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2$\bf\sf ^{nd}$ Slovenia-Japan Seminar

Center for Applied Mathematics and Theoretical Physics University of Maribor

28 June - 5 July 2003


Sponsors and Donators:

Ministry of Education, Science and Sport
of the Republic of Slovenia

Japan Society for the Promotion of Science

Nova Kreditna Banka Maribor

Directors and Organizers:

Marko Robnik (CAMTP)

Yoji Aizawa (Waseda University)

Phone +(386) (2) 2355 350 and 2355 351 $\bullet$ Fax +(386) (2) 2355 360 $\bullet$


Dear Colleagues, Dear Friends, Dear Participants,

Welcome to the 2nd Slovenia-Japan Seminar, which is taking place on 28 June through 5 July 2003 under the organization of CAMTP of the University of Maribor, and which is the follow-up meeting of the 1st Slovenia-Japan Seminar. It took place at the Waseda University in Tokyo, on 18-19 November 2002, under the direction of Professor Yoji Aizawa, and was was very successful. We had four participants from Slovenia and about 35 participants from Japan. The scientific programme was very rich and inspiring, and the hospitality of our Japanese colleagues and friends was really great and highly appreciated. I do hope that this 2nd meeting will be equally successful on the scientific side, and pleasant on the other side, as an extended social event. I have made every effort to make your stay in Slovenia interesting, comfortable and very much enjoyable. Apart from the main scientific stream of thought, we shall enjoy also cultural events (the chamber music concert by the String Quartet Feguš, Festival Lent), social events (conference dinner), touristic event (excursion to Lake Bled, Postojna Cave and Ljubljana), and even sportific events in the facilities of Terme Maribor.

Our Slovenia-Japan meetings are based on our joint Slovenia-Japan collaboration Programme, which always runs for two years, and is funded by the Ministry of Education, Science and Sport of the Republic of Slovenia, and by the Japan Society for the Promotion of Science. The principal investigator on the Japanese side is Professor Yoji Aizawa from Waseda University, Tokyo, and on the Slovenian side Professor Marko Robnik, director of CAMTP, University of Maribor. It provides funds also for the participation of Japanese colleagues - invited speakers - at our international Summer Schools and Conferences "Let's Face Chaos through Nonlinear Dynamics", of which the most recent one, the 5th one, was on 30 June - 14 July 2002 in Maribor, under the organization of CAMTP. Our Slovenia-Japan cooperation meanwhile became very intense, rich and fruitful. The scientific topics covered in our meetings and individual contacts have a broad scope of nonlinear dynamics of classical and quantum chaos, synergetics and theory of complex systems. It thus covers theoretical and experimental physics, mathematics, astronomy and astrophysics, chemistry, physiology, biology, medicine, engineering, economics and even sociology. Nevertheless, the dominant core of topics is in the domain of physics. Our joint scientific activities are thus very strongly interdisciplinary. We do hope that soon we shall start also the collaboration in joint research projects resulting in joint publications.

Slovenia appreciates science in Japan very highly and certainly benefits a lot from our mutual contacts. Conversely, we believe that Japanese colleagues will be stimulated by the results of the Slovenian scientists, and also that Slovenia can provide a kind of European base for further interactions of Japanese colleagues with European countries, especially from European Union, of which Slovenia will become a full member on 1 May 2004. At that important event new possibilities open up, the possibilities to find further cooperation ties also through the European dimensions, namely through the European institutions in Brussels. So we have some very clear vision, which we would like to convert into reality before too long.

I wish you enjoyable stay in Maribor, Slovenia, a successful symposium and pleasant cultural entertainment at our international Lent Festival in Maribor.

Professor Dr. Marko Robnik
-- Director of CAMTP --

Maribor, 23 June 2003

  Mon 30 June Tue 1 July Wed 2 July Thu 3 July Fri 4 July
Chairman Robnik Shudo Robnik Aizawa EXCURSION
09:00-10:00 Aizawa Romanovski Eckhardt Song-Ju Kim 08:30 departure
10:00-11:00 Shudo Kato Aizawa Romanovski 11:00 arr Bled
11:00-11:30 -------- Coffee & Tea ------------  
11:30-12:30 Stefanovska Stefanovska Paluš Prosen Robnik  
12:30-13:30 Miyake Bajec/Musizza Erzen/Bernjak Shudo Yamazaki  
13:30-15:30 --------- Lunch -------------- 13:30-14:30 lunch
Chairman Aizawa Stefanovska Romanovski Robnik 14:30 departure
15:30-16:30 Prosen Robnik Miyazaki Ruffing  
16:30-17:00 -------- Coffee & Tea ------------ 16:30 arr Postojna
17:00-18:00 Horvat Jamšek Umeno Znidaric Veble 17:00 Visit Cave
18:15-20:00 Dinner Dinner ---- Dinner 18:30 departure
20:00-24:00 Festival Festival Conference Dinner Slow Food and Festival Festival 19:30 arr Ljubljana 19:30-20:30 Old Town 20:30-22:00 dinner 24:00arr Maribor

Addresses of All Participants

Professor Dr. Yoji Aizawa
Department of Applied Physics
School of Science and Engineering
Waseda University
Okubo 3-4-1, Shinjuku-ku
Tokyo 1690072
Telefax +(81) (3) 3200 2457
Telephone +(81) (3) 3203 2457

Mr. Boštjan Bajec
Department of psychology, Faculty of arts
University of Ljubljana
Aškerceva 2
SI-1000 Ljubljana
Telefax +(386) (1) 42 59 301
Telephone +(386) (1) 24 11 184

Mr. Alan Bernjak
Skupina za nelinearno dinamiko in sinergetiko
Univerza v Ljubljani
Trzaška 25
1000 Ljubljana
Telefax +(386) (1) 4768 246
Telephone +(386) (1) 4264 630

Mr. Martin Horvat
Physics, Department, Faculty of Mathematics and Physics
University of Ljubljana
Jadranska 19
SI-1111 Ljubljana
Telefax +(386) (1) 2517 281
Telephone +(386) (1) 4766 588

Professor Dr. Bruno Eckhardt
Fachbereich Physik
Philipps-Universität Marburg
D-35032 Marburg
Telefax +(386) 6421 28 24511
Telephone +(49) 6421 28 21316

Mr. David Erzen, univ. dipl. fiz.
Jozef Stefan Institute
Jamova 39
SI-1000 Ljubljana
Telephone +(386) (41) 384 227

Mag. Janez Jamšek
Pedagoška fakulteta
Univerza v Ljubljani
Kardeljeva plošcad 16
SI-1000 Ljubljana
Telefax +(386) (1) 5892233 Telephone +(386) (1) 5892296

Lecturer Dr. Takeo Kato
Department of Applied Physics
Osaka-City University
3-3-138 Sugimoto, Sumiyoshi-ku
Osaka 558-8585
Telefax +(81) (6) 6605 2769
Telephone +(81) (6) 6605 2904

Dr. Song-Ju Kim
Chaos-based Cipher Chip Project, Presidential Research Fund,
Incorporated Administrative Agency, Communications Research Laboratory
4-2-1, Nukui-kitamachi, Koganei-shi, Tokyo 184-8795 Japan
Telefax: +81-42-327-6299
Telephone: +81-42-327-6556

Professor Dr. Yoshihiro Miyake
Dept. of Interdisciplinary Graduate School of Science and Engineering
Tokyo Institute of Technology
Midori, Yokohama, 226-8502
Telefax +(81) (45) 924 5646
Telephone +(81) (45) 924 5646

