INSTITUT D’ ÉTUDES SCIENTIFIQUES DE CARGESE 20130 Cargèse, Corse (France) http://cargese.univ-corse.fr

LABORATOIRE de PHYSIQUE THÉORIQUE de la MATIERE CONDENSÉE

NATO Advanced Study Institute International Summer School on

Chaotic Dynamics and Transport in Classical and Quantum Systems Cargèse (Corsica-France) - August 18-30 2003

L’I.E.S.C. est affilié au Centre National de la Recherche Scientifique et aux Universités de Corse et Nice-Sophia Antipolis. Il est subventionné par le MENESR, le CNRS et la CTC. Directeur : Élisabeth Dubois-Violette, tél. : (33) 1 69 15 61 01, [email protected] Institut : tél. : (33) 4 95 26 80 40 ou (33) 4 95 26 80 48, fax : (33) 4 95 26 80 45

NATO Advanced Study Institute Summer School on

Chaotic Dynamics and Transport in Classical and Quantum Systems

Scientific Organization : P. Collet (École Polytechnique, Paris) M. Courbage (Université Paris 7–Denis Diderot) S. Métens (Université Paris 7-Denis Diderot) A. Neishtadt (Space Research Institute, Moscow) G. Zaslavsky (New-York University) Director and Co-Director of the Nato-ASI : G. Zaslavsky and A. Neishtadt Coordination : M. Courbage Secrétariat : Evelyne Authier LPTMC - Université Paris 7 Case 7020 – 2, Place Jussieu 75251 Paris Cedex 05 Tel : (+33) 1 44 27 48 73 Fax : (+33) 1 46 33 94 01 e-mail : [email protected] The Summer-School is supported by : NATO-ASI 97914 European Science Foundation (Programm PRODYN) Université Paris 7 – LPTMC – MSC Ecole Polytechnique de Paris – Centre de Physique Théorique Collectivité Territoriale de Corse

COLLECTIVITE TERRITORIALE de Corse

Chaotic Dynamics and Transport Cargese – August 18 - 23 2003

8:30 - 9:20 9:30 -10:20

Monday 18 Registration 9:15 Opening Collet

10:30 -11:00 Coffee 11:00 -11:50 Laskar

Tuesday 19 Collet

Wednesday 20 Zaslavsky

Thurday 21 Courbage

Friday 22 Young

Gallavotti

Laskar

Gallavotti

Eckmann

Coffee Reichl

Coffee Neishtadt

Coffee Davidson

Coffee Shlesinger

LUNCH

and

Free

12:15 16:45-17:00 Coffee 17:00-17:50 Solomon

Coffee Ott

Coffee Rom Kedar

Coffee Reichl

Coffee Afraïmovich

18:15-19:05 Davidson

Bardou

Posters

Family

Leoncini Vasiliev Discussion

19:10-19:35 19:40-20:05 20 :15

Reception

Kaminski

Iomin Belyi Dinner

Chaotic Dynamics and Transport Cargese – August 23 – 30 2003

8:30 - 9:20

Monday 25 Afraïmovich

Tuesday 26 Zaslavsky

Wednesday 27 Gaspard

Thurday 28 Vulpiani

Friday 29 Strunz

9:30 -10:20

Young

Young

Courbage

Chaté

Zaks

Coffee Zeitlin

Coffee Zeitlin

Coffee Politi

Coffee Ciraolo

LUNCH

and

Free

Free Time

Coffee Strunz

Coffee POSTERS

10:30 -11:00 Coffee 11:00 -11:50 Neishtadt 12:15 16:45-17:00 Coffee 17:00-17:50 Gaspard

Coffee Eckmann

18:15-19:05 Vulpiani

Prants

Combescure

19:10-19:35 19:40-20:05 Discussion 20 :15

Vaienti

Discussion Dinner

NATO Advanced Study Institute International Summer School on

Chaotic Dynamics and Transport in Classical and Quantum Systems Cargèse (Corsica-France) - August 18-30 2003

List of Speakers Valentin Afraimovich Complexity, fractal dimensions and topological entropy in dynamical systems. Francois Bardou Levy flights, from atoms to molecules Viatcheslav Belyi Classical and Quantum Nonlocal Kinetic Equations for Plasma Cristel Chandre Time-frequency analysis of chaotic systems

Hugues Chaté Long-range order and effective temperature out of equilibrium Guido Ciraolo Control of chaotic diffusion in Hamiltonian dynamics Pierre Collet An ergodic theory refresher I- General concept and result II- Application to differentiable dynamics Monique COMBESCURE Méthodes semiclassiques rigoureuses pour le "chaos quantique" Maurice Courbage I; Basics of spectral theory of ergodic dynamical systems II. Decay of correlations and weak mixing of some area preserving maps with zero entropy Nir Davidson Ultra-cold atoms trapped inside atom-otpics billiards : dynamics and spectroscopy meet Jean-Pierre Eckmann I- Liapunov Multipliers and Decay of Correlations in Dynamical Systems, P. Collet and J.-P. Eckmann II- New methods for studying large networks - (work with E. Moses and D. Sergi) Fereydoon Family CONTROL OF CURRENT REVERSAL AND SEPARATION OF PARTICLES IN INERTIA RATCHETS Govanni Gallavotti I - SRB distributions for simple Anosov maps II - SRB distributions for chains of Anosov maps and space time chaos Pierre Gaspard Chaos, scattering and nonequilibrium statistical mechanics Alexander Iomin Quantum breaking time for chaotic systems with phase space structures Brunon Kaminski On the directional entropy of Z^2-actions on a Lebesgue probability space Jacques Laskar Frequency Map Analysis. Application to the study of chaotic dynamics in the Solar System and Particle AcceleratorI, II Xavier Leoncini Anomalous transport in two dimensional plasma turbulence Anatoly Neishtadt Destruction of adiabatic invariance and transport phenomena in systems with slow and fast motions-I, II Edward Ott The Onset of Coherent Behavior in Globally Coupled Systems Antonio Politi Heat transport in nonlinear lattices Serguei PRANTS Hamiltonian chaos in cavity quantum electrodynamics Linda Reichl Quantum Chaos-I -II Vered Rom-Kedar

