus yielding a finite unitary matrix by construction.

Lombardi's result for the classical approximation has ingredients that are 
essential for MQDT to become the exact solution, but beyond that he uses 
the approximation that the absolute value of the electron angular momentum 
does not change. The latter approximation is not always good though it is 
very convenient because the Poincar\'e section is then two-dimensional, 
{\it i. e. } we can actually draw it. Yet this approximation is not 
essential to the reinterpretation of MQDT as a QPM

The result has several interesting applications beyond the fact that it 
affords a new understanding  of MQDT. 

First it can serve as a paradigm for a QPM and may even give a hint,
when and to what extent we really have to use semi-classics to construct a QPM.

Second the interpretation as a QPM allows for a much more stable 
construction of eigen-values and eigen-functions of energy from the  
unitary matrix given by the QPM.

Third the evolution of the eigen-phases as a function of energy is essential
to the possible transfer of properties of the eigen-phase spectrum to the 
energy spectrum. This transfer is of basic interest for the understanding 
of the relation of chaotic classical dynamics and random matrix theory, as 
the latter is easier to understand for circular ensembles. 

\noindent{\bf References}
\\
M.J. Seaton Rep. Prog. Phys. {\bf 48} (1983) 167 (and refs. therein)
\\
E.B. Bogomolny, Nonlinearity {\bf 5} (1992) 805; Chaos {\bf 2} (1992) 5
\\
T. Prosen, Physica D {\bf 91} (1996) 244
\\
M. Lombardi, P. Labastie, M.C. Bordas and M. Broyer, J. Chem. Phys. 
{\bf 89} (1988) 3479
\\
F. Leyvraz, M. Lombardi, R. Mendez-Sanchez and T.H. Seligman (to be published)
\\
F. Leyvraz and T.H. Seligman,
in Proceedings of the fourth Wigner Symposium (1995); Ed. N. Atakishiev et al., 
(World Scientific, Singapore 1996) P 350; F. Leyvraz, R.A. 
Mendez-Sanchez and T.H. Seligman (chao-dyn/9902009)

\newpage

\begin{center}
\Large
{\bf Rings: Generic structures of rotating systems}
\normalsize

{\bf Thomas H. Seligman}

{\em Centro Internacional de Ciencias\\
and Centro de Ciencias F\'\i sicas, University of Mexico, UNAM,\\
Cuernavaca, M\'exico}
\end{center}

\normalsize
\noindent
From Galilieos discovery until a few years ago Saturns rings were a 
singular phenomenon in the sky. This has radically changed since we 
know that all major planets carry rings. This leads to the suspicion 
that such ring structures are universal phenomena. Yet it is not 
at all clear what universality class we look at.

In the present contribution we shall demonstrate that one of the two 
universal scenarios for a sudden appearance upon variation of a 
parameter of a localized manifold in 
a Hamiltonian scattering system, will always lead to ring structures 
if we deal with a system rotating around some point in space.

Consider free motion in a rotating frame: the particle will approach 
the center of rotation up to the closest approach on a spiral 
trajectory and then leave the central region on an outgoing spiral.
We now include a repulsive potential with convex potential contour lines 
rotating with frequency $ \omega $ about a center that lies outside 
the potential or at least its most repulsive part. The particle may 
hit this potential on the ingoing or outgoing trajectory. In the 
latter case the interaction may throw it onto another ingoing 
trajectory. This will give opportunity for a second interaction if
the absolute value of the Jacobi-integral (Hamiltonian in the rotating 
system) is not to large. This limitation results because otherwise the 
particle leaves the region, where such an interaction is possible 
before the center of interaction has swung around a sufficient angle.
Thus confined trajectories will appear as we reduce (or increase for 
negative values) the Jacobi-integral.
There are two generic ways this can occur as a function of a 
parameter. One is the sudden appearence of hyperbolic horseshoes with 
infinite exponents. The other and the one we are interested in is 
the saddle-center
bifurcation, which will give rise to stable periodic orbits in the 
rotating frame. As these orbits are made up of slightly distorted 
parts of a spiral they have ring shapes, and these non-circular 
rings will have a precession with the frequency of rotation of the 
system. 

We shall use the model of a hard disc rotating about a center outside 
of the disc to illustrate our general findings. This model has the 
particular advantage that it does not allow for the alternate generic 
scenario that we mentioned. A single disc will produce a wide ring 
with a very involved structure, but the introduction of a second disc
moving on an inner circle with an incommensurable frequency will be 
shown to produce narrow rings with structure.

\noindent{\bf References}
\\
{\it Planetary rings}, R. Greenberg and 
A. Brahic (eds.). The University of Arizona Press, Tucson (1984);
\\
B.A. Smith, and the Voyager imaging team, Science {\bf 212} 
163 (1981); {\bf 215} 504 (1982).
\\
L. Benet, T.H. Seligman (submitted for publication)
\\
 N. Meyer {\it et al.}, J. Phys A: Math. Gen.
{\bf 28} 2529 (1995).
\\
 M. Ding, {\it et al.}, 
Phys. Rev. A {\bf 42}, 7025 (1990); Phys Lett. A {\bf 153}, 21 
(1991).
\\
 T. T\'el, C. Grebogi and E. Ott, {\it Chaos} 
{\bf 3} (1993) 495.

\newpage

\begin{center}
\Large {\bf Complex trajectory description for\\ quantum tunnelling I }
\normalsize

{\bf Akira Shudo}

{\em Department of Physics, Tokyo Metropolitan University,\\
Tokyo, Japan}
\end{center}

\normalsize
\noindent
Tunnelling phenomenon is one of the most familiar quantum 
effects. The wavepacket in quantum mechanics can penetrate even into
the region where the classical trajectory cannot reach.
The barrier which prevents the transition to classically 
forbidden regimes is always formed by the energy potential 
in case of one degree of freedom system. 
However, in higher dimension, in addition to the energy, 
there sometimes appear dynamical barriers (KAM Tori) which are usually 
coexisting with chaotic components in the phase space.
The classical trajectories are also confined dynamically if they 
are placed in such regions initially. The quantum wavepacket can also 
go over such dynamical barriers, and tunnelling penetration out of 
dynamically formed barriers is called $\lq\lq$dynamical tunnelling"
(Heller $\&$ Davis 1981, Bohigas, Tomsovic $\&$ Ullmo 1993, Creagh 1998).

The purpose of our talk is to show how the essential 
difference between tunnelling in one dimension and that in more than 
one-degree-of-freedom systems is explicitly understandable in terms of 
the semiclassical methodology which has been elaborated extensively 
in recent studies of quantum chaos. The main tool we will use 
is the complex semiclassical theory in which not only usual real 
classical orbits but also complex classical trajectories play a 
key role in describing the purely quantum mechanical
phenomena such as tunnelling (Miller 1974).

We first introduce a simple model (kicked rotor), which is 
suitably designed to test the dynamical tunnelling as purely as 
possible and observe the time evolution of its wavefunction, 
especially by focusing on the tunnelling tail.
The complex semiclassical analysis is then performed, aimed at 
the interpretation of various features discovered in the tunnelling 
wavefunction in chaotic systems.
The procedure of the complex semiclassical analysis is almost the same 
as the real one, but in contrast with the real semiclassics, 
finding a definite rule to select {\it dominant complex 
orbits} becomes a crucial task, since in chaotic systems 
there appear vastly many number of candidate complex 
trajectories, all of which can contribute to the semiclassical propagator, 
but not all of which contribute equally in weights. 

The main message we will present in our first lecture is that 
it is indeed possible to select such important complex orbits under  
a certain rule, and the origin of various remarkable features 
observed in the tunnelling tail in chaotic systems, which are completely 
absent in integrable systems, can be explained by the group of complex 
manifolds forming the chain-like structure 
in the initial value representation of complex trajectories
(Shudo $\&$ Ikeda 1998).



\noindent{\bf References}
\\
Heller E.J. and Davis M.J. 1981 J. Chem. Phys. {\bf 75} 246.
\\
Creagh S.C. 1998 in {\it Tunnelling in Complex Systems} eds.
 S. Tomsovic (World Scientific)35.
\\
Bohigas, O., S. Tomsovic and D. Ullmo 1993 Phys. Rep. {\bf 223} 45.
\\
Miller W.H. and George T.F. 1974 Adv. Chem. Phys. {\bf 25} 69.
\\
Shudo A. and Ikeda K.S. 1998 Physica D {\bf 115} 234.
\\

\newpage
\begin{center}
\Large
{\bf Complex trajectory description for\\ quantum tunnelling II}
\normalsize

{\bf Akira Shudo}

{\em Department of Physics, Tokyo Metropolitan University,\\
Tokyo, Japan}
\end{center}

\normalsize
\noindent 
The subject of our second lecture is to present some theoretical backgrounds
for the semiclassical description of tunnelling in the presence of chaos, 
especially to discuss the {\it contributing and non-contributing problems} 
in the complex semiclassical approach. 

In applying the complex semiclassical method, 
we inevitably face the problem of how to select the contributing 
complex (tunnelling) paths from all the candidates which satisfy 
the saddle point condition in principle.
There would essentially be two distinct origins:

The first one is concerned with how to 
find the most largely weighted complex paths among vastly many 
complex candidates which should {\it a priori} be included 
in the semiclassical superposition. 
The weight of an individual contribution is almost determined by the 
imaginary part of the action along each classical trajectory, and 
the paths with large
imaginary actions may be negligible in the semiclassical sum.
This problem is purely classical, in particular it should be 
solved in the context of the {\it complex classical dynamics}
(Milnor 1990).

We will show that the most relevant complex classical trajectories 
in describing the tunnelling effect in chaotic systems are
very closely related to the {\it Julia set} in the corresponding 
complex dynamical system.
More precisely, it can be shown that the {\it Laputa chain}, which 
is the most important phenomenological object introduced 
to explain all the features of tunnelling in chaotic systems, 
is nothing but the intersection between the initial Lagrangian 
manifolds and the complex stable manifolds of saddles in the Julia set
(Shudo $\&$ Ikeda 1999). 