Dr. Syuji Miyazaki
Department of Applied Analysis and Complex Dynamical Systems
Graduate School of Informatics
Kyoto University
Yoshida-Honmachi, Sakyo-ku
606-8501 Kyoto
Telefax +(81) (75) 753 3388
Telephone +(81) (75) 753 3391
e-mail syuji/

Mr. Bojan Musizza
Group of Nonlinear Dynamics and Synergetics
Faculty of Electrical Engineering
University of Ljubljana
Trzaška 25
1000 Ljubljana

Dr. Milan Paluš
Institute of Computer Science AS CR
Pod vodarenskou vezi 2
182 07 PRAGUE 8
Czech Republic
Phone: +420 266 053 430
Fax: +420 286 585 789
E-mail: mp

Professor Dr. Tomaz Prosen
Department of Physics, Faculty of Mathematics and Physics
University of Ljubljana
Jadranska 19
SI-1000 Ljubljana
Telefax +(386) (1) 2517 281
Telephone +(386) (1) 4766 578

Professor Dr. Marko Robnik
CAMTP - Center for Applied Mathematics
and Theoretical Physics
University of Maribor
Krekova 2
SI-2000 Maribor
Telefax +(386) (2) 2355 360
Telephone +(386) (2) 2355 350

Professor Dr. Valery Romanovski
CAMTP - Center for Applied Mathematics
and Theoretical Physics
University of Maribor
Krekova 2
SI-2000 Maribor
Telefax +(386) (2) 2355 360
Telephone +(386) (2) 2355 361

Dr. Andreas Ruffing
Department of Mathematics
Munich University of Technology
Boltzmannstrasse 3
D-85747 Garching
EU, Germany
Telefax +(49) (89) 289-16837
Telephone +(49) (89) 289-16826
e-mail ruffing/

Professor Dr. Akira Shudo
Department of Physics
Tokyo Metropolitan University
Minami-Ohsawa 1-1, Hachioji-shi
Tokyo 192-0397
Telefax +(81) (426) 772 483
Telephone +(81) (426) 772 503

Professor Dr. Aneta Stefanovska
Group of Nonlinear Dynamics and Synergetics
Faculty of Electrical Engineering
University of Ljubljana
Trzaška 25
SI-1000 Ljubljana
Telefax +(386) (1)
Telephone +(386) (1) 476 8246

Dr. Ken Umeno
CCCP - Chaotic Cipher Chip Project
Communications Research Laboratory
4-2-1 Nukui-Kitamachi
Tokyo 184-8795
Telefax +(81) (42) 327 6299
Telephone +(81) (42) 327 6399

Dr. Gregor Veble
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor
Krekova 2
SI-2000 Maribor
Faculty of Mathematics and Physics, Department of Physics
University of Ljubljana
Jadranska 19
SI-1000 Ljubljana
Telefax +(386) (2) 2355 360
Telephone +(386) (1) 4766 578

Mr. Gregor Vidmar
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor
Krekova 2
SI-2000 Maribor
Telefax +(386) (2) 2355 360
Telephone +(386) (2) 2355 350

Mr. Marko Vranicar
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor
Krekova 2
SI-2000 Maribor
Telefax +(386) (2) 2355 360
Telephone +(386) (2) 2355 362

Dr. Yoshihiro Yamazaki
Department of Physics, Faculty of Science and Engineering
Waseda University
3-4-1 Okubo
Shinjuku-ku, Tokyo 169-8555
Telefax +81-3-3232-9746
Telephone +81-3-5286-8187

Mr. Marko Znidaric
Physics, Department, Faculty of Mathematics and Physics
University of Ljubljana
Jadranska 19
SI-1111 Ljubljana
Telefax +(386) (1) 2517 281
Telephone +(386) (1) 4766 588


Chaotic scattering by random potential

Yoji Aizawa

Department of Applied Physics, School of Science and Engineering
Waseda University, Tokyo, Japan

The origin of chaotic scattering is analysed by the Riemannian geometrization method, and the essential role of positive curvature effects will be elucidated. In the compact dynamical systems, the orbital instability is induced in the negative curvature region, but in open systems the positive curvature region plays more essential roles to generate chaotic scattering. When we consider integrable potentials accompanied by random perturbations, it is expect that the chaotic scattering will be enhanced by the perturbations. Actually, by carrying out with 2-dimensional systems, some universal aspects of the chaotic scattering which can be induced by random potentials are demonstrated.

Logarithmic scaling in the stationary/nonstationary chaos transition

Statistical features in the transition process from stationary to nonstationary chaos are studied by carrying out with the modified Bernoulli maps. The temporal behaviors of the transition process are analysed by use of renewal functions, and it is shown that the logarithmic scaling is universally induced by the nonadiabatic change of the intermittent parameter. The detailed structure of the logarithmic scaling regime is discussed by means of the finite-range statistics, and it is shown that the diversity in the fluctuation of transition paths can be characterized by the log-Weibull distribution.

Noise and oscillations in a sequence of reaction times

Boštjan Bajec$^1$, Valentin Bucik$^1$ and Aneta Stefanovska$^2$

University of Ljubljana, Ljubljana, Slovenia
$^1$Department of Psychology, Faculty of Arts
$^2$Group of Nonlinear Dynamics and Synergetics, Faculty of Electrical Engineering

The reaction time is the period that elapses between the presentation of a stimulus (e.g. visual, auditory) and the subject's reaction (e.g. manual, oral) to it. It is often used in psychological experiments in which the nature of the processes that underlie human behaviour is studied. In this kind of experiment, the fluctuations in reaction time are traditionally regarded as being random. Recent studies, by Gilden, Thornton, and Mallon (1996), Clayton and Bruhns-Frey (1997), Gilden (1997 and 2001), and Kelly, Heath, and Longstaff (2001) show, however, that the reaction times in different tasks are correlated in time.

In the present study we characterize the dynamics in time series consisting of reaction times. Its purpose was to establish whether the reaction times are related to the complexity of the presented stimuli, or to the rhythm of their presentation. Eight females participated in seven experiments. In each case, reaction times were recorded to 1188 stimuli. The stimuli used were visual, either letters or the positions of the hour hand on a clock. The participants were instructed to identify the letter, or the position of the hand, on a computer screen. The participant's reactions were verbal and the reaction time was that which elapsed between the presentation of the stimulus and the oral answer. In one experiment with letters as stimuli participants were offered to choose their own pace of stimuli presentation; in one experiments one-second intervals elapsed between stimuli, and in two experiments two-second intervals were used. In the experiments with positions of the pointer of the clock self-paced rhythm of presentations was used once, and two-second intervals between presentations twice.

The results of this study show that a time series of reaction times consists of both stochastic and deterministic components. Several rhythmical components were observed. The most dominant rhythms differ to some extent between participants, but some of them appear consistently in all individuals: frequencies that are apparently connected with the rhythm of breathing, with learning in the experimental situation, and with the rhythm of stimuli presentation.

Clayton, K., and Bruhns-Frey, B. (1997). Studies of mental "noise", Nonlinear Dynamics in Psychology and Life Sciences, 1 (3), 173-180.
Gilden, D.L. (1997). Fluctuations in time required for elementary decisions, Psychological Science, 8 (4), 296-301.
Gilden, D.L. (2001). Cognitive emissions of 1/f noise, Psychological Review, 108 (1), 33-56.
Gilden, D.L., Thornton, T., and Mallon, M.W. (1995). 1/f noise in human cognition, Science, 267 (5205), 1837-1839.
Kelly, A., Heath, R., and Longstaff, M. (2001). Response time dynamics: Evidence for linear and low-dimensional nonlinear structure in human choice sequences, Quarterly Journal of Experimental Psychology: Human Experimental Psychology, 54A (3), 805-840.