1. Soft billiard potentials. 2. Mixing and transport in fluid flows - geometrical approach. M. Shlesinger Sub and Supra Diffusion Including Stretched Times and Divergent Time Scales Tom Solomon The effects of Lagrangian chaos on multiphase processes and network dynamics

Walter Strunz I- Decoherence in quantum physics II- Stochastic approach to open quantum system dynamic Sandro VAIENTI Mesures invariantes pour des applications multidimensionelles non-uniformement dilatanttes Alexei Vasiliev Geometric and statistical properties induced by separatrix crossings Angelo Vulpiani TRANSPORT, DIFFUSION and FRONT PROPAGATION- I, II L.S. Young Strange attractors and periodically driven chaos- I, II, III

Michael Zaks Anomalous transport in steady plane flows of viscous fluids Georges Zaslavsky FRACTIONAL KINETICS IN DYNAMICAL SYSTEMS 1. FRACTIONAL KINETICS 2. PSEUDOCHAOS 3. COMPLEXITY AND ENTROPY FUNCTIONS (new approach) Vladimir Zeitlin Equatorial geophysical fluid dynamics and transport-I, II

ABSTRACTS

Valentin Afraimovich Complexity, fractal dimensions and topological entropy in dynamical systems.

The lectures are based on the notion of epsilon-complexity. The complexity, as a global characteristic of instability of orbits in dynamical systems, is closely related to dynamical chaos, and some of its features can be expressed in the complexity terms. The course contains 2 lectures: the first one "Dynamical chaos in terms of the epsilon-complexity" and the second one "Spatio-temporal characteristics of complexity". Francois Bardou Levy flights, from atoms to molecules

We will present recent theoretical advances on Levy flights in three simple microscopic systems, laser cooled atoms, tunnel junctions and RNA mutants. In these systems, non linear, or even singular, dependences of relevant physical variables on fluctuating parameters create long tails in the distribution of the physical quantities. These tails generate anomalous scaling behaviours through the inapplicability of the law of large numbers. For example, in laser cooling of atoms, a momentum random walk in the vicinity of a trap creates a renewal process with an infinite, mean waiting time that presents a non ergodic dynamics from which the cooling properties can be inferred. Maxim Barkov

Model of Ejection of Matter from Dense Stellar Cluster and Chaotic Motion of Gravitating Shells

It is shown that during the motion of two initially gravitationally bound spherical shells, consisting of point particles moving along ballistic trajectories, one of the shell may be expelled to infinity at subrelativistic speed Vexp ≤ 0.25 C. The problem is solved in Newtonian gravity. Motion of two intersecting shells in the case when they do not runaway shows a chaotic behaviour. We hope that this simple toy model can give nevertheless a qualitative idea on the nature of the mechanism of matter outbursts from the dense stellar clusters. Viatcheslav Belyi Classical and Quantum Nonlocal Kinetic Equations for Plasma A nonlinear kinetic equation, which is a generalization of the Balescu-Lenard equation, is derived for a spatially nonuniform multicomponent polarizable plasma. Explicit expressions of the collision integral and of the nonequilibrium pair correlation function are given in an approximation which takes into account the effects of spatial and temporal nonlocality as well as of the polarization. Balance equations for the momentum and energy densities are calculated in first order in nonlocality and potential contibutions to the fluxes due to polarization are obtained.--On the basic of the resolvent of the Hartree-Fock equation, a general expression for the pair correlation function of a quantum plasma in a non-Markovian approximation is derived taking into account polarization and exchange interaction of particles. The dielectric function including complete contribution of exchange interaction is introduced. In the Hubbard approximation, the collision integral and the internal energy are found in which the exchange interaction is taken into account in the particle distribution function, in the amplitude of scattering, and in the dielectric function. Cristel Chandre Time-frequency analysis of chaotic systems We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time-frequency decomposition of a trajectory using wavelets. The ridges of the time-frequency landscape of a trajectory, also called instantaneous frequencies, enable us to analyze the phase space structures. In particular, this method detects resonance trappings and transitions and allows a characterization of the notion of weak and strong chaos. We illustrate the method with the trajectories of the standard map and the hydrogen atom in crossed magnetic and elliptically polarized microwave fields. Hugues Chaté Long-range order and effective temperature out of equilibrium

Guido Ciraolo Control of chaotic diffusion in Hamiltonian dynamics With the aid of an original reformulation of the KAM theory, it is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose explicit form can be explicitly computed. In particular, it is shown that it is possible to control (reduce) the chaotic diffusion in the phase space of a 1.5 degrees of freedom Hamiltonian which models the diffusion of charged test particles in ``turbulent'' electric fields across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control anomalous transport in tokamaks is conceivable.

