(A Special Session at Maribor, Slovenia, 2 July 1999)

**1. Yoji Aizawa***Ergodic Properties of Non-stationary Chaos***2. Tomaz Prosen***Quantum Poincaré Mapping***3. Yoshiki Kuramoto***Turbulence with Multiscaling in Large Assemblies of Oscillators***4. Marko Robnik***Topics in Quantum Chaos of Generic Systems*Intermission for Lunch**5. Hiroshi Hasegawa***Information Theoretical Basis of Random Matrix Distributions***6. Janko Gravner***Growth Phenomena in Cellular Automata***7. Mitsugu Matsushita**-
*Formation of Colony Patterns by a Bacterial Cell Population* **8. Aneta Stefanovska***Topics in Nonlinear Dynamics in the Human Cardiovascular*

System

Waseda University, Tokyo, Japan

(1) |

(2) |

The significant point in Eq.(2) is that the divergence of does not obey the inverse power law but exhibits an essential singularity when 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],

(3) |

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 obeys another universal law,

(4) |

Aizawa Y 1989a

Aizawa Y 1989b

Nekhoroshev N N 1977

Waseda University, Tokyo, Japan

Aizawa Y 1989

Aizawa Y et al 1989

Aizawa Y 1995

Aizawa Y 1991

Kurosaki S and Aizawa Y 1997

Yuri M 1995

Yuri M 1996

Yuri M 1997

Aizawa Y 1989

Kikuchi Y and Aizawa Y 1990

Kikuchi Y and Aizawa Y 1990

Aizawa Y 1984

Aizawa Y and Kohyama T 1984

Aizawa Y 1989

Aizawa Y 1993

Aizawa Y 1998

Waseda University, Tokyo, Japan

[1]Makino H, Harayama T and Aizawa Y 1999

[2]Robnik M and Prosen T 1997

[3]Prosen T 1998

the parametrically excited pendulum

University College London, UK

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.

University College London, UK

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.

Center for Research and Applications of Nonlinear Systems,

University of Patras, Greece

J. Guckenheimer and P. Holmes :

S. Wiggins :

J. Murray :

A. S. Mikhailov :

G. Nicolis and I. Prigogine :

G. Nicolis :

T. Bountis :

T. Bountis, C. F. Starmer and A. Bezerianos :

Casati G and Prosen T 1999

Casati G, Maspero G and Shepelyansky D L 1999

and National Institute for Space Research, São José dos Campos, Brazil

Chian A C-L 1999 Order and chaos in nonlinear wave interactions in astrophysical and space plasmas,

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 Alfvén waves,

Chian A C-L, Lopes S R and Abalde J R 1996, Hamiltonian chaos in two coupled three-wave parametric interactions with quadratic nonlinearity,

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,

Chian A C-L, Borotto F A and Gonzalez W D 1998, Alfvén intermittent turbulence driven by temporal chaos,

Chian A C-L and Abalde J R 1999, Nonlinear coupling of Langmuir waves with whistler waves in the solar wind,

Chian A C-L, Borotto, F A, Lopes S R and Abalde J R 1999, Chaotic dynamics of nonthermal planetary radio emissions,

Lopes S R and Chian A C-L 1996, Controlling chaos in nonlinear three-wave coupling,

Lopes S R and Rizzato F B 1998, Chaos and energy redistribution in the nonlinear interaction of two spatio-temporal wave triplets,

Pakter R, Lopes S R and Viana R L 1997, Transition to chaos in the conservative four-wave parametric interactions,

Turing, A.N.

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,

T. B. Benjamin and J. E. Feir,

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence", Springer-Verlag, Berlin, 1984.

A.A Andronov, and L. Pontyagin.