Blood flow oscillations in congestive heart failure

Alan Bernjak$^1$, Peter Clarkson$^2$, Mark Lippett$^2$, Peter McClintock$^3$ and Aneta Stefanovska$^{1,3}$

$^1$University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Ljubljana, Slovenia

$^{2}$Cardiology Department, Royal Lancaster Infirmary, Ashton Road,
Lancaster LA1 4RP, UK

$^{3}$Lancaster University, Department of Physics, Lancaster LA1 4YB, UK

Congestive heart failure is a very common disease and is a pathophysiological state in which an abnormality of cardiac function is responsible for the failure of the heart to pump blood at a rate commensurate with the requirements of the metabolizing tissue. The heart is unable to meet the metabolic demands of the body either at rest or on exercise. Many patients are significantly impaired, with symptoms of fatigue and dyspn\oea. It is recognised that heart-rate variability (HRV) in such patients is significantly reduced, but the effect on the complex interactions among the other oscillators remains unknown.

We discuss changes in blood flow oscillations in patients with congestive heart failure, and evaluate the response following beta blockade.

Nonlinear dynamics and the transition to turbulence in shear flows

Bruno Eckhardt and Holger Faisst

Fachbereich Physik, Philipps Universität Marburg, D-35032 Marburg, Germany

Hagen-Poiseuille flow through a pipe of circular cross section belongs to the class of shear flows that does not become linearly unstable. The situation is similar to Taylor-Couette flow with the inner cylinder at rest as well as to Taylor-Couette in the limit of large radii where the system approaches plane Couette flow [1]. In these cases the transition to turbulence is not related to series of symmetry-breaking linear instabilities but rather to the formation of nonlinear 3-d flow states. These states originate in saddle node bifurcations, are all unstable and connect to form a chaotic saddle. It is the aim of the lecture to discuss this scenario for the transition to turbulence, to present the numerical evidence and to relate it to experimental observations.

For pipe flow we have identified a family of three-dimensional travelling waves [2]. Their topology is dominated by downstream vortices and streaks. They originate in saddle-node bifurcations at Reynolds numbers as low as $1250$, where $Re$ is based on the mean downstream velocity and the pipe diameter. All states are immediately linearly unstable at the bifurcation.

For the life time experiments [3] we appeal to the experiments of Darbyshire and Mullin [4] and keep the volume flux constant. We extended our numerical investigations to times of $2000$ or more (unit of time: mean streamwise velocity/radius), far exceeding the values accessible in the longest currently available laboratory experiment. As initial conditions we used a high amplitude uncorrelated superposition of spectral modes. The spatial structure is so rich and the amplitude so high that the probability to trigger turbulent dynamics is maximal for a wide range of Reynolds numbers.

Various conclusions can be drawn from our lifetime experiments: The minimum amplitude to trigger a long living turbulent dynamics decreases with Reynolds number. The life times depend very sensitively on the choice of parameters, resulting in large fluctuations even for fixed Reynolds number. The largest Lyapunov exponent is about $6.5\times10^{-2}$ at transitional Reynolds numbers and increases slowly with $Re$. This corresponds to an amplification by roughly $10^6$ over the $200$ time units of a typical nonlinear regeneration cycle. The regions of quickly decaying and long-living trajectories are separated by complicated, fuzzy stability borders. The results are in agreement with experiments by Darbyshire & Mullin [4]. The median of the life times of the turbulent states increases rapidly with Reynolds numbers and reaches the cut-off time of $2000$ at a Reynolds number of about $\approx 2200$. This may serve as a statistical definition of the transitional Reynolds number.

$[1]$ H. Faisst, B. Eckhardt Transition from the Couette-Taylor system to the plane Couette system , Phys. Rev. E, 61 (2000), 7227
$[2]$ H. Faisst, B. Eckhardt Travelling waves in pipe flow, /0304029
$[3]$ H. Faisst, B. Eckhardt Sensitive dependence on initial conditions in transition to turbulence in pipe flow , in preparation
$[4]$ A.G. Darbyshire, T. Mullin. Transition to turbulence in constant-mass-flux pipe flow.
J. Fluid Mech., 289, 83-114, (1995).

Synchronization of plasma potential relaxation oscillations by an external driving force
David Erzen$^{1}$, Tomaz Gyergyek$^{2,1}$, Milan Cercek$^{1}$ and Aneta Stefanovska$^{2}$
$^1$Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
$^2$University of Ljubljana, Faculty of Electrical Engineering, Trzaška 25, 1000 Ljubljana

If a current is drawn through a plasma various plasma instabilities can be excited. When electron current is drawn through a magnetized plasma column by a positive electrode electrostatic ion cyclotron (EICO) [1] or potential relaxation (PRO) [2] oscillations can be triggered.

In this work we discuss an experiment in which a planar anode is immersed in a weakly magnetized low pressure discharge plasma column with its surface perpendicular to the magnetic field lines. In order to have a better defined system we put additional grid into the plasma so that its surface is also perpendicular to the magnetic field lines. Both, the anode and the grid can be biased separately to arbitrary potentials with respect to the grounded vacuum vessel which is taken as the reference potential. When the anode is biased a few volts above the plasma potential and the grid is connected to the grounded vacuum vessel through a resistor, the electron saturation current to the anode starts to oscillate. The current modulation is caused by a traveling potential structure, which moves periodically from the grid to the anode. The mechanism of such PRO's is well described in the literature [2,3].

We have verified that the frequency of the oscillations is inversely proportional to the distance between the grid and the anode, which demonstrates that the frequency is determined by the transit time of ions from the grid to the anode. The spectrum of the electron current oscillations shows several higher harmonics. When a sinusoidal external voltage signal is applied to the grid, plasma oscillations can be synchronized to the external force even when the later has very low amplitude. We have observed synchronization of the plasma oscillations with the external force at the basic harmonic, higher harmonics (2:1), (3:1) and subharmonics (1:2) The corresponding Arnold tongues are presented. Experimental results are compared to approximate analytical solutions of the van der Pol equation with harmonical external force and good agreement is found. Phase diagrams of the experimental forces obtained by Hilbert transform method [4] are also presented. They show clear evidence that synchronization has indeed occurred.


[1] Rasmussen, J. J., and Schrittwieser, R. (1991). On the Current-Driven Electrostatic Ion-Cyclotron Instability: A Review, IEEE Trans. Plasma Sci. 19 , 457-501
[2] Gyergyek, T., Cercek, M., Jelic, N., and Stanojevic, M. (1993). Experimental Analysis of a Low Frequency Instability in a Magnetized Discharge Plasma, Contrib. Plasma Phys. 33 , 53-72
[3] Schrittwieser, R. W. (1993). The influence of electron/ion collisions on a low-frequency plasma instability, Int. J. Mass Spectr. Ion Proc.,129 , 205-213.
[4] Pikovsky, A., Rosenblum, M., and Kurths, J. Synchronization - A Universal Concept in Nonlinear Sciences, (Cambridge University Press, 2001).

Uni-directional transport properties of a serpent billiard

Martin Horvat and Tomaz Prosen

Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

We present a dynamical analysis of a classical billiard chain -- a channel with parallel semi-circular walls, which can serve as a prototype for a bended optical fiber. An interesting feature of this model is the fact that the phase space separates into two disjoint invariant components corresponding to the left and right uni-directional motions. Dynamics is decomposed into the jump map -- a Poincare map between the two ends of a basic cell, and the time function -- traveling time across a basic cell of a point on a surface of section. The jump map has a mixed phase space where the relative sizes of the regular and chaotic components depend on the width of the channel. For a suitable value of this parameter we can have almost fully chaotic phase space. We have studied numerically the Lyapunov exponents, time auto-correlation functions and diffusion of particles along the chain. As a result of a singularity of the time function we obtain marginally-normal diffusion after we subtract the average drift. The last result is also supported by some analytical arguments.