STELIANA CODREANU Synchronization of spatiotemporal nonlinear dynamical systems by an active control The approach we present in this work examines the synchronization of unidirectionally coupled nonlinear partial differential equations (PDEs), by an active control. It is a generalization of the method used by us to synchronize chaotic systems, described by one-, or twodimensional maps. The considered pair of PDEs are Fisher Kolmogorov's equations,which describe the dynamics of a field subjected to a diffusive transport and to a logistic growth, the synchronization of which we have studied both analitically and numerically. Pierre Collet An ergodic theory refresher I- General concepts and results II- Application to differentiable dynamics Pieter Collins Hyperbolicity and homoclinic orbits Symbolic dynamics for a surface diffeomorphism can be obtained from a knowledge of the homoclinic/heteroclinic orbits of the system, and yields a lower bound for the topological entropy. An important question is whether this computed entropy bound is optimal, and whether it can be realised. We show that under certain conditions, we can construct a uniformly hyperbolic diffeomorphism realising the entropy bound. In other cases, there is no entropy-minimising diffeomorphism, but the entropy bound can still be shown to be optimal by carefully constructing diffeomorphisms realising the entropy bound arbitrarily closely. Monique COMBESCURE Méthodes semiclassiques rigoureuses pour le "chaos quantique" Mini-course on semiclassical analysis Analyse semi-classique, états cohérents, théorie ergodique et un soupçon de "chaos quantique'': I propose to introduce, in a way as "pedestrian'' as possible, the so-called Semiclassical Analysis, as a passage, when the Planck constant can be considered small, between the formalisms of Classical and Quantum Mechanics. In this respect, I shall give an introduction to the "microlocal analysis'' (in phase space), of the symplectic transformations and their corresponding Operators in the quantum world : the unitary metaplectic group; of then semiclassical propagation of coherent states, of the Egorov theorem for the semiclassical evolution of quantum observables, and of the Weyl and Wigner transformations. In a second part, after a rapid summary of the foundations of classical Ergodic Theory, I will present important results of so-called "Quantum Chaos'', notably mathematical aspects of the Schnirelman Theorem, and of the Gutzwiller (or Balian-Bloch) semiclassical Trace Formulas.

Maurice Courbage I; Basics of spectral theory of ergodic dynamical systems Relationto mixing, weak mixing, correlations decay and K-systems II. Applications to some area preserving maps and billiards with zero entropyand to transport Nir Davidson Ultra-cold atoms trapped inside atom-otpics billiards : dynamics and spectroscopy meet TBA

Jean-Pierre Eckmann Liapunov Multipliers and Decay of Correlations in Dynamical Systems, P. Collet and J.-P. Eckmann The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in $C^1$, the essential decorrelation rate is never faster than what is dictated by the smallest unstable Liapunov multiplier. New methods for studying large networks - (work with E. Moses and D. Sergi) In the last few years many studies on large networks (WWW, proteins, etc) have been performed. In this talk, I want to explain the motivations for our own studies, which focus on local properties which we want to capture with geometric notions. Massimiliano ESPOSITO Quantum master equation for a system influencing its environment We derive a new perturbative quantum master equation for the reduced density matrix of a system interacting with an environment (with a dense spectrum of energy levels). The total system energy (system plus environment) is constant and finite. This equation takes into account the finite energy effects of the environment due to the total energy conservation. This equation is more general than the common perturbative equations used for describing a system in interaction with an environment (like the Redfield equation or the Cohen-Tannoudji one) because these last equations can be deduced from it in the limit of an infinitely large environment. We apply numerically this equation to the spin-GORM model. This model represents the interaction of a two-level system with an environment described by random matrices. We compare our equation with the exact von Neumann equation of the total system and show its superiority compared to the Redfield equation (in the Markovian and non-Markovian cases). Fereydoon Family CONTROL OF CURRENT REVERSAL AND SEPARATION OF PARTICLES IN INERTIA RATCHETS We have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of both the amplitude and the frequency of an external driving force. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. Control processes employing small perturbations on the frequency and the amplitude of the external force may be designed in view of the intermixed fractal nature of the domains of attraction of the mean velocity attractors. The range where each control parameter is capable to reverse the current is determined. The influence of the mass of the particle is also considered in order to design control techniques capable of separating particles of different masses. Stefano Galatolo Weak Chaos and information We consider the algorithmic information (also called Kolmogorov complexity) contained in symbolic orbits of Dynamical systems. This quantity is strongly related to the entropy of the system and to Lyapunov exponents. By the work of Brudno in a positive entropy dynamical system the information increases linearly with the time and is proportional to the entropy. When a system has zero entropy the information increaeses less than linearly.We consider zero entropy systems and show that the asymptotic behavior of this information increasing caracterizes the system under consideration. Moreover there are quantitative relations with the initial condition sensitivity of the system and with the Hausdorff dimension of the space. We also show some examples of numerical estimation of the information behavior by the use of compression algorithms. Govanni Gallavotti I - SRB distributions for simple Anosov maps II - SRB distributions for chains of Anosov maps and space time chaos Pierre Gaspard Chaos, scattering and nonequilibrium statistical mechanics In these two lectures, an overview will be given of recent works on transport processes in classical and quantum dynamical systems. The concept of hydrodynamic modes will be developed in order to describe the relaxation toward the thermodynamic equilibrium in spatially extended systems. The relaxation rate of the hydrodynamic modes can be understood in terms of Pollicott-Ruelle resonances in low-dimensional chaotic classical systems. The hydrodynamic modes of such systems are shown to present, in phase space, a fractal structure with a dimension related to the wavenumber of the mode, the transport coefficient, and the Lyapunov exponent. In the small-wavenumber limit, the hydrodynamic modes result into nonequilibrium steady states which are singular. The entropy production is shown to be tightly related to this singular character.