M. Argentina, P. Coullet,

M. Argentina et P. Coullet,

M. Argenina, P. Coullet and E. Risler, "Homoclinic instabilities in spatially extended systems", preprint INLN (1999)

M. Argentina, P. Coullet and L. Mahadevan,

M. Argentina, P. Coullet and V. Krinsky, ''Crossing of excitable waves in the Fizu-Nagumo model'', preprint (1998), submitted to

W. J. Firth and A. J. Scroggie,

Y. Pomeau,

C. Riera, P. Coullet and C. Tresser, "Localised structures in one space dimension", preprint INLN (1999)

and Niels Bohr Institute, Copenhagen, Denmark

Read chapter 1 and appendix A of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al.,

and Niels Bohr Institute, Copenhagen, Denmark

Read chapters 2 and 3 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al.,

and Niels Bohr Institute, Copenhagen, Denmark

Read chapters 7, 8 and 9 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al.,

and Niels Bohr Institute, Copenhagen, Denmark

Read chapters 9, 15 and 17 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al.,

and Niels Bohr Institute, Copenhagen, Denmark

Read chapter 22 and the take-home problem set for the next millennium in P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al.,

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.

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 . 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.

B. Eckhardt, S. Fishman, K. Müller and D. Wintgen, Phys. Rev. A

B. Eckhardt, O. Agam, S. Fishman, J. Keating, J. Main und K. Müller Phys. Rev. E

B. Eckhardt and J. Main, Phys. Rev. Lett.

B. Eckhardt, Physica D

B. Hüpper and B. Eckhardt, Phys. Rev. A

and on-off intermittency

Graduate School of Informatics, Kyoto University, Japan

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.
**References**

Fujisaka H and Yamada T 1985 *Prog. Theor. Phys.* **74** 918

Fujisaka H and Yamada T 1986 *Prog. Theor. Phys.* **75** 1087

Fujisaka H and Yamada T 1983 *Prog. Theor. Phys.* **69** 32

Yamada T and Fujisaka H 1983 *Prog. Theor. Phys.* **70** 1240

Platt N, Spiegel E.-A. and Tresser C 1993 *Phys. Rev. Lett.* **70**
279

Heagy J.-F., Platt N and Hammel S.-M. 1994 *Phys. Rev. E* **49** 1140

Ott E and Sommerer J.-C. 1994 *Phys. Lett. A* **188** 39

Lai Y.-C. and Grebogi C 1995 *Phys. Rev. E* **52** R3313

Harada T, Hata H and Fujisaka H 1999 *J. Phys. A* **32**

Fujisaka H, Matsushita S and Yamada T 1997 *J. Phys. A* **30** 5697

Fujisaka H, Ouchi K, Hata H, Masaoka B and Miyazaki S 1998 *Physica D*
**114** 237

Yamada T, Fukushima K and Yazaki T 1989 *Prog. Theor. Phys.
Suppl.* No.99 120

Cenys A, Namajunas A, Tamserius A and Schneider T 1996 *Phys. Lett. A*
**213** 259

Rodelsperger F, Cenys A and Benner H 1995 *Phys. Rev. Lett.* **75**
2594

Yamada T and Fujisaka H 1986 *Prog. Theor. Phys.* **76** 582

Hata H and Miyazaki S 1997 *Phys. Rev. E* **55** 5311

Miyazaki S and Hata H 1998 *Phys. Rev. E* **58** 7172

to on-off intermittency

Graduate School of Informatics, Kyoto University, Japan

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.

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.
**References**

Fujisaka H and Inoue M 1987 *Prog. Theor. Phys.* **77** 1334

Fujisaka H and Shibata H 1991 *Prog. Theor. Phys.* **85** 187

Fujisaka H 1992 in *From Phase Transitions to Chaos * eds. G Györgyi
et al (Singapore: World Scientific) pp 34-48

Beck C and Schlögl F 1993 *Thermodynamics of Chaotic Systems - An
Introduction -* (Cambridge: Cambridge University Press)

Ott E 1993 *Chaos in Dynamical Systems* (Cambridge: Cambridge University
Press)