Horvat M, Prosen T 2003 Uni-directional transport properties of a serpent billiard, preprint Dittrich T, Mehlig B, Schanz H, Smilansky U 1998 Signature of chaotic diffusion in band spectra Phys. Rev. E 57 (1) 359-365
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Metzler R, Klafter J 2000 The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep 339 1-77

Nonlinear cardio-respiratory interaction

Janez Jamšek$^{1,2,3}$, Aneta Stefanovska$^{1,3}$ and Peter V. E. McClintock$^3$

$^1$University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Trzaška 25, 1000 Ljubljana, Slovenia

$^{2}$University of Ljubljana, Faculty of Education,
Department of Physics and Technical Studies, Kardeljeva plošcad 16, 1000 Ljubljana, Slovenia

$^{3}$Lancaster University, Department of Physics, Lancaster LA1 4YB, UK

Most real systems are nonlinear and complex. In general, they may be regarded as a set of interacting subsystems; given their nonlinearity, the interactions can be expected to be nonlinear too.

The phase relationships between a pair of interacting oscillators can be obtained from bivariate data (i.e. where the coordinate of each oscillator can be measured separately) by use of the methods recently developed for analysis of synchronization, or generalized synchronization, between chaotic and/or noisy systems. Not only can the interactions be detected [1], but their strength and direction can also be determined [2]. The next logical step in studying the interactions among coupled oscillators must be to determine the nature of the couplings: the methods developed for synchronization analysis are not capable of answering this question.

Bispectral analysis, a technique based on high order statistics, was recently extended to encompass time dependence for the case of coupled nonlinear oscillators [3]. The method is applicable to univariate as well as to multivariate data obtained respectively from one or more of the oscillators.

The method will be first demonstrated for a generic model of interacting systems whose basic units are Poincaré oscillators. Their frequency and phase relationships are explored for different coupling strengths, both with and without Gaussian noise. The distinctions between additive linear or quadratic, and parametric (frequency modulated), interactions in presence of noise will be illustrated.

Then we will present results of bispectral analysis of blood flow signals and show that the cardiac and respiratory processes are apparently nonlinear phase coupled.


[1] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, A Universal Concept in Nonlinear Sciences, (Cambridge University Press, Cambridge 2001).

[2] Rosenblum, L. Cimponeiru, A. Bezerianos, A. Patzak, and R. Mrowka (2001). Identification of coupling direction: Application to cardiorespiratory interaction, Phys. Rev. E, 65 041909; M. Paluš and A. Stefanovska (in print). Direction of coupling from phases of interacting oscillators: An information-theoretic approach, Phys. Rev. E.

[3] J. Jamšek, A. Stefanovska, P.V.E. McClintock and I.A. Khovanov (in print). Time-phase bispectral analysis, Phys. Rev. E.

Classical and quantum mechanics
in SQUID and two confined hard disks

Takeo Kato

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

Chaos and its signature in quantum mechanics have been studied in various physical systems. I focus here on two physical systems; superconducting quantum interference devices (SQUID's) and two confined hard disks.

The SQUID system has recently attracted interest as a physical candidate of quantum computer (for a review, see the article by Makhlin, 2001). I first review fundamental features of this system and recent experimental studies of SQUIDs. After that, I show that this systems can also give an ideal stage for study of both classical conservative system and pure quantum system. As an example of classical dynamics, I show that in SQUID's there appears a deterministic diffusion, which can be categorized into two kinds; a anomalous diffusion and a normal diffusion (Tanimoto, 2002). By changing experimental parameters, one can also discuss quantum mechanics in SQUID's. As an example of it, I study the signature of chaos in quantum mechanics through spectral statistics (Kato, 2003).

Quantum chaos in billiard systems with one particle has been studied intensively, while the systems consisting of many particles have not been studied so much. As a simplest model including interacting particles, I study two hard disks confined by a circular hard wall. Classical dynamics shows dominantly a chaotic behavior, and spectral statistics of corresponding quantum system obeys the Wigner distribution. The avoid level crossings affect the pressure felt by the surrounding wall; there appears shoulders and dips in the pressure-volume plot. I conclude that this feature in pressures can be explained by the change in quantum correlation between two disks (Nakazono, 2003a, 2003b).

Makhlin Yu, Schön S and Shnirman A Phys. Mod. Phys. 73 357
Tanimoto K, Kato T, Nakamura K 2002 Phys. Rev. B 66 012507
Kato T, Tanimoto K and Nakamura K 2003 submitted to Phys. Rev. B
Nakazono N, Kato T and Nakamura K 2003a submitted to Prog. Theor. Phys. Suppl.
Nakazono N, Kato T and Nakamura K 2003b in preparation.

Randomness Evaluation and Hardware Implementation of Cellular Automaton-based Pseudo Random Number Generator

Song-Ju Kim

Chaos-based Cipher Chip Project, Presidential Research Fund,
Communications Research Laboratory, Incorporated Administrative Agency,
Tokyo 184-8795, Japan

We shall review the cellular automaton (CA) -based pseudorandom number generators (PRNGs), and evaluate the randomness of sequences generated by them using the statistical test suite of National Institute of Standards and Technology (NIST). We show that one of CA-based PRNGs has good randomness which is comparable to well-known good PRNGs such as SHA1, AES, and MUGI.

Because of its simple construction, CA is well known to be suitable for hardware implementation. We show that this CA-based PRNG is able to work up to the high clock frequency, and has 4.2Gbps speed (15K gates) in the case of Field-Programmable Gate Arrays (FPGA).


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S. J. Kim, K. Umeno, and A. Hasegawa, to be published in IEICE Technical Report [in Japanese].

Two Types of Anticipation in Sensory-Motor Coupling

Yoshihiro Miyake

Dept. of Computational Intelligence and Systems Science,
Tokyo Institute of Technology, Tokyo, Japan

The anticipatory timing control in sensory-motor coupling is indispensable to generate coordinative movement with dynamical environment, however its cognitive mechanism still remains obscure. In this study we used synchronization tapping task as a model system, and negative asynchrony phenomenon [1-4] where the tap onset precedes the stimulus onset was analyzed as an example of the anticipation. Especially, applying dual task method [5], the relationship between the anticipation mechanism and the higher brain function such as attention [6] and working memory [7] was investigated.

The results revealed two types of anticipatory timing control [8]. In the inter stimulus-onset interval (ISI) range of 450 to 1800ms, automatic anticipation that is not affected by attentional resources was observed and was based on feed forward process. In the 2400 to 3600ms range, the anticipation showed trade-off relationship in the allocation of attentional resources. Magnitude of synchronization error (SE) between tap onset and stimulus onset in this region was scaled by the ISI and the feed back process concerning ISI was suggested.

Furthermore, we used time-series analysis to clarify it in frequency response characteristics. As a result, it was shown that anticipatory behavior in sensory-motor coupling is composed of two different dynamics corresponding to the above two types of anticipatory timing control [9]. The former is characterized by the 1/f fluctuation between the power and the frequency, suggesting non-stationary process in unbounded variation [10]. The latter is characterized by the superimposition between white noise and the significant peak of periodic stimulus, suggesting stationary process in bounded variation.

Accordingly, anticipation dynamics in timing control was shown to be a dual processing between the attentional processing based on completeness and the embodied processing based on incompleteness. Not only psychophysical analysis but also some applications in the field of human-interface are mentioned in the lecture [11].