The second contributing and non-contributing problem arises from the 
{\it Stokes phenomenon} in multi-dimensions.
The Stokes phenomenon is the discontinuous change of the asymptotic 
solutions, and it occurs quite generically in the saddle point 
method or the differential equation with large parameters. 
Due to the Stokes phenomenon, not all of the complex 
classical trajectories necessarily contribute to the final semiclassical
propagator.
% and thus to find the method to deal with the 
%Stokes phenomenon in chaotic systems would be a crucial step in 
%carrying out the complex semiclassical approach.

Owing to the recent development of so-called {\it exact WKB analysis} which  
allows us to treat asymptotic expansions on the analytical basis via  
Borel-Laplace transform (Voros 1983), 
the Stokes phenomenon now becomes well defined concept.
We will give a prescription for analyzing the Stokes phenomenon 
in multi-dimensions, where the crossing of the Stokes curves is an essentially 
new event while it is absent in 2nd-order differential equations. 
It will be shown how the Riemann sheet structure of the Borel 
transform(or $\lq$adjacency' in another context (Berry $\&$ Howls 1991) ), 
which is a crucial information in our exact WKB understanding of 
the Stokes phenomenon, is determined by the {\it Stokes graph} 
(Shudo $\&$ Ikeda 1999). 



\noindent{\bf References}
\\
Milnor J. 1990 Introductory Lectures, SUNY at Stony Brook $\sharp$ 1990/5.
\\
Voros A. 1983 Ann. Inst. H. Poincar$\grave {\rm e}$ A {\bf 39} 211. 
\\
Berry M.V. and Howls. C.J. 1991 Proc. R. Soc. London {\bf 434} 657. 
%\\
%Shudo A. Ishii Y. and Ikeda K.S. 1999 to be published.
\\
Shudo A. and Ikeda K.S. 1999 to be published.
\\

\newpage

\begin{center}
\Large
{\bf Critical bifurcation  surfaces of discrete multi-dimensional
dynamics}
\normalsize

{\bf Michael Sonis}\\

{\em Dept. of Geography, Bar-Ilan University, Israel}\\
  
\end{center}                            

\normalsize
\noindent
The purpose of this lecture is to construct the three-dimensional
analytical representation of the general procedure of linear local
bifurcation analysis developed in Sonis, 1993,  on the basis of
classical Routh-Hurvitz conditions of asymptotic stability combined with
the movements of fixed point in the space of orbits. The bifurcation
phenomena are defined by the position of the boundaries of attraction of
the fixed point. It will be proven  that the domain of attraction of the
fixed point of 3D discrete dynamics is bounded by three critical
bifurcation surfaces: the divergence surface corresponding to the case
in which one of the eigenvalues of the Jacobi matrix of the linear
approximation of the dynamics equals to 1; the flip surface
corresponding to the existence of the eigenvalue -1, and the flutter
surface corresponding to the pair of complex conjugated eigenvalues with
absolute values equal to 1. The crossing of these surfaces by the movement
of the fixed point will generate the plethora of all possible
bifurcation phenomena. This scheme of local bifurcation analysis will be
demonstrated by numerous application to bifurcation analysis of the
Henon map, the different examples of Socio-Spatial discrete Dynamics and
the theoretical example of the bifurcation analysis of the T. Puu 
Dyopoly and Triopoly adjustment dynamics.
 
\noindent{\bf References}
\\
Dendrinos D S,  Sonis M 1990. {\em Chaos and Socio-Spatial Dynamics}, 
Springer Verlag Series of Applied Mathematics, vol. 86.
\\
Sonis M, 1992. {\em  ``Behavior of Iterational Processes near the Boundary
of Stability Domain, with applications to the Socio-Spatial  Relative
Dynamics".} in {\em Functional-Differential Equations}, eds M. Drachlin
and E. Litsin {\bf 1} pp 198-227.

\newpage

\begin{center}
\Large
{\bf Nonlinear socio-ecological  dynamics and first principles of
collective choice behavior of "Homo Socialis"}
\normalsize

{\bf Michael Sonis}

{\em Dept. of Geography, Bar-Ilan University, Israel}
\end{center}                            

\normalsize
\noindent
Socio-ecological dynamics emerged from the field of Mathematical Social
Sciences and opened up avenues for re-examination of classical problems
of collective behavior in Social and Spatial sciences.  The ``engine" of 
this collective behaviour is the subjective mental evaluation of level
of utilities in the future, presenting sets of composite
socio-economic-temporal-locational advantages. These dynamics present 
new laws of collective non-local multi-population behavior which are the
meso-level counterparts of the utility optimization in individual behavior.
The  central  core  of  the  socio-ecological choice dynamics includes 
the  following first principle of the collective choice behavior of
``Homo Socialis" based on the existence of ``collective consciousness":
the choice behavior of ``Homo Socialis" is a collective meso-level choice
behavior such that the relative changes in choice frequencies depend on
the distribution of innovation alternatives between adopters of
innovations. This hypothesis expands essentially the view point of the
social statistical mechanics by including into the consideration the
collective conscience of ``human molecules", arising from the social
interactions and informational mass media effects. 
The mathematical basis of the Socio-Ecological Dynamics includes two
mutually complemented analytical approaches both based on the use of
computer modeling as a theoretical and simulation tool. First approach
is the ``continuous approach" - the systems of ordinary and partial
differential equations reflecting the continuous time Volterra
ecological formalism in a form of antagonistic and/or cooperative
collective hyper-games between different sub-sets of choice
alternatives. Second approach is the ``discrete approach" - systems of
difference equations presenting a new branch of the non-linear discrete
dynamics - the Discrete Relative m-population/n-innovations
Socio-Spatial Dynamics (Dendrinos and Sonis, 1990). The generalization
of the Volterra mathematical formalism  leads further to the meso-level
variational principle of collective choice behavior determining the
balance between the resulting  cumulative social spatio-temporal
interactions among the population of adopters susceptible to the choice 
alternatives and the cumulative equalization of the power of elites
supporting different choice alternatives.  This balance governs the
dynamic innovation choice process and constitutes the dynamic meso-level
counterpart of the micro-economic individual utility maximization
principle. 
The above described law of elite competition can be extended to each 
many-dimensional iteration processes. Thus is the existence of the
extreme principle describing the discrete space-time transfer from the
past to the future. This principle represents the collective  
synergetic interactions between the constituencies of the iteration
process. Moreover, this principle is the discrete time analog of the 
law of meso-level variational principle of collective choice behavior
for the continuous time Socio-Ecological Dynamics.


\noindent{\bf References}
\\
Dendrinos D S,  Sonis M 1990. {\em Chaos and Socio-Spatial Dynamics}, 
Springer Verlag Series of Applied Mathematics, vol. 86.
\\
Sonis M, 1992. {\em ``Innovation Diffusion, Schumpeterian Competition and 
Dynamic Choice: a New Synthesis".} in {\em Journal of Scientific \&
Industrial Reseach, Special Issue on Mathematical Modelling of
Innovation Diffusion and Technological Change} {\bf 51, no.3} pp172-186.

\newpage


\begin{center}
\Large
{\bf  Impulses and waves in biological systems:\\
Exploring stability of membrane potential in excitable cells}
\normalsize

{\bf C. Frank Starmer}

{\em Medical University of South Carolina,
Charleston, USA}
\end{center}

\normalsize
\noindent
Excitable cells are either stable or oscillate.
Subthreshold stimulation produces a transient departure from equilibrium
while suprathreshold stimulation produces a major deparation from
equilibrium that requires some time to return to the rest state.
The major electrical features of an excitable cell are an insulating cell
membrane and gated channels that permeate the membrane and control the flow
of ions down concentration gradients.  A capacitor is formed by the cell
membrane surrounded by conducting fluid on either side while the gated
channels form non-linear resistors.  The parallel combination of capacitors
and Na and K channels define a cellular reaction process which controls
the electrical ``state" of a cell.  We explore the behavior of a simple
reaction diffusion process (Fitzhugh-Nagumo Model) of the form:
$$
        \frac{ \partial U}{ \partial t} = f(U) - W + \nabla ^2 U
$$
where $ f(U) = U(1 - U^2)  $ and
$$
        \frac{ \partial W}{ \partial t} = \alpha U - W + \gamma
$$
Phase plane analysis reveals the possibility of
two stable equilibria and 1 unstable equilibria, depending on medium properties.
The functions, f(U) and W are comparable to ionic currents that flow through
channels that penetrate the membrane of excitable cells.
By altering the channel conductances, the cell membrane potential will either be
stable or will oscillate.  Genetic diseases, that alter the properties of
membrane ion channels can lead to shifts in the equilibrium of excitable
cells from stable to unstable.  Epilepsy and the cardiac long QT syndrome
are two such examples.  Experimental studies of membrane potential in
cardiac cells display shifts in equilibrium when treated with drugs that
block the flow of ions in membrane ion channels.  This lecture will link
physical and mathematical models of excitable membranes and ion channel
blockade with the behavior of the {\em minimal} model (Fitzhugh-Nagumo)
and its associated phase
plane analysis of cellular stability and illustrate how genetic mutants of
ion channels lead to cardiac and cerebral arrhythmias that appear quite
complex.
 