Fujisaka H and Yamada T 1985 *Prog. Theor. Phys.* **74** 918

Fujisaka H and Yamada T 1986 *Prog. Theor. Phys.* **75** 1087

Platt N, Spiegel E.-A. and Tresser C 1993 *Phys. Rev. Lett.* **70**
279

Heagy J.-F., Platt N and Hammel S.-M. 1994 *Phys. Rev. E* **49** 1140

Ott E and Sommerer J.-C. 1994 *Phys. Lett. A* **188** 39

Lai Y.-C. and Grebogi C 1995 *Phys. Rev. E* **52** R3313

Yamada T and Fujisaka H 1986 *Prog. Theor. Phys.* **76** 582

Yamada T and Fujisaka H 1990 *Prog. Theor. Phys.* **84** 824

Fujisaka H and Yamada T 1993 *Prog. Theor. Phys.* **90** 529

Miyazaki S and Hata H 1998 *Phys. Rev. E* **58** 7172

Fujisaka H and Yamada T 1987 *Prog. Theor. Phys.* **77** 1045

Bonetto, F., Gallavotti, G.:

Bonetto, F., Gallavotti, G., Garrido, P.:

Bonetto, F., Gallavotti, G., Garrido, P.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G., Ruelle, D.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G., Cohen, E.G.D:

Gallavotti, G., Cohen, E.G.D:

Gallavotti, G.:

Cohen, E.G.D., Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Gallavotti, G.:

Most of the papers (and books) can be freely downloaded from:

and Fakultät Physik, Universität Göttingen, Germany

(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 of the spectrum and eigenfunctions.

In particular, the asymptotic decay of correlations is determined by the dimension of the spectral measure, while the diffusive spreading of wave packets is related to the ratio of the dimensions 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.

T. Geisel, R. Ketzmerick, G.Petschel, in

eds. G. Casati and B. V. Chirikov (Cambridge University Press, Cambridge 1995) p. 633.

R. Ketzmerick, K. Kruse, S. Kraut, T. Geisel,

R. Ketzmerick, K. Kruse, T. Geisel,

and Fakultät Physik, Universität Göttingen, Germany

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?

H.-U. Bauer, T. Geisel, K. Pawelzik, F. Wolf, in

H.-U. Bauer, T. Geisel, K. Pawelzik, F. Wolf,

F. Wolf, T. Geisel,

McGill University, Montreal, Quebec, Canada

L. Glass, M.C. Mackey.

L. Glass, J. Sun. Periodic forcing of a limit cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations.

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.

M.R. Guevera, L. Glass, A. Shrier. Phase-locking, period-doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells.

J. Keener, L. Glass. Global bifurcations of a periodically forced nonlinear oscillator.

A. Kunysz, L. Glass, A. Shrier. Overdrive suppression of spontaneously beating chick heart cell aggregates: Experiment and theory.

A. T. Winfree

McGill University, Montreal, Quebec, Canada

G. Bub, L. Glass. Bifurcations in a discontinuous circle map: A theory for a chaotic cardiac arrhythmia.

Courtemanche, L. Glass, J.P. Keener. Instabilities of a propagating pulse in a ring of excitable media.

L. Glass. Dynamics of cardiac arrhythmias.

L. Glass, A. Goldberger, J. Bélair. Dynamics of pure parasystole.

L. Glass, M.E. Josephson. Resetting and annihilation of reentrant abnormally rapid heartbeat.

A. L. Goldberger, E. Goldberger

T. Nomura, L. Glass. Entrainment and termination of reentrant wave propagation in a periodically stimulated ring of excitable media.

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.

McGill University, Montreal, Quebec, Canada

G. Bub, L. Glass, A. Shrier. Bursting calcium rotors in cultured cardiac myocyte monolayers.

A. Garfinkel, M. L. Spano, W. L. Ditto, J.N. Weiss. Controlling cardiac chaos.

L. Glass, W. Zeng. Bifurcations in flat-topped maps and the control of cardiac chaos.

K. Hall, L. Glass. Locating ectopic foci.

K. Hall, D. J. Christini, M. Tremblay, J. J. Collins, L. Glass, J. Billette. Dynamic control of cardiac alternans.

A. Kunysz, A. Shrier, L. Glass. Bursting behavior during fixed delay stimulation of spontaneously beating chick heart cell aggregates.