(1) Stevens L T 1886 Mind 11 393-404
(2) Fraisse P 1966 in Anticipation et comportement eds. J Requin (Paris: Centre National) pp233-257
(3) Mates J, Radil T, Mueler U and Poeppel E 1994 Journal of Cognitive NeuroScience 6 332-340
(4) Aschersleben G and Prinz W 1995 Perception and Psychophysics 57-3 305-317
(5) Baddeley A 1986 Working memory (Oxford: Oxford University Press)
(6) Kahnemann D 1973 Attention and efforts (Engelwood Cliffs, Prentice-Hall)
(7) Osaka N 2000 Brain and working memory (In Japanese) (Kyoto: Kyoto University Press)
(8) Miyake Y, Onishi Y and Poeppel E 2002 Transaction of SICE (In Japanese) 38 1114-1122
(9) Miyake Y, Komatsu T, Onishi Y and Poeppel E (in preparation)
(10) Aizawa Y 2000 Chaos, Soliton and Fractals 11 263-268
(11) Miyake Y (in press) Cognitive Processing

Crossover analyses between
anomalous diffusion and normal diffusion
-- Continuous-time random walk approach --

Syuji Miyazaki

Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics, Kyoto University, Kyoto, Japan

Two kinds of crossover phenomena between anomalous diffusion and normal diffusion are investigated. As the first topic, anomalous diffusion caused by modulational intermittency, which is also known as on-off intermittency, is studied on the basis of the continuous-time random walk (CTRW) approach. There exists a characteristic time scale $\tau$. For the time region $t \ll \tau$, anomalous diffusion is observed, which is followed by normal diffusion for $t \gg \tau$. Higher-order moments are analytically obtained by use of the saddle-point method, and it is found that they obey scaling relations that are reminiscent of extended self-similarity (ESS) and generalized extended self-similarity (GESS) found in turbulent behaviors. The results are compared with those obtained using the numerical inverse Laplace transform and from model simulations employing a coupled chaotic map. Good agreement between these results is obtained even for lower-order moments.

Anomalous diffusion found in fluid systems is studied as the second topic, inspired by experiments on soft-mode turbulence (SMT) by Tamura et al. and by numerical studies on oscillating convection flows by Sakaguchi. Diffusion constants and mean square displacements are analytically obtained based on the CTRW velocity model, and compared with those obtained based on the CTRW velocity model, and compared with those obtained from model simulations employing dissipative dynamics describing oscillating convection flows. Good agreement is also obtained.

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Neural and cardio-respiratory interactions during anaesthesia

Bojan Musizza$^{1,2}$, Peter V. E. McClintock$^3$, Milan Paluš$^4$,
Janko Petrovcic$^2$, Samo Ribaric$^5$ and Aneta Stefanovska$^{1,3}$

$^1$University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Trzaška 25, 1000 Ljubljana, Slovenia

$^{2}$Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
$^{3}$Lancaster University, Department of Physics, Lancaster LA1 4YB, UK
$^{4}$Academy of Sciences of the Czech Republic, Institute of Computer Science,
Pod vodárenskou vezí 2, 182 07 Prague 8, Czech Republic

$^{5}$University of Ljubljana, Faculty of Medicine,
Institute of Pathophysiology, Zaloška 4, Ljubljana, Slovenia

Little is known about the events that unfold during the temporary loss of consciousness corresponding to anaesthesia, so that there are still no reliable markers for depth of anaesthesia. Methods developed for studying interactions between nonlinear oscillators $[1,2,3]$ offer a promising way of furthering our understanding of the complex mechanisms that come in to play. It has been shown for example that the cardio-respiratory system passes through a sequence of different phase-synchronized states during deep anaesthesia $[4]$.

We now report results of the first simultaneous study of cardiac, respiratory and neural oscillations during anaesthesia in rats. We discuss the causalities of the interactions between these oscillations. This problem is approached from two different perspectives: first, we calculate the directionality of the couplings between the oscillators in question, using a method derived from information theory, i.e. we use a simple tool based on mutual information; and secondly, we try to approach the same problem by inferring the phase dynamics related to probe functions, again yielding information about the causal relationships.

The data analyzed indicate the presence of strong interactions between both the neural and respiratory oscillators, and also between cardio-respiratory oscillators. If the same events are at least partially reproducible in humans, this could then lead to the development of new markers for determining the depth of anaesthesia.


[1] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, A Universal Concept in Nonlinear Sciences, (Cambridge University Press, Cambridge 2001).

[2] Rosenblum, L. Cimponeiru, A. Bezerianos, A. Patzak, and R. Mrowka (2001). Identification of coupling direction: Application to cardiorespiratory interaction, Phys. Rev. E, 65 041909.

[3] M. Paluš and A. Stefanovska (in print). Direction of coupling from phases of interacting oscillators: An information-theoretic approach, Phys. Rev. E.

[4] A. Stefanovska, H. Haken, P.V.E. McClintock, M. Hozic, F. Bajrovic, and S. Ribaric (2000). Reversible transitions between synchronization states of the cardiorespiratory system, Phys. Rev. Lett, 85, 4831-4834.

Cardiorespiratory interactions

Milan Paluš$^1$, and Aneta Stefanovska$^{2}$

$^{1}$Academy of Sciences of the Czech Republic, Institute of Computer Science,
Pod vodárenskou vezí 2, 182 07 Prague 8, Czech Republic

$^2$University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Trzaška 25, 1000 Ljubljana, Slovenia

A directionality index based on conditional mutual information will be introduced and applied to the instantaneous phases of weakly coupled oscillators. Its abilities to distinguish unidirectional from bidirectional coupling, as well as to reveal and quantify asymmetry in bidirectional coupling, will be demonstrated using numerical examples of quasiperiodic, chaotic and noisy oscillators, as well as cardiorespiratory data.


[1] Paluš, M. & Stefanovska, A., (in print), Direction of coupling from phases of interacting oscillators: An information-theoretic approach, Phys. Rev. E.

[2] Stefanovska, Cardiorespiratory interactions, (2002) Nonlinear Phenom. Complex Syst., 5, 462-469.

Fidelity of quantum chaos

Tomaz Prosen

University of Ljubljana, Department of Physics
Jadranska 19, SI-1000 Ljubljana, Slovenia

Recently, we have witnessed a strong interest in the stability of quantum motion against small variations of the Hamiltonian characterized by fidelity. In this talk I will shortly review our theory on the behaviour of fidelity decay in different regimes characterized with several time-scales. It will be explained how the fidelity can be computed in terms of time-correlation function of the generator of perturbation. Interesting application of our ideas for the enhancement of the stability of quantum computation will be discussed in the end.