Vasiliy Govorukhin Numerical results on CABC-flows Results of numerical research of several three-dimensional compressible flows will be submitted. All these flows belong to a CABCclass which is a compressible analog of the ABC-flow [1,2]. The role of compressibility and symmetry in chaos of liquid particles movements will be discussed. The analysis of influence of compressed and incompressible perturbations on occurrence of chaos and stochastic webs will be carried out. [1] A. Morgulis, V.I. Yudovich, and G.M. Zaslavsky, Compressible helical flow Commun. Pure and Applied Math. XLVIII, 571-582 (1995). Seiichiro HONJO Breakup of Arnold web and global diffusion The diffusion properties of Hamiltonian dynamicalsystems are investigated in the frequency space. Structure of Arnold web and its partially breakup depending on parameters are visualized by long time calculation of 4-dimensional symplectic map. It is clearly shown that global diffusion are mainly achieved by diffusion across the overlapped higher-order resonances rather than diffusion along the lower-order resonances. Alexander Iomin Quantum breaking time for chaotic systems with phase space structures Quantized procedure breaks the applicability of semiclassical approximation with a corresponding breaking time t_h due to the uncertainty principle. A typical dynamical system is not ergodic. Two examples of both Hamiltonian and dissipative systems are considered. In these cases the quantum breaking time differs from t_h. The first example is quantum flights studied for the kicked rotor, and it is shown that temporal crossover from the classical to the quantum behavior is determined by the scaling properties of the phase space. The second example is a model of a quantum dissipative system which is considered in the regime when the classical limit corresponds to a chaotic attractor. Alexander Itin Resonant phenomena in classical dynamics of three-body Coulomb systems Classical dynamics of three-body Coulomb systems similar to a hydrogen molecular ion is considered (heavy-light-heavy charged particles). In the limit of infinitely heavy nuclei the system is reduced to the integrable two-center problem. When masses of heavy particles are finite, slow and fast motions appear. Averaging method predicts that actions of "fast" motions of the system with frozen nuclei are approximate integrals of the full system (adiabatic invariants). During slow evolution of the "heavy" subsystem certain resonance conditions can be satisfied. In the presentation the phenomena of capture into resonances and scattering on resonances are described and statistical properties of dynamics of adiabatic invariants are discussed. Brunon Kaminski On the directional entropy of Z^2-actions on a Lebesgue probability space The directional entropy is an interesting invariant which is useful to investigate Z^2-actions on a Lebesgue probability space with zero Conze-Katznelson-Weiss entropy. It has been introduced into the theory of dynamical systems by J.Milnor in 1986 and then investigated by several spectaculaires. Some basic properties of this notion have been proved by Boyle & Lind and Kaminski & Park. Recently Park has shown the continuity of the directional entropy for Z^2--actions generated by cellular automata which allowed Courbage and Kaminski to prove the formula of the directional entropy in the case of permutative automata. Janina KOTUS Geometry and Ergodic Theory of non-recurrent Elliptic Functions Alexandra Landsman Conditions for Resonance and Chaos in the FRC The dynamics of an ion are investigated inside the FRC under variation of angular momentum and elongation. FRC stands for Field Reversed configuration, a type of fusion reactor. Different types of resonances and the conditions under which they occur are classified according to the type of unperturbed Hamiltonian. The onset of strong chaos and limiting cases of large elongation oflarge parameters are discussed. Jacques Laskar Frequency Map Analysis. Application to the study of chaotic dynamics in the Solar System and Particle AcceleratorI, II

Xavier Leoncini Anomalous transport in two dimensional plasma turbulence Transport properties of particles evolving in a system governed by the Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a non linear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of coherent structures (vortices) within the flow. The characterization of anomalous transport is performed using a diagnostic inspired from "chaotic jets". In the vicinity of coherent structures the motion of tracers is quasi-ballistic. This phenomenon is known as stickiness and associated long time correlations. The fractional nature of the transport in this setting is discussed. Fabio Lepreti PARTICLE TRANSPORT AND ACCELERATION IN STOCHASTIC ELECTROMAGNETIC FIELDS We investigate the acceleration of particles occurring in plasma environments where turbulent electric and magnetic fields are present. Different models are used to build up complex field patterns where the transport of particles can be studied through test particle simulations. The radiation produced by the interactions of energetic particles with the ambient plasma and with the stochastic fields is included in our models, in order to perform a comparison with spectra obtained from the observations of astrophysical processes like, for example, solar flares. Emanuel Lima Chaotic Dissociation of Diatomic Molecules We present a classical and quantum study of the dissociation in diatomic molecules. As a moldel, we consider the one-dimensional morse potencial driven by an external electromagnetic field. The classical process is known to be chaotic and we search for its quantum counterpart. Helen Makarenko Dynamical properties of the bulk ZrTiCuNiBe metallic glass.

Amar Makhlouf Limit cycles of Lienard systems We study the existence of limit cycles of planar systems and especially the Lienar systems.We give an other proof of the theorem proved by L.Perko concerning the number of limit cycles of the system : x¢=y-F(x) y¢=-x with F(x)=e(a1x+...+a2m+1x2m+1), 0
Miguel MANNA A singular integrable equation from short capillary-gravity waves Competition between nonlinearity and dispersion at small scales is related to turbulence phenomena. The study of the Euler equations is a very hard task and great simplifications are obtained by considering intermediate asymptotic models. From the Euler equations with surface tension we derive in the short-wave approximation a new integrable 1+1 dimensional asymptotic model for the motion of the surface. This system has solutions which become multiple valued in finite time. We show that for a critical Bond number transition between stable and unstable regimes for the wave-train solution occurs. A very important open problem is the inclusion of viscosity (Navier-Stokes equations) which acts strongly over small scales and will affect the short-waves dynamics. Paul Manneville Subcritical transition to turbulence in the plane Couette flow The analogy between thermodynamic first order phase transitions and the transition to turbulence in the plane Couette flow is considered. Chaotic transients associated to the nucleation of turbulent spots as well as the decay of sustained turbulence below R=325 are discussed in view of this analogy.