\noindent{\bf References}
\\
Murray, J.D. 1993 {\em Mathematical Biology} 274-359, 1993. (Springer Verlag, Heidelberg) 
274-359
\\
Rinzel, J. and Terman, D.: Propagation phenomena in a bistable reaction-diffusion systems.  
{\em SIAM J. Appl. Math.} {\bf 42} 1111-1137, 1982.
\\
Starmer, C.F.: Theoretical characterization of ion channel blockade:  Ligand binding to periodically accessible receptors. 
{\em J. Theoretical Biology} {\bf 119} 235-249, 1986
\\
Curran, M.E. et. al.: A molecular basis for cardiac arrhythmia: HERG mutations cause long QT syndrome.  {\em Cell}
{\bf 80} 795-803, 1995.
\\
Schroeder, B.C. et. al.: Moderate loss of function of cyclic-AMP-modulated KCNQ2/KCNQ3 K+ channels
causes epilepsy.  {\em Nature} {\bf 396} 687-690, 1999.
\\

\newpage

\begin{center}
\Large
{\bf  Impulses and waves in biological systems:  \\
Exploring front formation and propagation}
\normalsize

{\bf C. Frank Starmer}

{\em Medical University of South Carolina,
Charleston,  USA}
\end{center}

\normalsize
\noindent
Isolated excitable cells are either stable or oscillate.  
Diffusive coupling between excitable cells introduces a new property: 
the spatial extent of the excitation must exceed a ``liminal" region that
is a function of media properties.  In a uniform medium, fronts initiated
by excitation smaller than the liminal region will collapse while fronts
initiated by excitation of a region larger than the liminal region will
expand. We explore the behavior of a simple reaction diffusion
process of the form:
$$
        \frac{ \partial U}{ \partial t} = f(U) - W + \nabla ^2 U 
$$
where $ f(U) = U(1 - U^2) $ and
$$
        \frac{ \partial W}{ \partial t} = \alpha U - W + \gamma
$$
This lecture will focus on a number of interesting questions related for
formation and propagation of waves in an excitable medium, specifically
wave patterns associated with cardiac arrhythmias.
In homogeneous and uniform excitable media in the rest state,
there are two responses to stimulation:
if the stimulus is less than a critical value, a collapsing wave is initiated while
if the stimulus is greater than a critical value, and continuous expanding wave is
initiated.  Under some conditions, it is possible to initiate a partial wave, i.e
a wave that expands in some directions and collapses in other directions.
Trailing a propagating wave is a region
known as the vulnerable region, where the non-uniformity
of the medium properties can result in incomplete front formation following
suprathreshold stimulation.  Depending on the time and
location of stimulation after the passage of a wave, stimulation
results in either: 1) a
decaying front; 2) an expanding front; and 3) a wave fragment (wavelet) that expands
in some directions and collapses in other directions.  In two and three
dimensional media,the wave fragment can evolve to a rotating spiral wave.
The duration of the period of vulnerability is
${\rm VP} = L / V $
where $L$ is the spatial extent of the suprathreshold stimulus
field and $V$ is the velocity of the {\em conditioning} wave .  We will show that drugs that reduce the excitability of
isolated cells (classified as antiarrhythmic drugs) will increase the VP,
leading to dramatic increases in the rate of spontaneous and potentially
fatal arrhythmias.  This new finding was confirmed in studies of
cardiac tissue.

\noindent{\bf References}
\\
Murray, J.D.  {\em Mathematical Biology} 274-359, 1993 (Springer Verlag, Heidelberg)
\\
Rushton, W.A.H. Initiation of the propagated disturbance.
{\em Proc R Soc Lond B} {\bf 124} 210-243, 1937.
\\
Wiener, N. and Rosenblueth, A. The mathematical formulation of the problem of conduction
of impulses in a network of connected excitable elements, specifically in cardiac muscle.
{\em Arch Inst Cardiol Mex} {\bf 16} 205-265, 1946.
\\
Starmer, C.F. et. al. Vulnerability in an excitable medium:  Analytical
and numerical studies of initiating unidirectional propagation.  {\em Biophysical
J.}  {\bf 65} 1775-1787, 1993,
\\
Starobin, J.M. et. al. Vulnerability in one-dimensional excitable media.
{\em Physica D} {\bf 70} 321-341, 1994.
\\
Starobin, J. and Starmer, C.F. A common mechanism links spiral wave meandering and
wavefront-obstacle separation. {\em Phys. Rev. E} {\bf 55} 1193-1196, 1997.

\newpage


\begin{center}
\Large
{\bf  Topics in nonlinear dynamics of the human cardiovascular system}
\normalsize

{\bf Aneta Stefanovska}

{\em Group of Nonlinear Dynamics and Synergetics,\\
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana,
Slovenia}
\end{center}

\normalsize
\noindent
We review the main results of analysis of time series measured
simultaneously at different sites of the human
cardiovascular system. Because our primary interest centres
on the dynamics of blood distribution, characteristics
during one cycle of blood through the system, which on average
takes about one minute, are considered.

First we discuss results of nonlinear time series
analysis with regard to system determinism and complexity. The
analysis in the time domain and in the phase space revealed
highly deterministic and almost conservative nature of cardiovascular
control system on time scale of minutes
(Stefanovska and Kro\v selj, 1997, Bra\v ci\v c and Stefanovska,
1998a, Stefanovska and Bra\v ci\v c, 1999a, 1999b).

Then we present results of time-frequency analysis.
The wavelet transform, a  method with logarithmic
frequency resolution, was used to study oscillations in
cardiovascular signals (Bra\v ci\v c and Stefanovska, 1998b,
Kvernmo, Stefanovska, Bra\v ci\v c, Kirkeboen and Kvernebo, 1998,
Stefanovska, Bra\v ci\v c and Kvernmo, in print).
The heart rate variability (HRV), derived from the signal resulting
from electrical activity of the heart (ECG), bears the variations
of instantaneous heart rate and RR intervals in the ECG. Several
central and peripheral oscillators contribute to the variations
between subsequent heart beats. Analysis of signals measured from
respiration, cardiac function and blood flow, all revealed the
existence of five almost periodic frequency components, illustrating
that all the cardiac periodicities are propagated
as traveling waves.

The physical description of the cardiovascular control system
depends essentially on the current understanding of physiological
origin of observed oscillations. The evidence that endothelial
activity is manifested as oscillations in blood flow with a
characteristic frequency of around 0.01Hz was recently
demonstrated (Stefanovska, Bra\v ci\v c and Kvernmo, in print).
The mode of changes in the characteristics of oscillations with
aging, physical activity, diabetes, after myocardial infarction,
or general anaesthesia also provides insight into the physiological
and physical characteristics of the system. In addition,
the strength of phase and frequency couplings among
the oscillations, modified in different states of the
cardiovascular system, is discussed.


\noindent{\bf References}
\\
Stefanovska A and Kro\v selj P 1997 {\em Open Sys. Information Dyn.}
{\bf 4}
457
\\
Bra\v ci\v c M and Stefanovska A 1998a  {\em Bull. Math. Biol.} {\bf 60} 417
\\
Stefanovska A and Bra\v ci\v c M 1999a {\em Contemporary Physics}
{\bf 40} 31
\\
Stefanovska A and Bra\v ci\v c M 1999b {\em Control Engineering
Practice}
{\bf 7} 161
\\
Bra\v ci\v c M and Stefanovska A 1998b  {\em Bull. Math. Biol.} {\bf 60} 919
\\
Kvernmo HD, Stefanovska A, Bra\v ci\v c M, Kirkeb\o en K-A and
Kvernebo K, 1998 {\em Microvascular Research} {\bf 56} 173
\\
Stefanovska A, Bra\v ci\v c M, Kvernmo HD {\em IEEE Trans. Biomed.
Eng.} In press.

\newpage

\begin{center}
\Large
{\bf    Cardiorespiratory interactions}
\normalsize

{\bf Aneta Stefanovska}

{\em Group of Nonlinear Dynamics and Synergetics,\\
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana,
Slovenia}
\end{center}

\normalsize
\noindent
Oscillators that have a standard waveform and amplitude to which
they return after small perturbation are known as limit-cycle
oscillators. They incorporate a dissipative mechanism to dump
oscillations that grow too large and source of energy to pump up
those that become too small. Coupled, they
continuously perturb each other and everyone of them permanently
tends toward a limit cycle with time-varying amplitude
and characteristic frequency.

\vspace{-1mm}
Signals derived from the
human cardiovascular system have been found to contain several periodic
components (Akselrod {\em et al}, 1981), thereby raising the question
of whether the blood distribution system and its regulatory mechanisms
can be described as a set of coupled oscillators
(Stefanovska and Bra\v ci\v c, 1999).
On minute time scale the peripheral blood flow was shown to contain five
characteristic frequencies (Bra\v ci\v c and Stefanovska, 1998). The
physiological origin of two of them is easy to evidence by simultaneous
measurements of the heart and respiratory activity. The heart rate at
rest
ranges from 0.6 Hz in sportsmen to 1.6 Hz in subjects with impaired
cardiovascular systems and the respiratory frequency from 0.15 Hz
to 0.35 Hz. As early as 1733, Hales observed that changes in heart
rate were related in a regular manner to the respiratory pattern,
and in 1847 Ludwig documented that the heart rate increased on
inspiration and decreased on expiration. The respiratory modulation
of the heart rate, known as ``respiratory sinus arrhythmia", raises
questions
not answered so far: how strongly those two systems are coupled, how
strongly they modulate each others amplitudes and characteristic
frequencies, does phase synchronisation exist at rest in healthy and
diseased states? Raschke (1987) analysed effects of mutual interactions
and demonstrated that during sleep inspiration starts preferably at a
certain phase of the cardiac cycle. Glass and Mackey (1988)
concluded that they are weakly coupled and generally not phase
locked. In contrast, Rosenblum {\em et al} (1998) revealed hidden
phase-locked regimes between the cardiac and the respiratory
rhythms.

\vspace{-1mm}
We discuss mutual frequency and amplitude modulation of the
heart and respiratory cycles.  The couplings are not symmetrical,
the respiratory system is imposing stronger influence to the cardiac
system than vice versa. The other three oscillatory processes
involved in the regulation of blood distribution system also influence
the heart and respiratory activity. Consequently, a stable phase
locking among those two oscillators can not be obtained in healthy
cardiovascular system. In addition, we present states of coupling
between the two oscillatory systems modified by aging, diabetes,
myocardial infarction, or general anaesthesia.


\noindent{\bf References}
\\
Axelrod S {\em et al} 1981 {\em Science} {\bf 213} 220
\\
Stefanovska A and Bra\v ci\v c M 1999 {\em Contemporary Physics}
{\bf 40} 31
\\
Bra\v ci\v c M and Stefanovska A 1998  {\em Bull. Math. Biol.} {\bf 60}
919
\\
Hales S 1773 {\em Statical Essays II, Haemastaticks}, (London:
Innings Manby)
\\
Ludwig C 1847 {\em Arch Anat Physiol} {\bf 13} 242
\\
Raschke F 1987 in {\em ``Temporal Disorder in Human Oscillatory
Systems"}, eds. L Rensing, U an der Heiden and MC Mackey,  (Berlin:
Springer) pp152-158
\\
Glass L and Mackey MC 1988 {\em From Clocks to Chaos: The Rhythms of
Life},
(Princeton: Princeton UP)
\\
Rosenblum M {\em et al} 1998 {\em IEEE Mag Eng Med Biol}
November/December 46

\newpage

\begin{center}
\Large
{\bf  Nonlinear dynamics, bifurcation, and symmetry}
\normalsize

{\bf Ian Stewart}

{\em Mathematics Institute,
University of Warwick \\ Coventry, UK}
\end{center}

\normalsize

\noindent

In many applications of nonlinear dynamics, the model system
possesses a degree of symmetry.   Symmetry affects the
`generic' behaviour of the system, and its presence
should be taken into account when analysing the predictions
of the model.
The course will provide a three-lecture introduction to
some basic
theory, including fundamental existence theorems for
the existence of symmetry-breaking
These include criteria for the existence of symmetry-breaking branches of
steady and periodic states, and some basic concepts related to
`symmetric chaos'.   We emphasize the role of symmetry as a
general framework for such problems.