Prosen T and Znidaric M 2002 Stability of quantum motion and correlation decay, Journal of Physics A: Mathematical & General 35 1455-1481

Prosen T and Znidaric M 2001 Can quantum chaos enhance the stability of quantum computation?, Journal of Physics A: Mathematical & General 34 L681-L687

Prosen T 2002 General relation between quantum ergodicity and fidelity of quantum dynamics, Physical Review E 65 036208

Prosen T and Seligman T H 2002 Decoherence of spin echoes, Journal of Physics A: Mathematical & General 35 4707-4727

Znidaric M and Prosen T 2003 Fidelity and purity decay in weakly coupled composite systems, Journal of Physics A: Mathematical & General 36 2463-2481

Prosen T, Seligman T H and Znidaric M 2003 Evolution of entanglement under echo dynamics, Physical Review A 67 042112

Quantum fidelity decay of classically integrable dynamics with vanishing time averaged perturbation

Tomaz Prosen

University of Ljubljana, Department of Physics
Jadranska 19, SI-1000 Ljubljana, Slovenia

We discuss quantum fidelity decay (also known as the quantum Loschmidt echo) of classically regular dynamics, in particular for an important special case of vanishing time averaged perturbation operator, i.e. vanishing expectation values of the perturbation in the eigenbasis of unperturbed dynamics. A complete semiclassical picture of this situation is derived in which we show that quantum fidelity of individual coherent initial states exhibits three different regimes in time: (i) first it follows the corresponding classical fidelity up to time $t_1 \sim \hbar^{-1/2}$, (ii) then it freezes on a plateau of constant value which can be semiclassically computed, (iii) and after a time scale $t_2 \sim \min\{\hbar^{1/2}\delta^{-2},\hbar^{-1/2}\delta^{-1}\}$ it exhibits fast ballistic decay as $\exp(-{\rm const}\,\delta^4 t^2/\hbar)$. All the constants are computed in terms of classical dynamics for sufficiently small effective value of Planck constant. A similar picture is worked out also for general initial states, and specifically for random initial states, where $t_1 \sim 1$, and $t_2 \sim \hbar^0\delta^{-1}$. Theoretical results are verified by numerical experiments on the quantized integrable kicked top.


Prosen T and Znidaric M 2002 Quantum fidelity decay of classically integrable dynamics with vanishing time averaged perturbation, preprint 2003

Quantum chaos in mixed type Hamiltonian systems

Marko Robnik

Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia

We shall review the basic aspects of complete integrability and complete chaos (ergodicity) in classical Hamiltonian systems, as well as all the cases in between, the generic, mixed type systems, where KAM Theory is applicable, and shall illustrate it using the billiard model systems.

Then we shall proceed to the quantum chaos and its stationary properties, that is the structure and the morphology of the solutions of the underlying Schroedinger equation which in case of 2-dim billiards is just the 2-dim Helmholtz equation. We shall discuss the statistical properties of chaotic eigenfunctions, the statistical properties of the energy spectra, and show arguments and results in support of the so-called universality classes of spectral fluctuations, where in the fully chaotic case the Random Matrix Theory (RMT) is applicable.

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.

We shall mention the rich variety of applications in the domain of physics.

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Randomness of classical chaotic motion

Marko Robnik

Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia

I shall discuss some new universal aspects of diffusion in classical deterministic and chaotic dynamical systems, especially in Hamiltonian systems. First ergodic (fully chaotic) systems will be discussed, and then the mixed type systems with a typical KAM scenario. Some generalizations by treating the correlations will be presented. Finally, I shall explain the relevance of these studies in the context of problems in stationary quantum chaos, namely the structure of stationary eigenfunctions and the statistical properties of the energy spectra.

Robnik M, Dobnikar J, Rapisarda A, Prosen T and Petkovšek M 1997 New universal aspects of diffusion in strongly chaotic systems Journal of Physics A: Mathematical & General 30 L803-L813.
Prosen T and Robnik M, General Poissonian model of diffusion in chaotic components 1998 Journal of Physics A: Mathematical & General 31 L345-L353.
Robnik M, Prosen T and Dobnikar J 1999 Multi-component random model of diffusion in chaotic systems,Journal of Physics A: Mathematical & General 32 1147-1162.
Robnik M 1998 Topics in quantum chaos of generic systems, Nonlinear Phenomena in Complex Systems (Minsk) 1 No. 1-22.
Prosen T and Robnik M 1999 Intermediate $E(k,L)$ statistics in the regime of mixed classical dynamics, Journal of Physics A: Mathematical & General 32 1863-1873.
Malovrh J and Prosen T 2002 Spectral statistics of a system with sharply divided phase space, Journal of Physics A: Mathematical & General 35 2483-2490.

Local limit cycles bifurcations in polynomial systems

Valery Romanovski

Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia

Consider systems of the form

\frac{dx}{dt}=P_{n}(x,y), \ \ \
\end{displaymath} (1)

where $P_{n}(x,y), Q_{n}(x,y)$ are polynomials of degree $n$, $x$ and $y$ are real unknown functions, and suppose that the coefficients of the polynomials $P_n,Q_n$ are from a parameter space ${\cal E}$. In the case, when the origin of (1) is a non-degenerate center or focus, a limit cycle bifurcates from the origin when the linearized system (1) changes its stability. This is the well-known Andronov-Hopf bifurcation. The limit cycles bifurcations which depend on nonlinear terms of system (1) (sometimes such bifurcations are called degenerate Andronov-Hopf bifurcations) are much less investigated, but there is a method for their study suggested by Bautin.

Following Bautin we say that the singular point $(x_0,y_0)$ of the system $E_{0}\in {\cal E}$ has cyclicity $k$ with respect to ${\cal E}$ if and only if any perturbation of $E_0$ in ${\cal E}$ has at most $k$ limit cycles in a neighborhood of $(x_0,y_0)$ and $k$ is the minimal number with this property. The problem of cyclicity is often called the local 16th Hilbert problem.

We present a method to investigate the cyclicity problem for polynomial systems (1). The method, which we use, is based on the algorithms of computational algebra. In particular, we show that the problem can be reduced to the algebraic problem of computing a basis of the ideal generated by the coefficients of the Poincaré map, and demonstrate how to find such a basis for some systems of the form (1), resolving, therefore, the cyclicity problem for this these systems.

Bautin N N 1952 Mat. Sb. 30 396-413 (in Russian); 1962 Amer. Math. Soc. Transl. Ser. 1 5 396-413.
Romanovski V G Rauh A 1998 Dynamic Systems and Applications 7, Issue 4, 529-552.
Roussarie R 1998 Bifurcation of planar vector fields and Hilbert's sixteenth problem Boston, Basel, Berlin : Birkhäuser.

Time-reversible Systems of Differential Equations

Valery Romanovski

Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia

Consider a system of differential equations

\frac{d\bf {x}}{dt}=P({\bf {x}}),

where ${\bf {x}}=(x_1,\dots,x_n)^T$, $P({\bf {x}})=
(P_1(x_1,\dots,x_n),\dots,P_n(x_1,\dots,x_n) )^T$ and $P_i$ are polynomials in $x_1,\dots,x_n$ with real or complex coefficients. Denote by $a$ the ordered vector of the coefficients of the polynomials $P_i({\bf {x}})$. Let $Q$ be a group of linear transformations of the phase space $E^n$ (usually $E= $ $\mathbb{R}$ or $\mathbb{C}$),

{\bf {y}}=q({\bf {x}}).

The above substitution transforms (1) into the system

\dot {\bf {y}}=\tilde P({\bf {y}}).

We denote by $b$ the vector of the coefficients of $\tilde P({\bf {y}})$. It is easily seen that every coefficient $b_i$ is a linear function of $a$. We will write this as


This relation defines the group $L_Q$ homomorphic to $Q$ or, in another words, defines a representation of the group $Q$.

Let $G$ be a matrix group acting on $u_1,\dots,u_n$. A polynomial $f({\bf {u}})\in k[{\bf {u}}]$ (where $k$ is any field) is called an invariant of $G$ if $f({\bf {u}})=f(A\cdot {\bf {u}})$ for any matrix $A$ from $G$.

We consider the case when $n=2$ and the group $Q$ is the group of rotation of the phase plane. We give an algorithm to compute a finite basis of invariants $J(a)$ of the group $L_Q$ and show that these invariants determine the set of all time-reversible systems within a given polynomial family (1). Moreover, they determine also the number of axes of symmetry of the phase picture of trajectories of system (1) and their location.

