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In investigating the complexity of dynamical systems, entropy and quantities connected with it play an important role. The nonlinear dynamics of a spatially extended physical, chemical or biological system is complex, as for example in the case of turbulent flows, unlike the simple motion of laminar fluids. The complexity of spatially extended dynamical systems has been described in many ways using several models. We will address issues related to the role of directional entropy in Lattice Dynamical Systems (LDS) and lifts of circle maps on the plane.

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.

This tutorial reviews a new important class of mathematical phenomenological models of neural activity generated by iterative dynamical systems: the so-called map-based systems. We focus on 1-D and 2-D maps for the replication of many features of the neural activity of a single neuron. It was shown that such systems can reproduce the basic activity modes such as spiking, bursting, chaotic spiking-bursting, subthreshold oscillations, tonic and phasic spiking, normal excitability, etc. of the real biological neurons. We emphasize on the representation of chaotic spiking-bursting oscillations by chaotic attractors of 2-D models. We also explain the dynamical mechanism of formation of such attractors and transition from one mode to another. We briefly present some synchronization mehanisms of chaotic spiking-bursting activity for two coupled neurons described by 1-D maps.

The problem of phase synchronization of coupled chaotic bursting oscillatory systems is studied using the probability distribution of return time to the quiscent state. It is shown that, in some range of the coupling strength, the continuous type distribution becomes discrete and is supported by few values around a preferred one. This shows that the shape and the amplitudes of the spikes are not important and the main aspect of coupling is the change of distribution of the units' time return difference.

The density of the measure-theoretic directional entropy for the lattice dynamical system is introduced and it is shown that for a lattice dynamical transformation commuting with the shift transformation the density coincides with the entropy of the ℤ2-action generated by these two transformations, i.e. it is a constant function with respect to the direction. It is also proved that in the noncommutative case this result fails to be true.

The deterministic dynamics of a spatially extended physical or chemical or biological system may be complex, as in the case of turbulent flows in contrast with the simple motion of laminar fluids. The complexity of extended dynamical systems has been described in many ways using several characteristics and several models. An important role, In investigating the complexity of dynamical systems, entropy and quantities connected with it plays an important role. In the smooth dynamics the Lyapunov exponents are quantities of this type.

The purpose of this paper is to present some notions in the theory of complexity in lattice models of extended dynamical systems.

We review some ideas and concepts on the irreversibility of deterministic dynamical systems that have been discussed during several years of collaboration with Ilya Prigogine and B. Misra.

We study the memory effects in time series generated by unstable non-chaotic dynamical systems and by non-Markovian stochastic sources. For this purpose we compare the decay rate of the Shanonn conditional entropy and the decay rate of the autocorrelations. These rates are completely distinct owing to the persistence of long range correlations. We obtain scaling behavior for the convergence of the block entropies.

Time operator describes the lifetime of quantum irreversible processes on the level of statistical states. The existence, interpretation, and properties of time operator are discussed and the master equation is derived. Time operator exists also in other dynamical systems with mixing properties. Models of non-Markovian processes with long memory having a time operator are discussed.

The necessary and sufficient conditions in order that a coarse graining with respect to a partition with finite number of cells leads to a Markovian evolution with a monotonically increasing entropy in abstract conservative dynamical systems are derived. These conditions are in particular verified when the symbolic dynamics associated to the partition induces a Markov chain. In this case the irreversible approach to equilibrium entails that the system should have a Bernoulli factor.

Entropy functionals are computed for non-stationary distributions of particles of Lorentz gas and hard disks. The distributions consisting of beams of particles are found to have the largest amount of entropy and entropy increase. The computations show exponentially monotonic increase during initial time of rapid approach to equilibrium. The rate of entropy increase is bounded by sums of positive Lyapounov exponents.

The purpose of this paper is to survey shortly some notions in the spectral theory of ergodic dynamical systems and their relevance to mixing and weak mixing. In addition, we present some dynamical systems of particles submitted to collisions with nondispersive obstacles and their ergodic and spectral properties. Transport is formulated in terms of random walk generated by deterministic dynamical systems and their moments.

Content: Part I : Theory P. Collet A SHORT ERGODIC THEORY REFRESHER M. Courbage Notes on Spectral Theory, Mixing and Transport V. Affraimovich, L. Glebsky: Complexity, Fractal Dimensions and Topological Entropy in Dynamical Systems G.M. Zaslavsky, V. Afraimovich: WORKING WITH COMPLEXITY FUNCTIONS G. Gallavotti SRB distribution for Anosov maps P. Gaspard DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY W.T. Strunz ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS E. Shlizerman, V. R. Kedar ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. M. Combescure Phase-Space Semiclassical Analysis.Around Semiclassical Trace Formulae Part II : Applications A. Kaplan et al ATOM-OPTICS BILLIARDS F. Family et al CONTROL OF CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS F. Bardou FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE MEAN WAITING TIMES X. Leoncini et al ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA TURBULENCE E. Ott et al THE ONSET OF SYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC AND PERIODIC DYNAMICAL UNITS A.Iomin, G.M. Zaslavsky QUANTUM BREAKING TIME FOR CHAOTIC SYSTEMS WITH PHASE SPACE STRUCTURES S.V.Prants HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM ELECTRODYNAMICS M. Cencini et al INERT AND REACTING TRANSPORT M. A. Zaks ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF VISCOUS FLUIDS J. Le Sommer, V. Zeitlin TRACER TRANSPORT DURING THE GEOSTROPHIC ADJUSTMENT IN THE EQUATORIALOCEAN A. Ponno THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC LIMIT

We introduce a new class of maps that describe the chaotic activity of spiking-bursting neurons observed in neurophysiological experiments. We show that, depending on the connection (diffusively or reciprocally synaptically), coupled maps demonstrate several modes of cooperative dynamics: i) weakly correlated chaotic pulsations, ii) anti-phase quasi-regular bursting activity and iii) chaotic synchronization. Such phenomena have been observed in recent experiments with central pattern generator chaotic neurons. Taking into account the fact that the shape and the amplitude of the spikes are not important for the organisation of such cooperative dynamics, we analyzed the timing of the bursts only. We showed that the regime of anti-phase regularisation is stable against noise.