Stefano Musacchio Two-dimensional turbulence of dilute polymer solutions We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. At sufficiently large elasticity the polymers react on the flow with manifold consequences: velocity fluctuations are drastically depleted, the velocity statistics becomes strongly intermittent and the distribution of finite-time Lyapunov exponents shifts to lower values, signalling the reduction of Lagrangian chaos. Anatoly Neishtadt Destruction of adiabatic invariance and transport phenomena in systems with slow and fast motions

There are many problems that lead to analysis of dynamical systems in which one can distinguish motions of two types: slow one and fast one. An averaging over fast motion is used for approximate description of the slow motion. First integrals of the averaged system are approximate first integrals of the exact system, i.e. adiabatic invariants. Resonant phenomena in fast motion ( passage through separatrix, passage through resonance, capture into resonance) lead to inapplicability of averaging, destruction of adiabatic invariance, dynamical chaos and transport in large domains in the phase space. In the talk some perturbation theory methods for description of these phenomena are outlined. As examples some problems from classical mechanics, hydrodinamics and plasma physics are considered. Edward Ott The Onset of Coherent Behavior in Globally Coupled Systems Plan of lecture: Systems of many coupled dynamical units are of great interest in a variety of scientific fields, including physics, chemistry and biology. In this lecture we will be interested in `global' coupling in which each element is coupled to all others. We begin with a review of coupled units whose uncoupled behavior is periodic, concentrating on the model of Kuramoto [1-3]. We then discuss the onset of synchronism in systems composed of units whose uncoupled dynamics is chaotic [4-7]. Following that, we present a general formalism [7] that simultaneously handles the case of both periodic and chaotic dynamics of the coupled units, including situations where both types of units are present in the same system. Numerical experiments are presented, and representative laboratory experiments are discussed (e.g., [8]). Most important references:

[1] Y. Kuramoto, in `International Symposium on Mathematical Physics,'edited by H. Araki, Lecture Notes in Physics, Vol. 39 (Springer, Berlin, 1975); and `Chemical Oscillators, Waves, and Turbulence' (Springer, Berlin, 1984). [2] E. Ott, `Chaos in Dynamical Systems,' second edition, Section 6.5 (Cambridge University Press, to be published in 2002 or 2003). [3] S. H. Strogatz, Physica D 143, 1 (2000) (this is a review article on the Kuramoto model). [4] A. S. Pikovsky, M. G. Rosenblum, and J. Kurths, Europhys. Lett. 34, 165 (1996). [5] H. Sakaguchi, Phys. Rev. E 61, 7212 (2000). [6] D. Topaj, W. -H. Kye, and A. S. Pikovsky, Phys. Rev. Lett. 87, 074101 (2001). [7] E. Ott, P. So, E. Barreto, and T. M. Antonsen, Physica D (submitted). [8] I. Z. Kiss, Y. Zhai, and J. L. Hudson, Phys. Rev. Lett. 88, 238301 (2002).

Séverine PACHE Motion of the enigmatic piston We consider the evolution of a systeme composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adaiabatic wall of mass M. The length of the container is a fixed paramater L wich can be finite or infinite. If the length L is infinite, the systeme evolves towards a stationary state with velocity V. If the length finite, we show the evolution proceeds in two stages. The first stage is a deterministic and adiabatic evolution to mechanical equilibrium with strong or weak damping. The second stage is a diathermic and stochastic evolution towards a thermal equilibrium. Antonio Politi Heat transport in nonlinear lattices Heat conductivity in low-dimensional systems is now known to diverge in the thermodynamic limit. I shall review the state of the art, by commenting both about nonequilibrium and equilibrium simulations. In the former context I will briefly discuss various algorithms to simulate heat baths. As for the theoretical aspects, I will discuss the case of harmonic (both homogeneous and disordered) chains with particular reference to the role of boundary conditions. Predictions of mode-coupling theory and coming from the relationship with Burgers equations will be compared with the results of accurate numerical simulations. The scenario arising in 2D systems will be also briefly introduced.

Antonio Ponno Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit

Serguei PRANTS Hamiltonian chaos in cavity quantum electrodynamics A review of recent results on nonlinear dynamics of the atom-photon interaction in high-quality cavities (cavity quantum electrodynamics) will be given. Main attention will be paid for manifestations of Hamiltonian chaos, fractals, intermittency, Levy flights and anomalous transport with atoms in micromasers and microlasers. Quantum-classical correspondence will be analyzed with different models in quantum and atomic optics.

Hamiltonian chaos, dynamical traps and fractals in a simple advection model with a topographical vortex M.V. Budyansky, S.V. Prants We investigate the dynamics of passive particles in a two-dimensional incompressible open flow composed of a fixed point vortex and a background current with a periodic component - a simple model of advection of passive particles (temperature, salinity, pollutants, plankton, etc.) by topographical vortices in the ocean [1]. Chaotic advection of tracers is proven to be of a homoclinic nature with transversal intersections of stable and unstable manifolds of the saddle point. In spite of simplicity of the flow, chaotic trajectories are very complicated alternatively sticking nearby boundaries of the vortex core and islands of regular motion and wandering in the mixing region. The boundaries act as dynamical traps for advected particles with a broad distribution of trapping times. This implies the appearance of fractal-like scattering function: dependence of the trapping time on initial positions of the tracers. It is confirmed numerically by computing a trapping map and trapping time distribution which is found to be initially Poissonian with a crossover to a power law at the PDF tail. The mechanism of generating the fractal is shown to resemble that of the Cantor set with the Hausdorff fractal dimension of the scattering function to be equal to approximately 1.84. Statistical characteristics of the particles advection in the same flow but with a noise added have been computed. Fractals have been found in the noise dynamics as well. 1. M.V. Budyansky, S.V. Prants, M.Yu. Uleysky. Dokl. Akad. Nauk. 2002. V.386 No.5. P.686-689. Hamiltonian ray chaos and clusters in inhomogeneous underwater sound channels D.V. Makarov, M.Yu. Uleysky, S.V. Prants The problem of impact of internal waves on long-range sound propagation in the deep ocean is examined. We introduce a realistic model of the background sound-speed profile [1] that permits a simple analytical description of ray nonlinear dynamics in terms of the canonical action-angle variables. In a range-dependent environment, provided by a wide spectrum of internal waves, ray trajectories may be chaotic in the sense of exponential sensitivity to small initial changes in the depth and/or grazing angles. We study Hamiltonian ray chaos theoretically and numerically. Our numerical study shows that sound rays can form clusters over arrival times at a given distance even in the chaotic regime. Different mechanismes of forming the clusters are considered. Ray clusters are caused by inhomogeneity of phase space and by existing of some zones of stability therein that take place both under deterministic and noise perturbations. Clustering depends both on nonlinear characteristics of sound-speed profile and on the spatial spectrum of the internal waves. 1. D.V.Makarov, S. V. Prants and M.Yu.Uleysky. Dokl. Akad. Nauk. V.382 N3 2002 394-396.