\section{Summary of Course Structure}
\label{coursestructure}

\begin{enumerate}

\item {\em Steady State Bifurcation}

Invariant functions and equivariant mappings.   Group actions,
isotropy subgroups, fixed-point spaces.   The isotropy
lattice.  Genericity of absolutely irreducible representations.
The Equivariant Branching Lemma.    

Examples: $\Otwo$-symmetric systems, $\D_n$-symmetric systems
$\SN$-symmetric systems.  

Applications: Speciation, convection, Taylor-Couette flow.

\item {\em Hopf Bifurcation to Periodic States}

Loop space, circle group action by phase shift, 
spatio-temporal symmetries.   Liapunov-Schmidt reduction.
Genericity of $\Gamma$-simple representations.
The Equivariant Hopf Theorem.    

Examples: $\Otwo$-symmetric systems, $\D_3$-symmetric systems.  

Applications: Oscillating hosepipe, animal locomotion, convection.

\item {\em Symmetric Chaos}

Pointwise and setwise symmetry of an attractor. Symmetry
on average.   Detectives.   Collisions of attractors.
Transverse Liapunov exponents.

Examples:   The cubic logistic map, $\D_3$-symmetric maps.

Applications:  Oscillator arrays, the Faraday Experiment.

\end{enumerate}

\section{Steady State Bifurcation}

Steady state bifurcation occurs when an equilibrium of a dynamical
system becomes unstable as a parameter $\lambda$ is varied.
In suitable circumstances, the generic result is the creation
of a branch of `new' equilibria.  We study such behaviour 
when the dynamical system has symmetry.  

Specifically, by ` has symmetry' we mean the following.
Let $\Gamma$ be a Lie group of linear transformations of $\R^n$.
We say that $f$ is
$\Gamma$-{\em equivariant} if
\begin{equation} \label{equivariance}
f(\gamma x,\lambda) = \gamma f(x,\lambda)
\end{equation}
for all $\gamma \in \Gamma$.
Consider a $\Gamma$-equivariant ODE
\begin{equation} \label{equiODE}
\frac{dx}{dt} + f(x,\lambda) = 0
\end{equation}
where $x \in \R^n$, $\lambda \in \R$.   For simplicity, assume that
$f(0,\lambda) \equiv 0$, so there exists a `trivial branch' of solutions
$x = 0$.   The {\em linearization} of $f$ is
\[
L_{\lambda} = \left.D_x f\right|_{0,\lambda}.
\]
Local bifurcation at $\lambda = 0$ occurs when the trivial branch undergoes a
change of linear stability, so that $L_0$ has eigenvalues on the imaginary
axis (often called {\em critical eigenvalues}).   

There are two cases:
\begin{itemize}
\item Steady-state bifurcation: $L_0$ has a zero eigenvalue.
\item Hopf bifurcation: $L_0$ has a complex conjugate pair of eigenvalues
$\pm \ii \omega$.
\end{itemize}

In the first lecture we look at steady-state bifurcation.  The
central result here is the Equivariant Branching Lemma,
which asserts the existence of branches of equilibria with
certain kinds of symmetry.

The basic examples are
 $\Otwo$-symmetric systems, $\D_n$-symmetric systems,
and
$\SN$-symmetric systems.   Here $\Otwo$ is the orthogonal
group in the plane, $\D_n$ is the dihedral group,
and $\SN$ is the symmetric group of degree $N$.

Applications include speciation in evolutionary
systems (modelled as a dynamic on phenotypic space), 
convection, and Taylor-Couette flow of a fluid 
between rotating cylinders.

\section{Hopf Bifurcation}

Hopf bifurcation leads to a branch of time-periodic
solutions.   It is analogous to steady-state
bifurcation, but now there is an
extra symmetry given by an action of the circle group.
Intuitively, this represents shifting the phase of a solution.
The problem becomes $\Gamma \times \SS$-equivariant.

The analogue of the Equivariant Branching Lemma is the
Equivariant Hopf Theorem.  It is proved by posing the
problem on
loop space --- a Banach space of periodic
functions --- upon which there is a circle group action by phase shift,
corresponding to
spatio-temporal symmetries.   

Applications include
the oscillating hosepipe of circular cross-section,
with both standing and rotating waves; animal locomotion,
where the patterns with which animals move their legs when
they walk, trot, run, and so on can be classified and
organised; and convection.


\section{Symmetric Chaos}

Symmetric systems can have chaotic attractors, and here
the symmetry of the attractor represents the symmetry
`on average' over long periods of time of the
corresponding states.  It is important to distinguish
between
pointwise and setwise symmetry of an attractor. 

The symmetry of an attractor can be found
experimentally by using the theory of `detectives'.   

Important phenomena include
collisions of attractors, in which conjugate versions
of a chaotic attractor can suddenly merge, with a change
of symmetry.

Examples include the cubic logistic map
\[
f(x,\lambda) = \lambda x (1-x^2)
\]
and
$\D_3$-symmetric maps.

Applications include chaotic dynamics
of oscillator arrays, synchronisation of
coupled chaotic oscillators, and the Faraday Experiment,
in which a dish of fluid is vibrated in the vertical direction.

\noindent{\bf References}\\
Ashwin P, Buescu J, Stewart I 1994   Bubbling of attractors and 
synchronization of chaotic oscillators, {\em Phys. Lett. A} {\bf 193}
126-139.
\\
Ashwin P, Buescu J, Stewart I 1996 From attractor to chaotic saddle: 
a tale of transverse instability, {\em Nonlinearity} {\bf 9}
703-737.
\\
Buescu J, Stewart I 1995  Liapunov stability and adding machines, 
{\em Ergod. Th. \& Dynam. Sys.} {\bf 15} 271-290.
\\
Cohen J, Stewart I 2000 Polymorphism viewed as phenotypic symmetry-breaking, 
in {\em Nonlinear Phenomena in Physical and Biological Sciences}
(ed. Malik S K), 
Indian National Science Academy, New Delhi 
(World Mathematical Year 2000), to appear.
\\
Collins J J, Stewart I 1993  Hexapodal gaits and coupled nonlinear 
oscillator models, {\em Biol. Cybern.} {\bf 68} 287-298.
\\
Collins J J, Stewart I 1993  Coupled nonlinear oscillators and the
symmetries of animal gaits, {\em J. Nonlin. Sci.} {\bf 3} 349-392.
\\
Dellnitz M, Golubitsky M, Stewart I, Hohmann A 1995 Spirals in 
scalar reaction-diffusion equations, {\em Internat. J. Bif. Chaos} 
{\bf 5} 1487-1501.
\\
Dionne B, Golubitsky M, Stewart I 1996 
Coupled cells with internal symmetry: 
I. Wreath products, {\em Nonlinearity} {\bf 9} 559-574.
\\
Dionne B, Golubitsky M, Stewart I 1996
Coupled cells with internal symmetry: 
II. Direct products, {\em Nonlinearity} {\bf 9} 575-599.
\\
Dionne B, Golubitsky M, Silber M, Stewart I 1995    Time-periodic 
spatially-periodic planforms in Euclidean equivariant PDE,
{\em Phil. Trans. Roy. Soc. London} A {\bf 352} 125-168.
\\
Golubitsky M, Field M 1992 {\em Symmetry in Chaos} Oxford
University Press, Oxford.
\\
Golubitsky M, Knobloch E, Stewart I 1999 Target patterns and spirals 
in planar reaction-diffusion systems, Research report UH/MD-256, 
University of Houston.
\\
Golubitsky M, Schaeffer D G 1985 {\em Singularities and Groups in
Bifurcation Theory} vol.1, Springer-Verlag, New York.
\\
Golubitsky M, Stewart I 1985  Hopf bifurcation in the presence of 
symmetry, {\em Arch. Rational Mech. Anal.} {\bf 87} 107-165.
\\
Golubitsky  M, Stewart I 1986  Symmetry and stability in Taylor-Couette 
flow, {\em SIAM J. Math. Anal.} {\bf 17} 249-288.
\\
Golubitsky M, Stewart I 1986  Hopf bifurcation with dihedral group symmetry: 
coupled nonlinear oscillators, in {\em Multiparameter Bifurcation 
Theory} (eds. Golubitsky M, Guckenheimer J), Proceedings of 
the AMS-IMS-SIAM Joint Summer Research Conference, July 1985,
Arcata; {\em Contemporary Math.} {\bf  56} Amer. Math. 
Soc. Providence RI 131-173.
\\
Golubitsky M, Stewart I, Collins J J, Buono L 1998 A modular network
for legged locomotion, {\em Physica} D {\bf 115} 56-72.
\\
Golubitsky M, Stewart I 1999  Symmetry and pattern formation in 
coupled cell networks, {\em Proceedings of IMA Conference on 
Pattern Formation 1998} to appear.
\\
Golubitsky M, Stewart I, Schaeffer D G 1988 {\em Singularities and Groups in
Bifurcation Theory} vol.2, Springer-Verlag, New York.
\\
King G, Stewart I 1992 Symmetric chaos,
in {\em Nonlinear Equations in the Applied
Sciences} (eds. Ames W F, Rogers C) Academic Press London
257-315.
\\ 
Stewart I 1987 Stability of periodic solutions in symmetric 
Hopf bifurcation, {\em Dynam. Stab. Sys.} {\bf 2} 149-166.
\\
Stewart I 1988 Bifurcations with symmetry, in {\em New Directions in Dynamical 
Systems} (eds. Bedford T, Swift J W), London Math. Soc. Lecture 
Notes {\bf 127} Cambridge Univ. Press  235-283.
\\
Stewart I 1992 Bifurcation theory old and new, in {\em Dynamics of Numerics and 
Numerics of Dynamics} (eds. Broomhead D S, Iserles A) 
IMA Conference Series {\bf 34} Oxford University Press 31-67.
\\
Stewart I 1992 Broken symmetry and the formation of spiral patterns in fluids, 
in {\em Spiral Symmetry} (eds. Hargittai  I, Pickover C A), 
World Scientific, Singapore 187-220.