Jarrah A, Laubenbacher R, Romanovski V G 2003 J. Symb. Comput. 35 577-89
Sibirsky K S 1982 An introduction to the algebraic theory of invariants of differential equations, Shtiintsa, Kishinev, (Russian, English translation: Manchester Univ. Press, New York, 1988).

Discrete Schrödinger Theory and
its Orthogonal Function Systems

Andreas Ruffing

Department of Mathematics, Munich University of Technology, Munich, Germany

Basic discrete analogs of Schrödinger's equation are investigated on a so-called $q$-linear grid or basic linear grid. A ladder operator formalism for a discrete harmonic oscillator analog is developed with a representation in the weighted Hilbert space $l^{2}(\bf Z)$ over the $q$-linear grid. The moment problem for the corresponding modified discrete $q$-Hermite polynomials of type II is revised. Conditions on the existence of a ladder operator formalism in connection with the considered moment problem are developed. The results are evaluated with respect to an application for purposes of discrete Schrödinger theory.


R. N. Álvarez, D. Bonatsos and Yu. F. Smirnov, $q$-Deformed vibron model for diatomic molecules, Physical Review A. 50 (1994) 1088-1095.
R. Askey, S. K. Suslov, The $q$-harmonic oscillator and the Al-Salam and Carlitz polynomials, Letters in Mathematical Physics 29 (1993) No. 2, 123-132.
N. M. Atakishiyev, S. K. Suslov, A realization of the $q$-harmonic oscillator, Theoret. and Math. Phys. 87 (1991) No. 1, 442-444.
C. Berg, M.E.H. Ismail, Q-Hermite polynomials and classical orthogonal polynomials, Can. J. Math. 48 (1996) 43-63.
C. Berg, A. Ruffing, Generalized $q$-Hermite Polynomials, Communications in Mathematical Physics 223 (2001) 1, 29-46.
K. Ey, A. Ruffing: Fixing the Ladder Operators on two Types of Hermite Functions, Dynamic Systems and Applications Vol. 12 (2003), 115-130.
A. Lorek, A. Ruffing, J. Wess: A Q Deformation of the Harmonic Oscillator, Zeitschrift für Physik C74 (1997), 369-378.
A. Ruffing, J. Lorenz, K. Ziegler, Difference Ladder Operators for a Harmonic Schrödinger Oscillator Using Unitary Linear Lattices, Journal of Computational and Applied Mathematics 153 (2003), 395-410.

Existence of unique invariant measure in complex
phase space and its manifestation in quantum mechanics

Akira Shudo

Department of Physics, Tokyo Metropolitan University,
Tokyo Japan

Recent studies on complex dynamical systems in more than one dimension have revealed there exists a unique ergodic measure $\mu$ in complex phase space which is (weakly) hyperbolic and gives maximal entropy of the dynamics. On the basis of convergency theorem of currents, a basic theorem in complex dynamical systems, this fact was first proved in case of the Hénon family (Bedford $\&$ Smillie 1991a, 1991b, 1992, Forn${\rm\ae}$ss $\&$ Sibony 1992), and recently extended to more generic settings (Dujardin 2003).

For hyperbolic systems, it is known that the support of an ergodic measure $\mu$ thus obtained is exactly the Julia set $J$ of the dynamical system. The orbits showing chaos in complex plane are limited to the ones on the Julia set $J$, and so one can say that chaos appears only on the Julia set. A remarkable fact is that a unique invariant measure $\mu$ exists even in mixed systems. In real phase space, mixed system have infinitely many ergodic components such as quasiperiodic or chaotic orbits and no specific measure attaining maximal entropy can be found in general. On the contrary, the above claim suggests that one can access arbitrary regions using the complex space in spite of the lack of ergodicity in real phase space.

In this talk, we will give sufficient conditions to realize such a situation. That is, assuming that supp$\mu$ = $J$, together with the fact that the 4-dimensional volume of filled-in Julia set $K$, which is defined as a set of bounded orbits in complex phase space, is zero, we can show that for arbitrary two points, each of which is confined on different KAM circles and separated on real plane, there always exist complex orbits connecting them.

This situation is suggestive in the semiclassical description of quantum mechanics because quantum tunneling exactly realizes such (real) classically forbidden processes, and it was indeed proved that the orbits on the Julia set are responsible for quantum tunneling in the presence of chaos(Shudo,Ishii $\&$ Ikeda 2002, 2003a). Here we will further discuss that it would be natural to formulate the semiclassical trace formula and also quantum ergodicity problem in terms of the unique invariant measure thus constructed, since unstable periodic orbits are densely distributed on it(Bedford $\&$ Smillie 1993). It appears a realizable project since Stokes phenomena in chaotic systems are now controllable(Shudo $\&$ Ikeda 2003b).

Bedford E and Smillie J 1991a Invent. Math. 103 69-99.
Bedford E and Smillie J 1991b J. Amer. Math. Soc. 4 657-679.
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Bedford E and Smillie J 1993 Invent. Math. 112 77-125;
Forn${\rm\ae}$ss J.E. and Sibony N, 1992 Duke Math. J. 65 345-380.
Dujardin R 2003 Hénon-like mappings in ${\bf C}^2$
Shudo A, Ishii Y and Ikeda K S 2002 J. Phys. A 35 L224-L231.
Shudo A, Ishii Y and Ikeda K S 2003a Julia set and chaotic tunneling, preprint.
Shudo A, Ishii Y and Ikeda K S 2003b Stokes geometry for quantized Hénon mapping, preprint.

Inside-outside duality and isospectrality of planar billiards

Akira Shudo

Department of Physics, Tokyo Metropolitan University,
Tokyo Japan

A famous question ``Can you hear the shape of a drum?" posed by Kac(Kac 1966) was negatively solved several years ago(Gordon, Webb $\&$ Wolpert 1992), that is, they found some concrete examples of non-congruent but isospectral pairs of planar billiard domains. In this talk we shall discuss some novel aspects of this issue:

1. Isospectrality of the planar domains which are constructed by successive unfolding of a fundamental building block is discussed in relation to iso-length spectrality of the corresponding domains. Although an explicit and exact trace formula such as Poisson's summation formula or Selberg's trace formula is not known to exist for such planar domains, equivalence between isospectrality and iso-length spectrality in a certain setting can be proved by employing the matrix representation of transplantation of eigenfunctions(Okada $\&$ Shudo 2001). As an application of this fact, transplantable pairs of domains, which are all isospectral pair of planar domains and therefore counter examples of Kac's question are numerically enumerated and it is found at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to Sunada(Sunada 1985).

2. The quantum billiard problem, that is the Dirichlet problem for the Helmholtz equation, can be rewritten as a Fredholm integral equation of the second kind and the eigenenergies can be specified as the zeros of the Fredholm determinant on the real axis. However the Fredholm determinant also has complex zeros corresponding to the resonances when the billiard table is regarded as a scatterer against the exterior wave function. More precisely, the Fredholm determinant admits factorization into the interior and exterior contributions, where the former has zeros at interior Dirichlet eigenenergies and the latter at resonances of the Neumann scattering(Tasaki, Harayama $\&$ Shudo 1997). This fact naturally leads us, instead of the Kac's original one, a new question ``can one determine the shape of billiard table through the interior eigenenergies and exterior resonances, i.e., all zeros of the Fredholm determinant?'' We here discuss the possibility to distinguish isospectral pairs using information of outside scattering problem(Okada, Shudo, Harayama $\&$ Tasaki, 2003).

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Okada Y, Shudo A, Harayama T and Tasaki S, 2003 preprint.

How useful is the variability in cardiovascular dynamics?