Saar Rahav Effective Hamiltonians for periodically driven systems

Linda Reichl Quantum Chaos-I -II

Vered Rom-Kedar 1. Soft billiard potentials. 2. Mixing and transport in fluid flows - geometrical approach. M. Shlesinger Sub and Supra Diffusion Including Stretched Times and Divergent Time Scales History of Random Processes Montroll-Weiss Continuous Time Random Walks Equivalent Master Equations Levy Probabilities : Fractal Space and Fractal Time Long Correlation Memory Sub-diffusive Transport Defect Diffusion Stretch Exponential Relaxation Glass Transition Vogel-Fulcher Law for Glasses (Temperature) Bendler-Fontanella-Shlesinger Law for Glasses (Temperature/Pressure) Supra-Diffusion in Turbulence

Relativistic Diffusion

Eli Shlizerman PARABOLIC RESONANCES AND THE PERTURBED NONLINEAR SHRÖDINGER EQUATION Eli Shlizerman and Vered Rom-Kedar We examine the existence of parabolic resonances and various regimes of instabilities in the perturbed Nonlinear Shrödinger equation (NLS). A two-mode Fourier truncation of the NLS pde, presented by A.R. Bishop et al, is considered as a basic model. Such truncation provides two d.o.f integrable Hamiltonian system, which represents the homoclinic structures of the full pde. Parabolic resonances in the truncated dynamical system can be identified by understanding its global structure. To obtain analytical results and understanding under which circumstances such instabilities arise we use tools such as energy-momentum bifurcation diagram (EMBD) and Fomenko graphs. Tom Solomon The effects of Lagrangian chaos on multiphase processes and network dynamics Ion STROE CHAOTIC MOTION OF SATELLITES IN AN ELLIPTIC ORBITS

The motion of rotation a satellite with respect to center of mass is the subject of many scientific works. In some papers the chaotic character of the motion of rotation with respect to normal axis to orbital plane is proved. Nonlinear three-degree-of-freedom attitude motion is analyzed in this paper. Considering a central inverse square and the change of gravitational force over the distributed mass of the rigid body, the gravity gradient torque is used to analyze the general motion of the gravitational stabilized spacecraft in an elliptic orbit.

Walter Strunz Decoherence in quantum physics The superposition principle is fundamental to quantum interference effects. Due to environmental influences, however, quantum superpositions may lose their coherence and decay into a mixture of states. Various important aspects of this decoherence are discussed: we derive the astonishing decoherence time scale, we identify robust states, and we describe corresponding experiments. Stochastic approach to open quantum system dynamic Stochastic Schroedinger equations may be employed to describe the dynamics of open quantum systems. We give a microscopic derivation that allows us to generalize the stochastic approch to open quantum systems to non-Markovian situations. Vladimir Ten Normal Distribution of Velocities of Deterministic Systems

Sandro VAIENTI Mesures invariantes pour des applications multidimensionelles non-uniformement dilatanttes In this work, joint with Huyi Hu, we propose a technique to construct absolutely continuous invariant measures for a wide class of non uniformly expanding maps of a compact set in R^n. THis maps could have a non-bounded distorsion property and they could have an ergodic decomposition of the acim with both finite and \sigma-finite components with the parabolic point in the the same closure Jiri Vanicek Semiclassical evaluation of fidelity in the Fermi-golden-rule and Lyapunov regimes

Alexei Vasiliev Geometric and statistical properties induced by separatrix crossings

Sebastien VISCARDY Viscosity and chaos in a two hard disk model

Angelo Vulpiani TRANSPORT, DIFFUSION and FRONT PROPAGATION FIRST LECTURE:

We study transport of passive scalar, in particular the anomalous Diffusion and non-ideal cases, i.e. when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. In such a situation usual asymptotic quantities do not give relevant information about the transport mechanisms. On the other hand the Finite Size Lyapunov Exponent appears to be rather powerful. SECOND LECTURE Front propagation in two dimensional steady and unsteady cellular flows is investigated in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. We provide an analytical approximation for the front speed as a function of the stirring intensity. Piotr Waz Theories of the Motion of the Martian Satellites Tatiana Yelenina Semi-analytical model of the force-free magnetic field in the system "star-accretion disc" TBA L.S. Young Strange attractors and periodically driven chaos Michael Zaks Anomalous transport in steady plane flows of viscous fluids In time-independent two-dimensional fluid motions there is neither room for chaotic fluid motions, nor place for chaotic advection. Nevertheless under certain circumstances the transport properties of such flows can be notrivial. We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of isolated vortices or past lattices of solid obstacles. Due to the repeated passages through the stagnation regions, dynamics of a tracer particle acquires features which are not quite usual in the context of ordered laminar motions : autocorrelation for a Lagrangian observable decays in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. Time of the passage through a unit cell of the flow diverges near certain streamlines. At the isolated stagnation points such divergence is logarithmic, whereas near the edge of the solid body it has a stronger, power-like singularity. We demonstrate that spreading of a droplet of tracers in such flows is anomalous; in a flow with stagnation points, this is a slow subdiffusive transport, whereas in a flow past the array of the solid obstacles, the droplet is stretched at a superdiffusive rate. Georges Zaslavsky FRACTIONAL KINETICS IN DYNAMICAL SYSTEMS 1. FRACTIONAL KINETICS --Hamiltonian chaos. Recurrences. Anomalous transport. Sticky domains. Levy flights. Fractional kinetics. Renormalization group for kinetics. Log-periodicity. Persistent fluctuations. Applications. 2. PSEUDOCHAOS --Systems with zero Lyapunov exponent. Polygonal billiards. Kinetics and transport. Maxwell Demon. Persistent fluctuations. 3. COMPLEXITY AND ENTROPY FUNCTIONS (new approach) --Coplexity and entropy. Phase space analysis. Epsilon-separation. Epsilon, t- separation. Time-space coupling. Polynomial Compexity and entropy. Directional complexity and entropy. --Entropy for systems with zero Lyapunov exponents. Vladimir Zeitlin Equatorial geophysical fluid dynamics and transportI,II.