\newpage


\begin{center}
\Large
{\bf Microwave studies of chaotic billiards and disordered systems }
\normalsize

{\bf Hans-J\"urgen St\"ockmann}
  
{\em Fachbereich Physik der Philipps-Universit\"at,\\
Marburg, Germany}
\end{center}                            

\normalsize
\noindent

Chaotic billiards are ideal model systems to study the quantum mechanical 
properties of classically chaotic systems. Billiards with hard walls are 
conceptionally particularly simple. The classical trajectories can be 
calculated elementarily, and the Schr\"odinger equation reduces to the free 
wave equation with Dirichlet boundaries at the walls. Nevertheless the 
calculations soon become elaborately, in particular, if spectra or wave 
functions are to be determined in dependence of one external parameter, 
usually the billiard length or the position of an inserted scatterer. Here 
analogous experiments are an interesting alternative. They make use of the 
equivalence of the stationary Schr\"odinger equation, and the 
time-independent wave equation, the Helmholtz equation. In two-dimensional 
microwave billiards this equivalence is complete, including the boundary 
conditions.

After a short introduction into the problematic of the quantum mechanics of 
classically chaotic systems, shortly termed `quantum chaos', different 
types of billiard experiments are introduced. The main part of the talk is 
devoted to microwave billiards. Results on global and local billiard 
spectral level dynamics are presented as well as measurements in 
three-dimensional chaotic microwave resonators. Here the equivalence 
between Schr\"odinger equation and electromagnetic wave equation does no 
longer hold. The talk ends with a discussion of transmission and 
localization in disordered systems.

\noindent{\bf References}\\
St\"ockmann H J and Stein J 1990
{\em Phys. Rev. Lett.} {\bf 64} 2215\\
Sridhar S 1991
{\em Phys. Rev. Lett.} {\bf 67} 785\\
Stein J and St\"ockmann H J 1992
{\em Phys. Rev. Lett.} {\bf 68} 2867\\
Gr\"af H D {\em et al.} 1992
{\em Phys. Rev. Lett.} {\bf 69} 1296\\
So P {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 74} 2662\\
Stein J {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 75} 53\\
Kudrolli A {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 75} 822\\
Ellegaard C {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 75} 1546\\
St\"ockmann H J 1999
{\em Quantum Chaos - An Introduction}, Cambridge university press

\newpage

\begin{center}
\Large
{\bf Microwave billiards as scattering systems}
\normalsize

{\bf Hans-J\"urgen St\"ockmann}
  
{\em Fachbereich Physik der Philipps-Universit\"at,\\
Marburg, Germany}
\end{center}                            

\normalsize
\noindent


It is impossible to study a system without disturbing it by the measuring 
process. To determine, e. g., the spectrum of a microwave billiard, we have 
to drill a hole into its wall, introduce a wire, and irradiate a microwave 
field. Due to the presence of the  antenna a rectangular microwave cavity, 
e. g., is no longer integrable, but has become pseudointegrable. 

The measurement thus unavoidably yields an unwanted combination of the 
system's own properties and those of the measuring apparatus. The 
mathematical tool to treat the coupling between the system and the 
environment is provided by scattering theory, which has originally been 
developed in nuclear physics. Later this theory has been successfully applied to mesoscopic systems and microwave billiards as well.

In this talk scattering theory will be applied to billiard systems, resulting 
in the billiard equivalent of the Breit-Wigner function, well-known from 
nuclear physics for many years. The consequences of the coupling for the 
spectral statistics will be discussed, as well as for resonance depths and 
widths. In the last part results on transmission through billiards with open 
channels are presented. Here we have a close correspondence to mesoscopic 
systems, which can be linked to scattering theory via the Landauer formula 
expressing the conduction through mesoscopic devices in terms of 
transmission probabilities. 

\noindent{\bf References}\\
Lewenkopf C H and Weidenm\"uller H A 1991
{\em Ann. Phys.} {\bf 212} 53\\
Alt H {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 74} 62\\
Stoffregen U {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 74} 1995\\
Stein J {\em et al.} 1995
{\em Phys. Rev. Lett.} {\bf 75} 53\\
Lehmann N {\em et al.} 1995
{\em Nucl. Phys. A} {\bf 583} 223\\
Alt. H {\em et al.} 1996
{\em Phys. Lett. B} {\bf 366} 7\\
Haake F {\em et al.} 1996
{\em J. Phys. A} {\bf 29} 5745\\
St\"ockmann H J and \v{S}eba P 1998
{\em J. Phys. A} {\bf 31} 3439\\
St\"ockmann H J 1999
{\em Quantum Chaos - An Introduction}, Cambridge university press

\newpage


\begin{center}
\Large
{\bf  Phase resetting in medicine and biology -- 
\\Stochastic approach and applications}
\normalsize

{\bf Peter A. Tass}
  
{\em Department of Neurology,\\
Heinrich-Heine-University, D\"usseldorf, Germany}
\end{center}                            

\normalsize
\noindent
Synchronization processes are of great importance in several branches of 
science, e.g., in biology, medicine and physics. 
We present a new theoretical approach to phase resetting and 
stimulation-induced synchronization and desynchronization in a population 
of interacting oscillators in the presence of noise (Tass 1999). 
\\[0.2cm] The theoretical investigations of spontaneously emerging dynamics of 
populations of interacting oscillators have revealed numerous significant
results (Kuramoto 1984). From the standpoint of neuroscience, however,  
there is still an enormous need for theoretical studies addressing 
stimulation-induced transient synchronization and desynchronization processes. 
The impact of stimuli on synchronized neuronal oscillators is of great 
importance for the study of brain functioning 
(Steriade, Jones, Llin\'{a}s 1990, Singer, Gray 1995, Hari, Salmelin 1997)
and of thera\-peutic stimulation techniques in neurology and neurosurgery 
(Benabid et al. 1991, Blond et al. 1992).
\\[0.2cm] In Winfree's (1980) pioneering topological approach to phase 
resetting the influence of noise and synchronizing couplings was neglected. 
For this reason with methods from synergetics (Haken 1983) and statistical 
mechanics we, first, study patterns of synchrony emerging in the presence 
of noise (Kuramoto 1984, Tass 1999). Second, the impact of periodic and, 
in particular, single pulsatile stimuli on different synchronized states 
is investigated in detail (Tass 1999).
\\[0.2cm] We encounter a variety of characteristic stimulation-induced dynamical
features, for instance, different sorts of transient desynchronization and 
resynchronization processes. Moreover, data analysis tools are presented 
which enable us to apply the theory to design and evaluate stimulation 
experiments. Finally, we propose improvements to stimulation 
techniques as used by neurologists and neurosurgeons in the context of
Parkinson's disease (Tass 1999).

\noindent{\bf References}
\\
Benabid A L et al. 1991 {\em The Lancet} {\bf 337}, 403
\\
Blond S et al. 1992 {\em J. Neurosurg.} {\bf 77}, 62
\\
Haken H 1983 {\em Advanced Synergetics},
(Berlin: Springer)
\\
Hari R, Salmelin R 1997 {\em TINS} {\bf 20}, 44
\\
Kuramoto Y 1984 {\em Chemical Oscillations, Waves, and Turbulence}
(Berlin: Springer)
\\
Singer W, Gray C M 1995 {\em Annu. Rev. Neurosci.} {\bf 18}, 555
\\
Steriade H, Jones E G, Llin\'{a}s R 1990  
{\em Thalamic Oscillations and Signaling} (New York: John Wiley \& Sons)
\\
Tass P A 1999 {\em Phase Resetting in Medicine and Biology -- Stochastic 
Modelling and Data Analysis} (Berlin: Springer) (will appear in May 1999)
\\
Winfree A T 1980 {\em The Geometry of Biological Time}
(Berlin: Springer)

\newpage

\begin{center}
\Large 
{\bf Detection of cerebral synchronization processes}
\end{center}                            
\normalsize

\begin{center}
{\bf Peter A. Tass}
  
{\em Department of Neurology,\\
Heinrich-Heine-University, D\"usseldorf, Germany}
\end{center}                            


\noindent

Synchronization of neuronal oscillatory activity is important under both 
physiological and pathological conditions. For instance, according to animal 
experiments synchronization appears to be a basic mechanism for coordinating 
different neuronal populations during complex tasks like visual pattern 
recognition (Singer, Gray 1995) and motor control (Roelfsema et al. 1997). By 
means of magnetoencephalography (MEG) and electroencephalography (EEG)
the cerebral activity can noninvasively be measured in humans 
(H\"am\"al\"ainen et al. 1993). To study synchronization processes of this 
kind in humans two data analysis tools (Tass et al. 1998, Tass 1999) were 
developed which are applied to MEG data or EEG data or directly to the 
cerebral current density obtained by suitable inverse methods (cf. 
H\"am\"al\"ainen et al. 1993).

The first method makes it possible to detect $n:m$ phase synchronization
in noisy non-stationary data (Tass et al. 1998). It is designed for detecting 
self-synchronization in terms of an adjustment of the rhythms of ongoing 
oscillations. To this end a sliding window analysis is performed, where one 
needs a window length corresponding to at least eight to ten periods of the 
oscillation. This method was developed based on the concept of phase 
synchronization of chaotic oscillators (Rosenblum, Pikovsky, Kurths 1996, 
Pikovsky, Rosenblum, Kurths 1996). 

The second method aims at analyzing short-term synchronization processes which 
are induced by a stimulus (Tass 1999). Accordingly, it is designed to detect
so-called stimulus-locked $n:m$ transients which are stimulus locked epochs
displaying a stereotyped time course of the $n:m$ phase difference (with and 
without conduction delay) (Tass 1999). Its time resolution reaches down to 
the millisecond range, and, hence, this method can cope with short synchronous
epochs which are important in the context of cerebral information processing. 
This data analysis tool originates from the theoretical approach to 
phase resetting and stimulation-induced synchronization and desynchronization 
in clusters of interacting oscillators which was presented in the first lecture. 