J. Parkinson, C. Papp. Repetitive paroxysmal tachycardia

J. Sun, F. Amellal, L. Glass, J. Billette. Alternans and period-doubling bifurcations in atrioventricular nodal conduction.

University of California, Davis

Aizenman M and Lebowitz J 1988

Bramson M and Neuhauser C 1994

Bramson M, Griffeath D, and Lawler G 1992

Chopard B and Droz M 1999

Fisch R, Gravner J, and Griffeath D 1993

Gravner J 1999

Gravner J and Griffeath D 1996

Gravner J and Griffeath D 1998

Gravner J and Griffeath D 1998

Griffeath D 1994. In

Packard N and Wolfram S 1985

Toffoli T and Margolus N 1987

Vichniac G 1984

Willson S 1984

Bergé P, Pomeau Y and Vidal C 1986

Boberg L and Brosa U 1988

Eckhardt B, Marzinzik K and Schmiegel A 1998 in

eds. J Parisi, S C Müller and W Zimmermann

Gebhardt T and Grossmann S 1993

Gebhardt T and Grossmann S 1994

Grossmann S in

Grossmann S 1999

Koschmieder E L 1993

Schuster H G 1988

Trefethen L N, Trefethen A E, Reddy S C and Driscol T A 1993

global performance-global scaling

Since heat current and roll velocity are global responses of the fluid to the driving temperature difference only global arguments turn out to determine the scaling behaviors with and with . 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 and allows to derive the various scaling laws valid in the different regions of the parameter space . 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).

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).

Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S, Wu X Z, Zaleski S and Zanetti G 1989

Chavanne X, Chilla F, Castaing B, Hebral B, Chaboud B and Chaussy J 1997

Cioni S, Ciliberto S and Sommeria J 1997

Grossmann S and Lohse D 1998

Koschmieder E L 1993

Shraiman B I and Siggia E D 1990

Siggia E D 1994

Wu X Z and Libchaber A 1991

rigorous and less rigorous insights

In this lecture, after introductory estimates of the dissipation coefficient , 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 . 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

Busse F H 1970

Busse F H 1978

Doering C R and Constantin P 1992

Doering C R and Constantin P 1994

Grossmann S 1995

Hopf E 1941

Howard L N 1972

Kerswell R R 1997

Lohse D 1994

Nicodemus R, Grossmann S and Holthaus M 1997

Nicodemus R, Grossmann S and Holthaus M 1997

Nicodemus R, Grossmann S and Holthaus M 1997

Nicodemus R, Grossmann S and Holthaus M 1998

Nicodemus R, Grossmann S and Holthaus M 1998

Nicodemus R, Grossmann S and Holthaus M 1999 "Towards lowering dissipation bounds"

Sreenivasan K R 1984

H. Haken,

J.A.S. Kelso,

McCulloch and W. Pitts,

H.R. Wilson and J.D. Cowan,

V.K. Jirsa and H.Haken,

Y. Kuramoto and I. Nishikawa,

P. Tass and H. Haken,

C. Uhl,

R.E. Mirollo and S.H. Strogatz,

U. Ernst, K. Pawelzik and T. Geisel,

P.L. Nunez, Oxford University Press, Oxford (1995)

V.K. Jirsa and H.Haken,

Anderson localization

Atomic Energy Research Institute, Nihon University, Kanda-Surugadai, Tokyo

1. B.L.Al'tshuler,

Montambaux, J.-L. Pichard, and J. Zinn-Justin(ELSEVIER, Amsterdam1995).