Aneta Stefanovska

University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Trzaška 25, 1000 Ljubljana, Slovenia

The cardiovascular system (CVS) continuously adjusts the cardiac output to match the need of the whole organism. The cardiac output is a product of the stroke volume, i.e. the amount of blood expelled in each cycle by the heart, and the heart rate. The stroke volume and the heart rate both vary in time.

The variations of the heart rate are known as heart rate variability (HRV). It consists of the frequency of cardiac oscillations at each instant in time. The stroke volume contains information about the corresponding amplitude.

Why should the frequency vary? Several physiological processes are involved in the regulation of cardiac output, acting on different time-scales. The physiological processes underlying them are of different characters (chemical, mechanical, electrical, or a combination), and are spatially distributed. In the blood distribution system, oscillations with frequencies spanning from 0.01 Hz to 1 Hz are involved. The cardiac frequency then varies as a result of couplings with the other oscillatory processes.

The CVS is a very complex system. The oscillators that govern its dynamics are coupled and as a consequence, not just the heart rate, but all the characteristic frequencies vary in time. In practice the time of observation of CVS dynamics is inevitably limited. Consequently, it is difficult to distinguish clearly between the deterministic and the stochastic components.

The variability of the cardiac frequency is of paramount importance and it demonstrates the adaptability of the CVS. We discuss the association between changes in HRV and cardiovascular disease, and how specific diseases can be characterized by modifications in specific oscillatory component(s) and their couplings.

Introduction to Chaotic Codes for Digital Communications

Ken Umeno

Chaotic Cipher Chip Project,
Communications Research Laboratory, Tokyo, Japan

We shall review the recent developments of spread spectrum communication systems with chaotic codes, achievements of the optimal CDMA systems with chaotic codes, and chaotic cipher(symmetric and asymmetric), as well as basic theory of exactly solvable chaos, Lebesgue spectrum analysis for chaotic sequence and recently proven theorems about digital chaos (chaotic mapping over module \(2^n\)) .

We discuss various scientific issues about chaotic codes and their practical applications which link classical ergodic theory with modern digital signal processing technology.

Umeno K 1997 Phys. Rev. E 55 5280
Umeno K 1998 Phys. Rev. E 58 2644
Umeno K and Kitayama K 1999 Electron. Lett. 25 1999
Umeno K 2000 Jpn. J. Appl. Phys. 39A 1442
Umeno K 2001 J. Nonlnear Analaysis. 47 5753
Umeno K and Yamaguchi A 2002 IEICE Trans. Fund. E85-A 849

Fidelity of Perturbed Classical Evolution

Gregor Veble$^{1,2,3}$, Giuliano Benenti$^{3}$, Giulio Casati$^{3}$

$^{1}$ Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia
$^{2}$ Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
$^{3}$Center for Nonlinear and Complex Systems, Università dell' Insubria, Como, Italy

The stability of the evolution of Hamiltonian systems under the influence of a static perturbation attracted substantial attention recently, especially in the field of quantum computing. The main question is, how does the type of dynamics affect the divergence of the evolution of two slightly different systems. In our work we focused on classical dynamics. We compared the Liouville evolution of classical phase space densities in the original and the perturbed system, starting from the same initial condition. The overlap of the two evolved phase space densities is what is called classical fidelity.

We studied both chaotic and integrable systems. In chaotic systems the initial decay of the classical fidelity is exponential and given by the Lyapunov exponent. In our work, however, we focused on the asymptotic behaviour. It turns out that the asymptotic decay in chaotic systems can be related to the decay of correlations. This means that asymptotically the decay of fidelity can be either algebraic or exponential, where in the latter case the decay rate is not necessarily related to the Lyapunov exponent.

In the integrable case the initial decay of fidelity turns out to be mildly surprising as it can sometimes be even faster than exponential, while in the other case the initial decay is algebraic in nature. In our work we analytically showed that the type of decay depends on the shape of the perturbation but not its size. The fast type of decay can be related to the change of the frequency and therefore results in a ballistic divergence of the evolutions in the two systems, while the algebraic decay can be related to the perturbation of the shape of the tori.

Benenti G, Casati G and Veble G 2003 Phys. Rev. E 67 055202(R)
Benenti G, Casati G and Veble G 2003 Decay of the classical Loschmidt echo in integrable systems preprint nlin.SI/0304032

Cardiovascular variability and ageing

Katja Azman-Juvan$^1$, Robert Mavri$^2$, Dušan Štajer$^{1,3}$, Aneta Stefanovska$^2$ and Gregor Vidmar$^2$

$^1$University Clinical Centre, Ljubljana, Slovenia

$^2$University of Ljubljana, Faculty of Electrical Engineering,
Group of Nonlinear Dynamics and Synergetics, Ljubljana, Slovenia

$^3$University of Ljubljana, Medical Faculty, Ljubljana, Slovenia

We discuss changes with ageing in cardiovascular oscillations and their couplings. Analyses of heart rate variability, the propagation of cardiac oscillations within the system, cardio-respiratory couplings, and the direction of coupling, will be presented for a group of healthy subjects of all ages between 14 and 84 years.

Dynamical-Morphological Property of Adhesive Tape in Peeling

Yoshihiro Yamazaki

Department of Physics, Waseda University Japan

We experimentally investigated the dynamical behavior of an adhesive tape in peeling at a constant speed with the emphasis on the emergence of slow and fast peeling motion. Especially focusing our attention on the whole pattern formed by the peeled adhesive on the tape, the dynamical-morphological phase diagram as a function of peel speed and spring constant was obtained. The spatio-temporal patterns turn out to be classified into four types: low-speed pattern, high-speed pattern, oscillatory pattern, and spatio-temporal intermittent pattern. We expect that we can understand the the dynamical property in peeling process adhesive tape by investigating the formation process of the spatio-temporal patterns.

Yamazaki Y and Toda A
Gay C and Leibler L 1999 Physics Today 52 48
Urahama Y 1989 J. Adhesion 31 47
Persson B N J 1998 Sliding Friction (Springer-Verlag, Berlin)

Anomalous Diffusion and Dynamical Localization in Polygonal Billiards

Marko Znidaric

Faculty of Mathematics and Physics,
University of Ljubljana, Ljubljana, Slovenia

As the Schrödinger equation is linear, it cannot posses exponential instability in time as do the classical equations of motion. Quantum time evolution is dynamically stable and all Lyapunov exponents are strictly zero. Still, the quantum evolution mimics classical exponential instability up to so-called log-time $\log{1/\hbar}$. In some quantum systems there is another time scale, usually larger than log-time, connected with the phenomena of dynamical localization. This is purely quantum interference effect and results in a quantum suppression of classical diffusion. For times smaller than this localization time, quantum evolution follows the classical diffusion. Afterwards, the quantum system ``notices'' its discrete spectrum and the diffusion stops, resulting in a localized state. Since the localization time is usually much larger than the log-time, exponential instability may not be relevant for dynamical localization, although all known examples of dynamical localization, from kicked rotors and maps to quantum billiards, take place in classically chaotic systems. In view of that, it is important to see whether exponential sensitivity is at all necessary for dynamical localization. We will study numerically classical and quantum dynamics of a piecewise parabolic area preserving map on a cylinder which emerges from the bounce map of elongated triangular billiards. The classical map has no exponential sensitivity and exhibits anomalous diffusion. Quantization of the same map results in a system with dynamical localization and pure point spectrum.

Prosen T and Znidaric M 2001 Phys. Rev. Lett. 87 114101
Casati G and Prosen T 1999 Phys. Rev. Lett. 83 4729
Casati G and Prosen T 2000 Phys. Rev. Lett. 85 4261

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Gregor Veble 2003-06-24