The two methods provide us with complementary information concerning
spontaneously emerging synchronization as well as stimulus-induced 
synchronous epochs. Analyzing the two types of synchronization in MEG and
EEG data opens up new and promising possibilities for the study of brain 
functioning. Theory and physiological applications are presented in detail. 

\noindent{\bf References}
\\
H\"am\"al\"ainen  M et al. 1993 {\em Rev. Mod. Phys.} {\bf 65}, 413
\\
Pikovsky A S, Rosenblum M G, Kurths J 1996 
{\em Europhys. Lett.} {\bf 34}, 165
\\
Roelfsema P R et al. 1997 {\em Nature} {\bf 385}, 157
\\
Rosenblum M G, Pikovsky A S, Kurths J 1996 {\em Phys. Rev. Lett.}
{\bf 76}, 1804 
\\
Singer W, Gray C M 1995 {\em Annu. Rev. Neurosci.} {\bf 18}, 555
\\
Tass P A 1999 {\em Phase Resetting in Medicine and Biology -- Stochastic 
Modelling and Data Analysis} (Berlin: Springer) (will appear in May 1999)
\\
Tass P et al. 1998 {\em Phys. Rev. Lett.} {\bf 81}, 3291

%*******************************************************************
%*******************************************************************

\newpage

\newcommand{\blp}{\begin{minipage}{17cm} \noindent}
\newcommand{\elp}{\end{minipage}}
\newcommand{\bsp}{\begin{minipage}{8.3cm} \small \noindent}
\newcommand{\esp}{\end{minipage}}

{\parindent=0cm
\Large
\begin{center}
{\bf SHORT REPORTS}\\
\end{center}
\normalsize

\vspace{0.3in}
\blp
{\bf Dr. Arantxa Alonso}\\
{\it Departament de F\' \i sica Aplicada, UPC, Barcelona, Spain}\\
Secondary bifurcation and transition to chaos in a O(2)-symmetric convective 
system for low Prandtl numbers\\
\elp

\blp
{\bf Dr. Nils Berglund}\\
{\it Georgia Tech, Atlanta, USA}\\
Dynamic bifurcations: hysteresis, scaling laws, and feedback control\\
\elp

\blp
{\bf  Mr. Marcel Clerc}\\
{\it  Institut Non Lin\' eaire de Nice, France}\\
Lorenz bifurcation and quasireversible bifurcations\\
\elp

\blp
{\bf Prof. Dr. Vladimir Damgov}\\
{\it Space Research Institute, Sofia, Bulgaria}\\
`Quantized' oscillations and irregular behaviour of a class of kick-excited
self-adaptive dynamical systems\\
\elp

\blp
{\bf Mr. Karl Mario Garcia-Ruiz}\\
{\it UNAM, Mexico City, Mexico}\\
Stochastic study of the Lorenz system\\
\elp

\blp
{\bf Dr. Thomas Gorin}\\
{\it Centro Internacional de Ciencias A.C., Cuernavaca, Mexico}\\
Regular scattering systems and the statistical S-matrix approach\\
\elp

\blp
{\bf Dr. Toma\v z Gyergyek}\\
{\it Faculty of Electrical Engineering, University of Ljubljana, Slovenia}\\
Observation of synchronization, periodic pulling and period doubling 
bifurcations of the plasma oscillations triggered by a harmonically forced 
anode type double layer in a discharge plasma\\ 
\elp

\blp
{\bf Dr. Takahisa Harayama}\\
{\it ATR Adaptive Communications Research Laboratories, Kyoto, Japan}\\
Whispering gallery mode laser\\
\elp

\blp
{\bf Dr. George Krylov}\\
{\it CAMTP, University of Maribor, Slovenia}\\
Path integral approach and canonical transformations in quantum mechanics\\
\elp

\blp
{\bf Prof. Dr. Israel Laulicht}\\
{\it Bar-Ilan University, Ramat-Gan, Israel}\\
On the transient and stationary parametric excitation of spin waves\\
\elp

\blp
{\bf Prof. Dr. Francois Alain Leyvraz}\\
{\it Centro de Ciencias Fisicas, Cuernavaca, Mexico}\\
Structural invariance: A connection between classical dynamics and random
matrix theory\\
\elp

\blp
{\bf Prof. Dr. Vjera Lopac}\\
{\it University of Zagreb, Croatia}\\
Dynamics of some bouncing ball systems\\
\elp

\blp
{\bf Dr. Philippe Marcq}\\
{\it University of Potsdam, Germany}\\
A Langevin equation for intermittent velocity fluctuations in fully developed
turbulence\\ 
\elp

\blp
{\bf Prof. Dr. Anatoly Bronislavovich Rinkevich}\\
{\it Institute of Metal Physics, Ekaterinburg, Russia}\\
Pulse current instability and turbulence in metals\\
\elp

\blp
{\bf Prof. Dr. Valeri Romanovski}\\
{\it CAMTP, University of Maribor, Slovenia}\\
On WKB series for the radial Kepler problem\\
\elp

\blp
{\bf Dr. Andreas Ruffing}\\
{\it Center of Mathematical Sciences, M\" unchen, Germany}\\
Comparing discretized and continuous quantum models: An unexpected possibility
of checking phase space structure\\
\elp

\blp
{\bf Dr. Luca Salasnich}\\
{\it University of Milano, Italy}\\
Chaotic vibrations in finite quantum systems: Atomic nuclei and trapped
Bose-Einstein condensates\\
\elp

\blp
{\bf Mr. Gregor Veble}\\
{\it CAMTP, University of Maribor, Slovenia}\\
Study of regular and irregular states in generic systems\\
\elp

\blp
{\bf Dr. Karol Zyczkowski}\\
{\it Smoluchowski Institute of Physics, Krakow, Poland}\\
Monge distance between quantum states---toward quantum Lyapunov exponent\\ 
\elp
  
\newpage

\Large
\begin{center}
{\bf POSTERS}\\
\end{center}
\normalsize

\vspace{0.3in}
\blp
{\bf Professor Dr. Hosny Ali Abdusalam}\\
{\it Cairo University, Egypt}\\
Modeling the failure of a fractal tree and Julia sets\\
\elp

\blp
{\bf Mr. Michael Barth}\\
{\it University of Marburg, Germany}\\
Global versus local billiard level dynamics: The limits of universality\\
\elp

\blp
{\bf Mr. Thomas Bartsch}\\
{\it Institut f\" ur theoretische Physik, Stuttgart, Germany}\\
Significance of ghost orbit bifurcations in semiclassical spectra\\
\elp

\blp
{\bf Mr. Michael Baune}\\
{\it University of Bremen, Germany}\\
Investigation on the complex behaviour of a platinum electrode in various
aqueous solutions\\
\elp

\blp
{\bf Miss Maide Bucolo}\\
{\it Universit\` a degli Studi di Catania, Italy}\\
Identification of nonlinear dynamics by using neuro-fuzzy technique\\
\elp

\blp
{\bf Lic. Gabriel G. Carlo}, 
Eduardo Vergini and %??? 
Alejandro J. Fendrik\\
{\it Comisi\' on Nacional de Energia At\' omica, Buenos Aires, Argentina}\\
Numerical verification of Percival's conjecture in a quantum billiard\\ 
\elp

\blp
{\bf Lic. Gabriel G. Carlo} 
and Eduardo Vergini\\
{\it Comisi\' on Nacional de Energia At\' omica, Buenos Aires, Argentina}\\
Semiclassical quantization with short periodic orbits\\
\elp

\blp
{\bf Miss Corinne Chevallard}\\
{\it Institut Non Line\' eaire de Nice, France}\\
Zig-zag instability of an Ising well in liquid crystals\\
\elp

\blp
{\bf Professor Dr. Gamal Edrees}\\
{\it Mansoura University, Egypt}\\
The effect of adrenergic blockers on ethanol mediated metabolic and
hematologic changes in mammals\\
\elp

\blp
{\bf Dr. Halina Grushevskaya (Krylova)}\\
{\it Belarusian State University, Minsk, Belarus}\\
Deterministic chaos in quantum two-level systems with singular perturbation: 
Application to oscillatory models of Hopkin-Huxley neuron\\
\elp

\blp
{\bf Dr. Takahisa Harayama}\\
{\it ATR Adaptive Communications Research Laboratories, Kyoto, Japan}\\
Semiclassical limit of Fredholm determinant for quantized 
strongly chaotic billiards\\
\elp

\blp
{\bf Mr. Christian Hoeppler}\\
{\it Eberhard-Karls-University, T\" ubingen, Germany}\\
Quantum fluctuations in the cohesive force of metallic nanowires\\
\elp

\blp
{\bf Dr. Vladimir Leonidovich Kobelev}\\
{\it Institute of Metal Physics, Ekaterinburg, Russia}\\
Fractional Langevin equation to describe anomalous diffusion\\
\elp

\blp
{\bf Prof. Dr. Vjera Lopac}\\
{\it University of Zagreb, Croatia}\\
Dynamics of some bouncing ball systems\\
\elp

\blp
{\bf Mr. Hironori Makino}\\
{\it Waseda University, Tokyo, Japan}\\
Energy level statistics of oval billiards\\
\elp

\blp
{\bf Dr. M\' aria Marko\v sov\' a}\\
{\it Slovak University of Technology, Bratislava, Slovakia}\\
Universality classes for the ricepile SOC model with adhesive properites\\
\elp

\blp
{\bf M.Sc. Carlos Roberto Mejia Monasterio}\\
{\it Centro de Ciencias Fisicas UNAM, Cuernavaca, Mexico}\\
Low-lying states in a diffusive quantum dot. Is there Fock-space 
localization?\\
\elp

\blp
{\bf Mr. Matej Mencinger}\\
{\it University of Maribor, Slovenia}\\
Systems of differential equations and algebra\\
\elp

\blp
{\bf Miss Cruz M\' endez Valverde}\\
{\it Universidad de Salamanca, Spain}\\
Periodic orbits and trace formula in the logarithmic billiard\\
\elp