2. T. Guhr, A. Müller-Groeling,and H. A. Weidenmüller,

3. M. Toda, R. Kubo,and N. Sait

4. B. L. Al'tshuler and B. I. Shklovskii, Sov. Phys. JETP

5. B. L. Al'tshuler, I. Kh. Zharekeshev, S. A. Kotochigova and B. I. Shklovskii, Sov. Phys. JETP

6. V. E. Kravtsov, I. V. Lerner,B. L. Al'tshuler and A. G. Aronov, Phys. Rev. Lett.

7. M. L. Mehta and F. J. Dyson, J. Math. Phys.

8. A. G. Aronov, V. E. Kravtsov and I. V. Lerner, Phys. Rev. Lett.

9. A. G. Aronov and A. D. Mirlin, Phys. Rev.

10. J. T. Chalker, I. V. Lerner and R. Smith, Phys. Rev. Lett.

11. J. T. Chalker, V. E. Kravtsov and I. V. Lerner, JETP Lett.

Anderson Localization II

Atomic Energy Research Institute, Nihon University, Kanda-Surugadai, Tokyo

12. F. Leyvraz and T. H. Seligman, J. Phys. A: Math. Gen.

13. T. Guhr, Phys.Rev.Lett.

14. H. Kunz and B. Shapiro, Phys. Rev.

15. K. M. Frahm, T. Guhr and A. Müller-Groeling, Ann.Phys.(N.Y.)

16. R.A.Jalabert, J.-L.Pichard and C.W.J. Beenakker, Europhys. Lett.

17. D. Weinmann and J.-L.Pichard, Phys.Rev.Lett.

18. M. Gaudin, Nucl. Phys.

19. H. Hasegawa and J.-Z. Ma, J. Math. Phys.

20. H. Hasegawa

21. J.-L. Pichard and B. Shapiro, J.Phys.I (France)

22. P.J. Forrester, Phys. Lett.

23. D. Ruelle,

Florida Atlantic University, Boca Raton, USA

Braitenberg V and Schüz A (1991)

Haken H (1996)

Jirsa V K and Haken H (1996)

Jirsa V K, Fuchs A and Kelso J A S (1998)

Kelso J A S (1995)

Kistler W M, Gerstner W and van Hemmen J L (1997)

Nunez P L (1974)

Nunez P L (1995)

Wilson H R and Cowan J D (1972)

Wilson H R and Cowan J D (1973)

Florida Atlantic University, Boca Raton, USA

Braitenberg V and Schüz A (1991)

Cross M C and Hohenberg P C (1993)

Felleman D J and Van Essen D C (1991)

Haken H (1975)

Haken H (1983)

Jirsa V K and Haken H (1996)

Kelso J A S (1995)

Markram H, Lübke J, Frotscher M and Sakmann B (1997)

Murray J D (1993)

J. Keener and J. Sneyd.

Pumir, A., Plaza, F., Krinsky , V.

A. Pumir, V. Krinsky.

V. Krinsky and A.Pumir

V. Krinsky

and different specific forms --

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.

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.
**References**

Mori H and Kuramoto Y 1998 *Dissipative Structures and Chaos*
(Berlin Heidelberg: Springer-Verlag) Ch. 5

Chapman S and Cowling T. G. 1970 *The Mathematical Theory of
Nonuniform Gases* 3rd. Ed. (Cambridge: Cambridge Univ. Press)

Kuramoto Y 1984 *Chemical Oscillations, Waves, and Turbulence*
(Berlin: Springer-Verlag)

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.
**References**

Winfree A T 1967 *J. Theor. Biol.* **16** 15

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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.
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The Technical University of Denmark, Denmark

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The Technical University of Denmark, Denmark

Yu. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde,

Yu. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde,

E. Mosekilde,

The Technical University of Denmark, Denmark

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E. Mosekilde,

Department of Applied Biochemistry, Osaka City University, Osaka, Japan

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mesoscopic 3-dimensional billiards:

orbital magnetism and persistent current

Ma J and Nakamura K,

Nakamura K 1993

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91191 Gif sur Yvette cedex, France

R.A. Jalabert, J.-L. Pichard and C.W.J. Beenakker, 1994,

H. Baranger and P.A. Mello, 1994,

J.-L. Pichard, 1991, in

A.D. Stone, P.A. Mello, K.A. Muttalib and J.-L. Pichard, 1991, in

C.W.J. Beenakker, 1997,

T. Guhr, A. Müller-Groeling and H. Weidenmüller, 1998,

II. Chaotic mixing of the one body states by electron-electron interactions and delocalization in one dimension.

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III. Two dimensional gases of charges with Coulomb repulsions: A new metal between the Fermi glass and the Wigner crystal.