PARTICIPANTS Valentin Afraimovich IICO UASLP. Communication Optica. Universitet Autonomny de San Luis. Karakorum 1470, Iomas, 4eme section. CP78200 San Luis Potosi. SLP Mexico

[email protected] Omar Al Hammal Institute "Carlos I" for Theoretical and Computational Physics. Universidad de Granada Avda. Fuente Nueva, s/n, 1 18071 Granada Spain

[email protected] Athanasios Arvanitidis Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Geece

[email protected] Francois Bardou IPCMS-CNRS, 23, rue du Loess, BP 43 67037 Strasbourg Cedex 2 France

[email protected] Maxim Barkov Space Plasma Physics-Space Research InstituteProfsoyuznaya str. 84/32 117997 Moscow Russie

[email protected] Jacopo Bellazzini Universita' di Pisa - Dipartimento Ingegneria AerospazialeVia Caruso 120 56100 Pisa Italie

[email protected] Viatcheslav Belyi Theoretical department-Russian Academy of Sciences IZMIRAN 142190 Troitsk, Russia

[email protected] Cristel Chandre CPT -- CNRS-Centre de Physique Théorique Campus de Luminy - case 907 13288 Marseille, France

[email protected] Hugues Chaté SPEC - CEA - Saclay, 91191 Gif-sur-Yvette, France

[email protected] Guido Ciraolo Centre de Physique Theorique,CNRS case 907 13288 Marseille cedex 9, France

[email protected] Steliana Codreanu Department of Theoretical Physics-Babes-Bolyai University Kogalniceanu str. 1 2400 CLUJ-NAPOCA Roumanie

[email protected] Pierre Collet Centre de physique theorique-CNRS Ecole Polytechnique, route de Saclay 91128 Palaiseau cedex France

[email protected] Pieter Collins Control and computation-CWIKruislaan 413 1098 SJ Amsterdam Nederland

[email protected]

Monique Combescure Institut de Physique Nucléaire de Lyon, CNRS-Bât Paul Dirac, 4 rue Enrico Fermi 69622 VILLEURBANNE France

[email protected] Maurice Courbage LPTMC case 7020-Université de Paris 7, 2 place Jussieu 75231 Paris Cedex 05 France

[email protected] Giampaolo Cristadoro Center for Nonlinear and Complex Systems and Dipartimento di Scienze Chimiche, Fisiche e Matematiche-Universita' dell'Insubria (sede di Como)-via Valleggio,11 22100 Como Italie

[email protected] Nir Davidson Dept. of Physics of Complex Systems-Weizmann Institute of Science Rehovot 76100 Israel

[email protected] Filippo De Lillo Dipartimento di Fisica Generale-University of TorinoVia Giuria,1 10125 Torino Italie

[email protected] Jean-Pierre Eckmann Departement de Physique Theorique and Section de Mathematiques Universite de Geneve-32, Bld D'Yvoy 1211 Geneva 4 Suisse

[email protected] Massimiliano Esposito Service de Chimie Physique CP 231-Universite Libre de Bruxelles Boulevard du Triomphe B-1050 Bruxelles Belgique

[email protected] Fereydoon Family Physics Department-Emory UniversityPhysics Department, Emory University GA Atlanta USA

[email protected] Stefano Galatolo Dipartimento di matematica applicata-Universita di PisaVia Bonanno Pisano 25b 56126 Pisa Italie

[email protected] Govanni Gallavotti Fisica -Univ. Roma 1P.le Moro 2 00185 Roma Italie

[email protected] Pierre Gaspard Center for Nonlinear Phenomena and Complex SystemsCampus Plaine, CP 231 B-1050 Brussels Belgique

[email protected] Alessandro Giuliani Physics department-Universita' di Roma "La Sapienza"- Via Ivanoe Bonomi, 92, 00139 Roma, Italy

[email protected] Vasiliy Govorukhin Rostov State University - Computational mathematics Zorge str. 5 344090 Rostov-on-Don Russie

[email protected]

Seiichiro Honjo University of Tokyo, Graduate School of Arts and Sciences-Department of Basic Science, Kaneko LaboratoryKomaba 3-8-1, Meguroku 153-8902 Tokyo Japon

[email protected] Alexander Iomin Department of Physics, Technion 32000 Haifa Israel

[email protected] Alexander Itin Lab. 627 Space Research Institute-Profsoyuznaya str. 84/32 117997 Moscow Russia

[email protected] Brunon Kaminski Faculty of Mathematics and Informatics Nicholas Copernicus University, ul Chopina 12/18, 87-100 Torun, POLAND

[email protected] Janina Kotus Warsaw Univ.of Technology-Department of MathematicsPlac Politechniki 1 00-661 Warsaw Pologne

[email protected] Alexandra Landsman Princeton University, 235 Thunder Circle PA 19020 Ben Salem USA