\blp
{\bf Professor Dr. Osama Nagy}\\
{\it Faculty of Education, Phys. \& Chem. Dept., Kafr El-Sheikh, Egypt}\\
Vibration of energy levels of $N_2^+$ molecular cation and vibronic
excitation cross sections by electron impact\\
\elp

\blp
{\bf Mr. Slaven Pele\v s}\\
{\it University of Manitoba, Winnipeg, Canada}\\
Analysis of chaos in periodically driven mechanical system\\
\elp

\blp
{\bf Mr. Christophe Riera}\\
{\it Institut Non Lin\' eaire de Nice, France}\\
The chaotic dripping faucet\\
\elp

\blp
{\bf Prof. Dr. Anatoly Bronislavovich Rinkevich}\\
{\it Institute of Metal Physics, Ekaterinenburg, Russia}\\
High frequency electromagnetic autosoliton in tungsten and aluminium\\ 
\elp

\blp
{\bf Prof. Dr. Valeri Romanovski}\\
{\it CAMTP, University of Maribor, Slovenia}\\
Groebner bases and local limit cycles bifurcations in polynomial systems\\
\elp

\blp
{\bf Mr. Hendrik Schanze}\\
{\it University of Marburg, Germany}\\
Transmission studies through a cosine-shaped billiard\\
\elp

\blp
{\bf Dr. Susumu Shinohara}\\
{\it Waseda University, Tokyo, Japan}\\
The breakup of KAM curves in non-twist area-preserving maps\\
\elp

\blp
{\bf Lic. Fernando P. Simonotti}\\
{\it Comision Nacional de Energia At\' omica, Buenos Aires, Argentina}\\
Semiclassical Husimi representation for billiard maps\\
\elp

\blp
{\bf Mr. St\' ephan Thiberge}\\
{\it Institut Non Lin\' eaire de Nice, France}\\
Localized structures in cholesteric liquid crystals\\
\elp

\blp
{\bf Mr. Nicolas Vandenberghe}\\
{\it Institut Non Lin\' eaire de Nice, France}\\
Limit cycle instability: An application to Sine Gordon equation\\
\elp

\blp
{\bf Mr. Gregor Veble}\\
{\it CAMTP, University of Maribor, Slovenia}\\
Study of spectral statistics of classically integrable systems\\
\elp

\blp
{\bf Mr. Marko Vrani\v car}\\
{\it CAMTP, University of Maribor, Slovenia}\\
WKB corrections to the energy splitting in double well potentials\\
\elp

\twocolumn[\Large \begin{center}
{\bf THE LIST OF PARTICIPANTS}\end{center}\small
\vspace*{0.3in}
]

\noindent
\bsp
{\bf Prof. Dr. Hosny Ali Abdusalam}\\
Department of Mathematics\\
Faculty of Science, Cairo University\\
Giza\\
Egypt\\
e-mail hosny at math-sci.cairo.eun.eg\\
\esp

\bsp
{\bf Mr. Jacobo Aguirre}\\
Hermosilla 125\\
28009 Madrid\\
Spain\\
e-mail faguirre at abonados.cplus.es\\
\esp

\bsp
{\bf Dr. Arantxa Alonso}\\
Departament de F\' \i sica Aplicada,
UPC.\\
Modulo B5 (Campus Nord)\\
C/ Jordi Girona Salgado s/n\\
08034 Barcelona\\
Spain\\
e-mail arantxa at benard.upc.es\\
\esp

\bsp
{\bf Prof. Dr. Drago Bajc}\\
Pedago\v ska fakulteta Maribor\\
Koro\v ska c. 160\\
2000 Maribor\\
Slovenia\\
\esp

\bsp
{\bf Mr. Michael Barth}\\
Universit\" at Marburg\\
Fachbereich Physik, AG NFP\\
Renthof 5\\
D-35032 Marburg\\
Germany\\
e-mail
michael.barth at physik.uni-marburg.de\\
\esp

\bsp 
{\bf Mr. Thomas Bartsch}\\
Institut f\" ur theoretische Physik\\
Teilinstitut 1\\
Pfaffenwaldring 57/4\\
70550 Stuttgart\\
Germany\\
e-mail
 bartsch at theo1.physik.uni-stuttgart.de\\
\esp

\bsp
{\bf Mr. Michael Baune}\\
Universit\" at Bremen\\
Leobener Str. NW II, B 1040\\
28359 Bremen\\
Germany\\
e-mail mbaune at uni-bremen.de\\
\esp

\bsp
{\bf Dr. Nils Berglund}\\
School of Physics\\
Georgia Tech\\
Atlanta, Georgia 30332-0430\\
USA\\
e-mails 
 berglund at wias-berlin.de,\\
 berglund at epfl.ch\\
\esp

\bsp
{\bf Mr. Gordan Bosanac}\\
Mariborska 44\\
10000 Zagreb\\
Croatia\\
e-mail gbosanac at phy.hr\\
\esp

\bsp
{\bf Miss Maja Bra\v ci\v c}\\
Faculty of Electrical Engineering\\
Tr\v zaska 25\\
1000 Ljubljana\\
Slovenia\\
e-mails 
 maja.bracic at fe.uni-lj.si,\\
 maja at osc.fe.uni-lj.si\\
\esp

\bsp
{\bf Miss Maide Bucolo}\\
Dipartimento Eletrico Eletronico
e Sistemistico\\
Universit\` a degli Studi di Catania\\
Viale Andrea Doria No. 6\\
95125 Catania\\
Italy\\
e-mail mbucolo at dees.unict.it\\
\esp

\bsp 
{\bf Mr. Hrvoje Buljan}\\
Vukovarska 240\\
10000 Zagreb\\
Croatia\\
e-mail hbuljan at doxy.phy.hr\\
\esp

\bsp
{\bf Lic. Gabriel Gustavo Carlo}\\
Laboratorio Tandar-
Depto. de F\' \i sica\\
Comisi\' on Nacional de Energia At\' omica\\
Av. del Libertador 8250 \\
(1429) Buenos Aires\\
Argentina\\
e-mail carlo at tandar.cnea.gov.ar\\
\esp

\bsp
{\bf Miss Corinne Chevallard}\\
Institut non Lin\' eaire de Nice\\
1361 Route des Lucioles\\
06560 Valbonne\\
France\\
e-mail chevalla at inln.cnrs.fr\\
\esp

\bsp
{\bf Mr. Marcel Clerc}\\
Institut non Lin\' eaire de Nice\\
1361 Route des Lucioles\\
06560 Valbonne\\
France\\
e-mail clerc at inln.cnrs.fr\\
\esp

\bsp
{\bf Prof. Dr. Vladimir Nikolov Damgov}\\
Space Research Institute\\
Bulgarian Academy of Sciences\\
P.O. Box 799\\
6 Moscowska Str.\\
1000 Sofia\\
Bulgaria\\
e-mail vdagmov at bas.bg\\
\esp

\bsp
{\bf Prof. Dr. Gamal Edrees}\\
Mansoura University\\
Faculty of Sciences\\
Mansoura\\
Egypt\\
e-mail sinfac at mum.mans.eun.eg\\
\esp

\bsp
{\bf Mrs. Serra Marina Francesca}\\
Milano, Italy\\
\esp

\bsp
{\bf Mr. Karl Mario Garcia-Ruiz}\\
Facultad de Quimica, UNAM\\
Cd. Universitaria\\
Coyoacan D.F.\\
CP 04510 Mexico City\\
Mexico\\
e-mail karl at eros.pquim.unam.mx\\
\esp

\bsp
{\bf Mr. Francesco Giulio Ginelli}\\
Universit\` a degli Studi di Firenze\\
Physics Department\\
Largo E. Fermi 2\\
Florence\\
Italy\\
e-mail ginelli at fi.infn.it\\
\esp

\bsp
{\bf Dr. Thomas Gorin}\\
Centro Internacional de Ciencias\\
Avenida Universidad s/n\\
Col. Chamilpa C.P. 62131\\
A.P. 6 101\\
Curenavaca, Morelos\\
Mexico\\
e-mail gorin at cicc.unam.mx\\
\esp

\bsp
{\bf Dr. Halina Grushevskaya
(Krylova)}\\
Physics Department\\
Belarusian State University\\
4 Fr. Skorina Ave.\\
Minsk 220050\\
Republic of Belarus\\
e-mail grushevskaja at phys.bsu.unibel.by\\
\esp

\bsp
{\bf Dr. Toma\v z Gyergyek}\\
Faculty of Electrical Engineering\\
Tr\v za\v ska 25\\
1000 Ljubljana\\
Slovenia\\
and\\
In\v stitut Jo\v zef Stefan\\
Jamova 39\\
1000 Ljubljana\\
Slovenia\\
e-mails tomaz.gyergyek at fe.uni-lj.si\\
tomaz.gyergyek at ijs.si\\
\esp

\bsp
{\bf Dr. Takahisa Harayama}\\
ATR Adaptive Communications\\
Research Laboratories\\
2-2 Hikaridai Seika-cho Soraku-gun\\
Kyoto 619-02\\
Japan\\
e-mail harayama at acr.atr.co.jp\\
\esp

\bsp
{\bf Mr. Christian Hoeppler}\\
Institute of Applied Physics (IAP)\\
Eberhard-Karls-University\\
Auf der Morgenstelle 10\\
D-72076 T\" ubingen\\
Germany\\
e-mail
christian.hoeppler at uni-tuebingen.de\\
\esp

\bsp
{\bf Mr. Szymon Maksymilian
 Kielbasa}\\
Instytut Fizyki UJ\\
Zaklad Optyki Atomowey\\
ul. Reymonta 4\\
30-059 Krakow\\
Poland\\
\esp

\bsp
{\bf Dr. Vladimir Leonidovich
 Kobelev}\\
Institute of Metal Physics\\
Ural Division of RAS\\
S. Kovalevskaya st. 18\\
Ekaterinburg 620219\\
Russia\\
e-mail vkobelev at imp.uran.ru\\
\esp

\bsp
{\bf Mr. Jochen Kronj\" ager}\\
Wendelgasse 2\\
35037 Marburg\\
Germany\\
e-mail kronjaeg at stud-mailer.uni-marburg.de\\
\esp

\bsp
{\bf Dr. George Krylov}\\
CAMTP, University of Maribor\\
Krekova 2\\
SI-2000 Maribor\\
Slovenia\\
e-mail george.krylov at uni-mb.si\\
\esp