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(1) Bogomolny E B 1992

(2) Doron E and Smmlansky U 1992

(3) Prosen T 1995

(4) Prosen T 1996

University of Ljubljana, Slovenia

(1) Prosen T 1998

(2) Jona-Lasinio G and Presilla C 1996

(3) Castella H, Zotos X, and Prelovšek P 1995

(4) Prosen T 1998 `A Map from 1d Quantum Field Theory to Quantum Chaos on a 2d Torus', preprint cond-mat/9809211

and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy

Anteneodo C and Tsallis C 1998

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Atalmi A, Baldo M, Burgio G F and Rapisarda A 1998

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Latora V, Rapisarda A and Ruffo S 1999

Torcini A and Antoni M 1999

Yawn K R and Miller B N 1997

From electron billiards to Coulomb blockade

Dresden, Germany

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Richter, K.,

Aleiner I.L. and A.I. Larkin, Chaos, Solitons & Fractals

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Simmel, F., D. Abusch-Magder, D.A. Wharam, M.A. Kastner, and J.P. Kotthaus (cond-mat 9901274).

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Cuernavaca, México

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T. Prosen, Physica D

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F. Leyvraz, R.A. Mendez-Sanchez and T.H. Seligman (chao-dyn/9902009)

and Centro de Ciencias Físicas, University of Mexico, UNAM,

Cuernavaca, México

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Dendrinos D S, Sonis M 1990.

Sonis M, 1992.

Exploring stability of membrane potential in excitable cells

where and

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

Murray, J.D. 1993

Rinzel, J. and Terman, D.: Propagation phenomena in a bistable reaction-diffusion systems.

Starmer, C.F.: Theoretical characterization of ion channel blockade: Ligand binding to periodically accessible receptors.

Curran, M.E. et. al.: A molecular basis for cardiac arrhythmia: HERG mutations cause long QT syndrome.

Schroeder, B.C. et. al.: Moderate loss of function of cyclic-AMP-modulated KCNQ2/KCNQ3 K+ channels causes epilepsy.

Exploring front formation and propagation

where and

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 where is the spatial extent of the suprathreshold stimulus field and is the velocity of the

Murray, J.D.

Rushton, W.A.H. Initiation of the propagated disturbance.

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.

Starmer, C.F. et. al. Vulnerability in an excitable medium: Analytical and numerical studies of initiating unidirectional propagation.

Starobin, J.M. et. al. Vulnerability in one-dimensional excitable media.

Starobin, J. and Starmer, C.F. A common mechanism links spiral wave meandering and wavefront-obstacle separation.

Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia

Stefanovska A and Krošelj P 1997

Bracic M and Stefanovska A 1998a

Stefanovska A and Bracic M 1999a

Stefanovska A and Bracic M 1999b

Bracic M and Stefanovska A 1998b

Kvernmo HD, Stefanovska A, Bracic M, Kirkebøen K-A and Kvernebo K, 1998

Stefanovska A, Bracic M, Kvernmo HD

Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia

Signals derived from the human cardiovascular system have been found to contain several periodic components (Akselrod

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.

Axelrod S

Stefanovska A and Bracic M 1999

Bracic M and Stefanovska A 1998

Hales S 1773

Ludwig C 1847

Raschke F 1987 in

Glass L and Mackey MC 1988

Rosenblum M

Coventry, UK

*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: -symmetric systems, -symmetric systems -symmetric systems. Applications: Speciation, convection, Taylor-Couette flow.*Hopf Bifurcation to Periodic States*Loop space, circle group action by phase shift, spatio-temporal symmetries. Liapunov-Schmidt reduction. Genericity of -simple representations. The Equivariant Hopf Theorem. Examples: -symmetric systems, -symmetric systems. Applications: Oscillating hosepipe, animal locomotion, convection.*Symmetric Chaos*Pointwise and setwise symmetry of an attractor. Symmetry on average. Detectives. Collisions of attractors. Transverse Liapunov exponents. Examples: The cubic logistic map, -symmetric maps. Applications: Oscillator arrays, the Faraday Experiment.

for all . Consider a -equivariant ODE

where , . For simplicity, assume that , so there exists a `trivial branch' of solutions . The

Local bifurcation at occurs when the trivial branch undergoes a change of linear stability, so that has eigenvalues on the imaginary axis (often called

- Steady-state bifurcation: has a zero eigenvalue.
- Hopf bifurcation: has a complex conjugate pair of eigenvalues .

and -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.