[email protected] Jacques Laskar Astronomie et Systemes Dynamiques-IMC77, Av. Denfert-Rochereau F-75014 PARIS France

[email protected] Xavier Leoncini PIIM-Université de Provence, Centre de St Jerome, case 321 13396 Marseille Cedex 20 France

[email protected] Fabio Lepreti Section of Astrophysics, Astronomy, and Mechanics-Department of Physics, Aristotle University of ThessalonikiDepartment of Physics, Aristotle University of Thessaloniki 54124 Thessaloniki Grèce

[email protected] Emanuel Lima Instituto de Física de São Carlos-Universidade de São PauloMajor Júlio Salles 870, Vila Pureza, 13561-010 16 São Carlos Brazil

[email protected] Helen Makarenko Department of Physics-V.Karazin National University4 Svobody square 61077 Kharkiv Ukraine

[email protected] Amar Makhlouf Université de Annaba-Labo de Mathématiques, 14 Rue Zighoud Youcef, DREAN 36 ELTARF, ALGERIE

[email protected] Miguel Manna Physique Mathematique et Theorique CNRS-UMR5825-Universite Montpellier 2Place Eugene Bataillon 34095 Montpellier, France

[email protected]

Paul Manneville Laboratoire d'Hydrodynamique-CNRS-Ecole PolytechniqueEcole polytechnique 91128 Palaiseau, France

[email protected] Stéphane Métens LPTMC, case 7020-Université Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05, France

[email protected] Stavros Muronidis Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Grèce

[email protected] Stefano Musacchio Dipartimento di Fisica Generalevia Pietro Giuria 1 10125 Torino, Italie

[email protected] Anatoly Neishtadt Space Research Institute -Russian Academy of Sciences, Profsoyuznaya 84/32 Moscow 117997, Russia

[email protected] Tali Oliker Technionyuvalim 56 20142 d.n. misgav, Israel

[email protected] Edward Ott I.R.E.A.P.-University of Maryland, City COLLEGE PARK 20742 MARYLAND, USA

[email protected] Séverine Pache ITP-EPFL, Ecublens 1015 Lausanne, Suisse

[email protected] Antonio Politi ISTITUTO NAZIONALE DI OTTICA APPLICATA, LARGO E. FERMI 6 50125 FIRENZE, Italy

[email protected] Antonio Ponno Dipartimento di Matematica-Universita' di Milano, Via Saldini 50 20133 Milano, Italie

[email protected] Serguei Prants Institution Pacific Institute of the Russian Academy of Sciences 43 BALTIISKAYA St. 690041 VLADIVOSTOK RUSSIA

[email protected] Saar Rahav Technion-Department of Physics 32000

Haifa Israel

[email protected] LindaReichl Department of Physics-Center for Statistical Mechanics and Complex SystemsThe University of Texas at Austin, USA

[email protected] Vered Rom-Kedar Weizmann InstituteP.O. Box 26-Department of Computer science and applied mathematics 76100 Rehovot, Israel

[email protected] Marc Senneret LPTMC, case 7020-Université de Paris 72 place Jussieu 75251 PARIS CEDEX 05 France

[email protected] Michael Shlesinger Office of Naval Research, 800 n°Quincy Str., Arlington, VA22217-5660, USA

[email protected]

Eli Shlizerman Computer Science and Applied Mathematics - Dynamical Systems-Weizmann Institute of Science 76100 Rehovot Israel

[email protected] Dominique Simpelaere Université Paris 6-Pierre et Marie Curie, Lab. De Probabilié, 4, Place Jussieu 75252 Paris cedex 05, France

[email protected] Tom Solomon Bucknell University Lewisburg, PA 17837, USA

[email protected] George Stilogiannis Dept. of Mathematics Aristotle University of Thessaloniki 54006 Thessaloniki, Grèce

[email protected] Ion Stroe Mechanics-POLITEHNICA University of Bucharest Splaiul Independentei 313 RO-77206 Bucharest ROMANIA [email protected] Walter Strunz Physikalisches Institut, Universitat Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany [email protected] Vladimir Ten Dept of Mathematics and Mechanics Moscow State University-MSU, Vorobevy gory 119899 Moscow Russia [email protected] Sandro Vaienti University of Toulon and Centre de Physique TheoriqueCase 907 3288 Marseille Cedex 09 France [email protected] Jiri Vanicek Jefferson Physical LaboratoryMSRI, 1000 Centennial Drive CA 94720 Berkeley USA [email protected] Alexei Vasiliev Laboratory of Chaotic Dynamics Space Research Institute-Profsoyuznaya 84/32 117997 Moscow Russie [email protected] Sebastien Viscardy Universite Libre de Bruxelles-Service de Chimie-PhysiqueCampus Plaine, CP 231 1050 Brussels Belgium [email protected] Thérèse Vivier UMR5584 (CNRS) - Institut Mathématiques de Bourgogne BP47970 21078 Dijon Cedex France [email protected] Angelo Vulpiani Dipartimento di Fisica, Universita` di Roma La Sapienza P.le A. Moro 2 I-00185 Roma Italie [email protected] Piotr Waz Center for Astronomy-Nicholaus Copernicus University, Gagarina 11 87-100 Torun Pologne [email protected] Tatiana Yelenina Numerical Simulation of Electrodynamics and Magnetohydrodynamics-Keldysh Institute of Applied Mathematics of RAS4, Miusskaya sq. 125047 Moscow Russie [email protected] L.S. Young Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA [email protected] Michael Zaks Humboldt University of Berlin-Dept of Stochastic Processes, Newtonstr. 15 D-12489 Berlin Allemagne [email protected] Georges Zaslavsky

Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA [email protected] Vladimir Zeitlin Laboratoire de Meteorologie Dynamique, E N S, 24 Rue Lhomond 75231 PARIS CEDEX 05 France zeitlin @lmd.ens.fr