\bsp
{\bf Mr. Mladen Latkovi\' c}\\
Department of Theoretical Physics\\
University of Zagreb\\
Bijeni\v cka c. 32\\
10000 Zagreb\\
Croatia\\
laci at phy.hr\\
\esp

\bsp
{\bf Prof. Dr. Israel Laulicht}\\
Dep. of Physics\\
Bar-Ilan University\\
Ramat-gan 52900\\
Israel\\
e-mails 
 laulicht at physnet.ph.biu.ac.il\\ 
lavlii at mail.biu.ac.il\\
\esp

\bsp
{\bf Miss Irena Lepsikova}\\
Paris, France\\
\esp

\bsp
{\bf Prof. Dr. Francois Alain Leyvraz}\\
Centro de Ciencias Fisicas\\
Av. Universidad s/n\\
Col. Chamilpa\\
Cuernavaca Morelos\\
Mexico\\
e-mail leyvraz at ce.fis.unam.mx\\
\esp

\bsp
{\bf Prof. Dr. Vjera Lopac}\\
University of Zagreb\\
FKIT, Maruli\' cev Trg 19\\
HR-10000 Zagreb\\
Croatia\\
e-mail vlopac at marie.fkit.hr\\
\esp

\bsp
{\bf Mr. Hironori Makino}\\
Department of Applied Physics\\
School of Science and Engineering\\
Waseda University\\
Okubo 3-4-1, Shinjuku-ku\\
Tokyo 169-0072\\ 
Japan\\
e-mail
 makino at aizawa.phys.waseda.ac.jp\\
\esp

\bsp
{\bf Dr. Philippe Marcq}\\
Lehrstuhl Nichtlineare Dynamik\\
Institut f\" ur Physik\\
Universit\" at Potsdam\\
Am Neuen Palais, Postfach 601553\\
D-14415 Potsdam\\
Germany\\
e-mail
 marcq at agnld.uni-potsdam.de\\
\esp

\bsp
{\bf Dr. M\' aria Marko\v sov\' a}\\
Department of Computer Science\\ and Engineering\\
Faculty of Electrical Engineering \\
and Information Technology\\
Slovak University of Technology\\
Ilkovi\v cova 3\\
Mlynska dolina\\
81219 Bratislava\\
Slovakia\\ 
e-mail mark at elf.stuba.sk\\
\esp

\bsp
{\bf M.Sc. Carlos Roberto
Mejia Monasterio}\\
Centro de Ciencias Fisicas UNAM\\
Apartado Postal 48-3\\
Cuernavaca Morelos\\
C.P. 62251\\
Mexico\\
e-mail mejia at fis.unam.mx\\
\esp

\bsp
{\bf Mr. Matej Mencinger}\\
Smetanova 17, F6\\
2000 Maribor\\
Slovenia\\
e-mail matej.mencinger at uni-mb.si\\
\esp

\bsp
{\bf Miss Cruz M\' endez Valverde}\\
F\' \i sica Te\' orica, Facultad de Ciencias\\
Universidad de Salamanca\\
E-37008 Salamanca\\
Spain\\
e-mail cruz at sonia.usal.es\\
\esp

\bsp
{\bf Prof. Dr. Igor Mozeti\v c}\\
CAMTP, University of Maribor\\
Krekova 2\\
2000 Maribor\\
Slovenia\\
e-mail igor.mozetic at uni-mb.si\\
\esp

\bsp
{\bf Mr. Nicola Murgia}\\
Strada 55 No. 10\\
Poggio dei Pini\\
I-09012 Capoterra (CA)\\
Italy\\
e-mail nicola.murgia at dsf.unica.it\\
\esp

\bsp
{\bf Prof. Dr. Osama Zaid Nagy}\\
Faculty of Education\\
Phys. \& Chem. Dept.\\
Kafr El-Sheikh\\
Egypt\\
e-mail onagy at dec1.tanta.eun.eg\\
\esp

\bsp
{\bf Mr. Slaven Pele\v s}\\
University of Manitoba, Physics Dpt.\\
502 Allen Bldg.\\
Winnipeg, MB R3T 2N2\\
Canada\\
e-mail peles at theory.uwinnipeg.ca\\
\esp

\bsp
{\bf Mr. Mikhail Pronine}\\
Universit\" at Bremen\\
FB1 Physik\\
Postfach 330440\\
28334 Bremen\\
Germany\\
e-mail pronin at physik.uni-bremen.de\\
\esp

\bsp
{\bf Mr. Vanja Ratkovi\v c}\\
R. Austrije 3 (c/o Vasili\v c)\\
10000 Zagreb\\
Croatia\\
e-mail vanja at phy.hr\\
\esp

\bsp
{\bf Mr. Christophe Riera}\\
Institut non Lin\' eaire de Nice\\
1361 Route des Lucioles\\
06560 Valbonne\\
France\\
e-mail riera at inln.cnrs.fr\\
\esp

\bsp
{\bf Prof. Dr. Anatoly
Bronislavovich\\ Rinkevich}\\
Institute of Metal Physics\\
Ural Division of\\
Russian Academy of Sciences\\
18 S. Kovalevskaya St., GSP-170\\ 
Ekaterinenburg 620219\\
Russia\\
e-mail rin at imp.uran.ru\\
\esp

\bsp
{\bf Prof. Dr. Valeri Romanovski}\\
CAMTP, University of Maribor\\
Krekova 2\\
2000 Maribor\\
Slovenia\\
e-mail valery.romanovsky at uni-mb.si\\
\esp

\bsp
{\bf Dr. Andreas Ruffing}\\
Center of Mathematical Sciences\\
Chair of Applied Mathematics\\
Arcisstrasse 21\\
D-80333 M\" unchen\\
Germany\\
e-mail ruffing at appl-math.tu-muenchen.de\\
\esp

\bsp
{\bf Dr. Luca Salasnich}\\
INFM, UdR Milano Universit\` a\\
Dipartimento di Fisica\\
Universit\` a di Milano\\
Via Celoria 16\\
20133 Milano\\
Italy\\
e-mail salasnich at mi.infm.it\\
\esp

\bsp
{\bf Mr. Rudi Sch\" afer}\\
Universit\" at Marburg\\
Fachbereich Physik, AG NFP\\
Renthof 5\\
D-35032 Marburg\\
Germany\\
e-mail
rudi.schaefer at physik.uni-marburg.de\\
\esp

\bsp
{\bf Mr. Hendrik Schanze}\\
Universit\" at Marburg\\
Fachbereich Physik, AG NFP\\
Renthof 5\\
D-35032 Marburg\\
Germany\\
e-mail
 h.schanze at physik.uni-marburg.de\\
\esp

\bsp
{\bf Dr. Susumu Shinohara}\\
Department of Applied Physics\\
Waseda University\\
3-4-1, Okubo Shinjuku-ku\\
Tokyo 169-8555\\
Japan\\
e-mail
 susumu at aizawa.phys.waseda.ac.jp\\
\esp

\bsp
{\bf Lic. Fernando P. Simonotti}\\
Laboratorio Tandar-
 Departamento de F\' \i sica\\
Comision Nacional de \\
Energia At\' omica\\
Av. del Libertador 8250 (1429)\\
Ciudad de Buenos Aires\\
Argentina\\
e-mail
 simonotti at tandar.cnea.gov.ar\\
\esp

\bsp
{\bf Mr. Erik Sinde}\\
Max-Planck-Institut f\" ur \\
Physik komplexer Systeme\\
N\" othnitzer Str. 38\\
D-01187 Dresden\\
Germany\\
e-mail erik at mpipks-dresden.mpg.de\\
\esp

\bsp
{\bf Mr. Daniel Sven\v sek}\\
Fakulteta za matematiko in fiziko, \\
oddelek za fiziko\\
Univerza v Ljubljani\\
Jadranska 19\\
1000 Ljubljana\\
Slovenia\\
e-mail daniel at fiz.uni-lj.si\\
\esp

\bsp
{\bf Mr. Josip Tav\v car, dipl. ing.}\\
Slovensko dru\v stvo za sistemsko raziskovanje\\
Ulica Heroja \v Slandra 9\\
2000 Maribor\\
Slovenia\\
e-mail josip.tavcar at uni-mb.si\\
\esp

\bsp
{\bf Mr. St\' ephan Thiberge}\\ 
Institut non Lin\' eaire de Nice\\
1361 Route des Lucioles\\
06560 Valbonne\\
France\\
e-mail thiberge at inln.cnrs.fr\\
\esp

\bsp
{\bf Mr. Nicolas Vandenberghe}\\
Institut non Lin\' eaire de Nice\\
1361 Route des Lucioles\\
06560 Valbonne\\
France\\
e-mail vandenbe at inln.cnrs.fr\\
\esp

\bsp
{\bf Mr. Gregor Veble}\\
CAMTP, University of Maribor\\
Krekova 2\\
2000 Maribor\\
Slovenia\\
e-mail gregor.veble at uni-mb.si\\
\esp

\bsp
{\bf Mr. Radim Vocka}\\
DSM/SPEC,  CEN Orme des Merisiers\\
91191 Gif-sur-Yvette Cedex\\
France\\
e-mail radim at spec.saclay.cea.fr\\
\esp

\bsp
{\bf Mr. Marko Vrani\v car}\\
CAMTP, University of Maribor\\
Krekova 2\\
2000 Maribor\\
Slovenia\\
e-mail mark.vranicar at uni-mb.si\\
\esp

\bsp
{\bf Mr. Vinko Zlati\' c}\\
Bukova\v cka cesta 12a\\
10000 Zagreb\\
Croatia\\
e-mail icetinic at croatica.botanica.hr\\
\esp 

\bsp
{\bf Dr. Karol Zyczkowski}\\
Smoluchowski Institute of Physics\\
Jagiellonian University\\
ul. Reymonta 4\\
30-059 Krakow\\
Poland\\
e-mail karol at tatry.if.uj.edu.pl\\
\esp

\bsp
{\bf Mr. Marko \v Znidari\v c}\\
Ulica 9. maja 26\\
2204 Miklav\v z\\
Slovenia\\
e-mail marko.znidaric at fmf.uni-lj.si\\
\esp

\onecolumn
}

\end{document}