Ashwin P, Buescu J, Stewart I 1994 Bubbling of attractors and synchronization of chaotic oscillators,

Ashwin P, Buescu J, Stewart I 1996 From attractor to chaotic saddle: a tale of transverse instability,

Buescu J, Stewart I 1995 Liapunov stability and adding machines,

Cohen J, Stewart I 2000 Polymorphism viewed as phenotypic symmetry-breaking, in

Collins J J, Stewart I 1993 Hexapodal gaits and coupled nonlinear oscillator models,

Collins J J, Stewart I 1993 Coupled nonlinear oscillators and the symmetries of animal gaits,

Dellnitz M, Golubitsky M, Stewart I, Hohmann A 1995 Spirals in scalar reaction-diffusion equations,

Dionne B, Golubitsky M, Stewart I 1996 Coupled cells with internal symmetry: I. Wreath products,

Dionne B, Golubitsky M, Stewart I 1996 Coupled cells with internal symmetry: II. Direct products,

Dionne B, Golubitsky M, Silber M, Stewart I 1995 Time-periodic spatially-periodic planforms in Euclidean equivariant PDE,

Golubitsky M, Field M 1992

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

Golubitsky M, Stewart I 1985 Hopf bifurcation in the presence of symmetry,

Golubitsky M, Stewart I 1986 Symmetry and stability in Taylor-Couette flow,

Golubitsky M, Stewart I 1986 Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, in

Golubitsky M, Stewart I, Collins J J, Buono L 1998 A modular network for legged locomotion,

Golubitsky M, Stewart I 1999 Symmetry and pattern formation in coupled cell networks,

Golubitsky M, Stewart I, Schaeffer D G 1988

King G, Stewart I 1992 Symmetric chaos, in

Stewart I 1987 Stability of periodic solutions in symmetric Hopf bifurcation,

Stewart I 1988 Bifurcations with symmetry, in

Stewart I 1992 Bifurcation theory old and new, in

Stewart I 1992 Broken symmetry and the formation of spiral patterns in fluids, in

Marburg, Germany

Stöckmann H J and Stein J 1990

Sridhar S 1991

Stein J and Stöckmann H J 1992

Gräf H D

So P

Stein J

Kudrolli A

Ellegaard C

Stöckmann H J 1999

Marburg, Germany

Lewenkopf C H and Weidenmüller H A 1991

Alt H

Stoffregen U

Stein J

Lehmann N

Alt. H

Haake F

Stöckmann H J and Šeba P 1998

Stöckmann H J 1999

Stochastic approach and applications

Heinrich-Heine-University, Düsseldorf, Germany

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, Llinás 1990, Singer, Gray 1995, Hari, Salmelin 1997) and of therapeutic stimulation techniques in neurology and neurosurgery (Benabid et al. 1991, Blond et al. 1992).

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).

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).

Benabid A L et al. 1991

Blond S et al. 1992

Haken H 1983

Hari R, Salmelin R 1997

Kuramoto Y 1984

Singer W, Gray C M 1995

Steriade H, Jones E G, Llinás R 1990

Tass P A 1999

Winfree A T 1980

Heinrich-Heine-University, Düsseldorf, Germany

Hämäläinen M et al. 1993

Pikovsky A S, Rosenblum M G, Kurths J 1996

Roelfsema P R et al. 1997

Rosenblum M G, Pikovsky A S, Kurths J 1996

Singer W, Gray C M 1995

Tass P A 1999

Tass P et al. 1998