We discuss a possible construction of Markov evolutions with Boltzmann entropy, associated, through a coarse — graining, to deterministic systems which are not as strongly unstable as K — systems. Yet, this extension implies that the dynamical system should have an absolutely continuous component in its spectrum.

We study a deterministic conservative dynamic model inspired by the Kac ring model. We show that some initial probability distributions with long range correlation, which go to equilibrium asymptotically, can be transformed, through a transition to new dissipative and stochastic dynamics, into states with damped correlations.

The present work is devoted to the following question: What is the relation between the deterministic laws of dynamics and probabilistic description of physical processes? It is generally accepted that probabilistic processes can arise from deterministic dynamics only through a process of “coarse graining” or “contraction of description” which inevitably involves a loss of information. In this work we present an alternative point of view toward the relation between deterministic dynamics and probabilistic descriptions. Speaking in general terms, we demonstrate the possibility of obtaining (stochastic) Markov processes from deterministic dynamics simply through a “change of representation” which involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. The fundamental implications of this finding for statistical mechanics and other areas of physics are discussed. From a mathematical point of view, the theory we present is a theory of invertible, positivity preserving and necessarily nonunitary similarity transformations that convert the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. We explicitly construct such similarity transformations for the so-called Bernoulli systems. This construction illustrates also the construction of the so-called Lyapounov variables and the operator of “internal time” which play an important role in our approach to the problem of irreversibility. The theory we present can also be viewed as a theory of entropy increasing evolutions and their relation to deterministic dynamics. The possibility of obtaining entropy increasing evolutions from deterministic dynamics through a nonunitary similarity transformation is concretely illustrated in the present work.

A procedure of analytic continuation of the resolvent of Liouville operators for quantum statistical systems is discussed. When applied to the theory of irreversible processes of the Brussels School, this method supports the idea that the restriction to a class of initial conditions is necessary to obtain an irreversible behaviour. The general results are tested on the Friedrichs model.

We discuss the problem of nonunitary equivalence, via positivity-preserving similarity transformations, between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes. Dynamical systems admitting such nonunitary equivalence with stochastic Markov processes are said to be intrinsically random. In a previous work, it was found that the so-called Bernoulli systems (discrete time) are intrinsically random in this sense. This result is extended here by showing that a more general class of dynamical systems---the so-called K systems and K flows---are intrinsically random. The connection of intrinsic randomness with local instability of motion is briefly discussed. We also show that Markov processes associated through nonunitary equivalence to nonisomorphic K flows are necessarily nonisomorphic.

We continue our previous work on dynamic “intrinsically random” systems for which we can derive dissipative Markov processes through a one-to-one change of representation. For these systems, the unitary group of evolution can be transformed in this way into two distinct Markov processes leading to equilibrium for either t → + ∞ or t → - ∞. To lift the degeneracy, we first formulate the second principle as a selection rule that is meaningful in intrinsically random systems. For these systems, this excludes a set of unrealizable states. As a result of this exclusion, permitted initial conditions correspond to a set of states that is not invariant through velocity inversion. In this way, the time-reversal symmetry of dynamics is broken and these systems acquire a new feature we may call “intrinsic irreversibility.” The set of admitted initial conditions can be characterized by an entropy displaying the amount of information necessary for their preparation. The initial conditions selected by the second law correspond to a finite amount of information, while the initial conditions that are rejected correspond to an infinite amount of information and are therefore “impossible.” We believe that our formulation permits a microscopic formulation of the second law of thermodynamics for well-defined classes of dynamical systems.

We give the mathematical details and various extensions of the results stated in previous work of Courbage and Prigogine. “Intrinsic random systems” are deterministic and conservative dynamical systems for which we can associate two dissipative Markov processes through a one-to-one “change of representation”, the first leading to equilibrium for t→+∞ and the second for t→-∞. The microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t→+∞. The set of admitted initial conditions D+ is then characterized by a non-equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions. Thus, rejected initial states correspond to an infinite amount of information. To realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to Markov processes to such measures. Owing to the non-invariance of D+ under the time inversion, the evolution of these states in the new representation can only be given by one of the two possible Markov processes.

We discuss the question of the dynamical meaning of the second law of thermodynamics in the framework of quantum mechanics. Previous discussion of the problem in the framework of classical dynamics has shown that the second law can be given a dynamical meaning in terms of the existence of so-called Lyapounov variables—i.e., dynamical variables varying monotonically in time without becoming contradictory. It has been found that such variables can exist in an extended framework of classical dynamics, provided that the dynamical motion is suitably unstable. In this paper we begin to extend these results to quantum mechanics. It is found that no dynamical variable with the characteristic properties of nonequilibrium entropy can be defined in the standard formulation of quantum mechanics. However, if the Hamiltonian has certain well-defined spectral properties, such variables can be defined but only as a nonfactorizable superoperator. Necessary nonfactorizability of such entropy operators M has the consequence that they cannot preserve the class of pure states. Physically, this means that the distinguishability between pure states and corresponding mixtures must be lost in the case of a quantal system for which the algebra of observables can be extended to include a new dynamical variable representing nonequilibrium entropy. We discuss how this result leads to a solution of the quantum measurement problem. It is also found that the question of existence of entropy of superoperators M is closely linked to the problem of defining an operator of time in quantum mechanics.