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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.

Chaos in the rhythmic activity is a major issue that has been discussed in many studies of neuroscience and physiology, and especially in the respiratory air flow. Here, we present the results of two studies concerning the activity and the connectivity of the respiratory neural network in healthy humans and patients with obstructive lung disease. Our results show an increase in the dynamic chaos of airway flow in patients, focusing on expiratory flow. We then sought the reasons for this augmentation in analyzing the activity of neural centers involved in respiratory rhythmogenesis, using functional brain imaging of the automatic neural networks, the first group generating inspiration (pre-Botzinger complex) and the second in charge of expiration (the parafacial group). Brain imaging reveals in healthy humans a significant activation of the pre-Botzinger complex linked to a high active inspiration while patients have a higher expiratory parafacial excitability leading to an active expiration. We also propose a theoretical model of respiratory rhythmogenesis which reproduces the synchronized respiratory neural network from two chaotic pacemakers, the first modelling the pre-Botzinger complex and the second modelling the expiration. Our model reveals how the synchronized chaotic activity of this network reproduced the experimental data of the activity of the respiratory nerve centers both in healthy humans and the patients. We are able to reproduce fMRI signal after hemodynamic convolution of the simulated synchronized neural network. Besides, the respiratory neural network comprises the automatic brainstem and voluntary cortical network. The extension of the study to other important aspects as functional connectivity and Granger causality allow to better understand the communication within the network with the aim to develop new therapeutic strategies involving the modulation of brain oscillation (Hess et al., PLoS One 8:e75740, 2013; Yu et al., Hum. Brain Mapp. 37:2736–2754, 2016).

International audience; We investigate transport properties of an ensemble of particles moving inside an infinite periodic horizontal planar barrier billiard. A particle moves among bars and elastically reflects on them. The motion is a uniform translation along the bars' axis. When the tangent of the incidence angle, alpha , is fixed and rational, the second moment of the displacement along the orthogonal axis at time n , , is either bounded or asymptotic to K n2 , when n -->∞ . For irrational alpha , the collision map is ergodic and has a family of weakly mixing observables, the transport is not ballistic, and autocorrelation functions decay only in time average, but may not decay for a family of irrational alpha 's. An exhaustive numerical computation shows that the transport may be superdiffusive or subdiffusive with various rates or bounded strongly depending on the values of alpha . The variety of transport behaviors sounds reminiscent of well-known behavior of conservative systems. Considering then an ensemble of particles with nonfixed alpha , the system is nonergodic and certainly not mixing and has anomalous diffusion with self-similar space-time properties. However, we verified that such a system decomposes into ergodic subdynamics breaking self-similarity.

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 quantum description of the unstable systems is investigated starting from the Schrodinger equation and using Hamiltonian describing discrete levels interacting with a continuum. This approach is applied to kaons decay processes by using a simple Hamiltonian model. Then, CP-violation and decoherence properties are displayed and studied. (C) 2009 Elsevier B.V. All rights reserved.

We study the decay probability distribution and the survival probability of unstable quantum systems using an explicit formula of the spectral projections of the time operator in the statistical Liouville description for solvable Hamiltonians. We apply this formula to the one-level Friedrichs model to study the decay distribution of the excited decaying state under coupling with a continuum of degrees of freedom. Then we show that this formula eliminates the Zeno effect for short-time decay. We also show that the long-time asymptotic of the survival probability is a sum of an algebraically decaying term and an exponentially decaying one.

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.

Breathing involves a complex interplay between the brainstem automatic network and cortical voluntary command. How these brain regions communicate at rest or during inspiratory loading is unknown. This issue is crucial for several reasons: (i) increased respiratory loading is a major feature of several respiratory diseases, (ii) failure of the voluntary motor and cortical sensory processing drives is among the mechanisms that precede acute respiratory failure, (iii) several cerebral structures involved in responding to inspiratory loading participate in the perception of dyspnea, a distressing symptom in many disease. We studied functional connectivity and Granger causality of the respiratory network in controls and patients with chronic obstructive pulmonary disease (COPD), at rest and during inspiratory loading. Compared with those of controls, the motor cortex area of patients exhibited decreased connectivity with their contralateral counterparts and no connectivity with the brainstem. In the patients, the information flow was reversed at rest with the source of the network shifted from the medulla towards the motor cortex. During inspiratory loading, the system was overwhelmed and the motor cortex became the sink of the network. This major finding may help to understand why some patients with COPD are prone to acute respiratory failure. Network connectivity and causality were related to lung function and illness severity. We validated our connectivity and causality results with a mathematical model of neural network. Our findings suggest a new therapeutic strategy involving the modulation of brain activity to increase motor cortex functional connectivity and improve respiratory muscles performance in patients. Hum Brain Mapp 37:2736–2754, 2016. © 2016 The Authors Human Brain Mapping Published by Wiley Periodicals, Inc.

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.

WestudytheasymptoticlimitdistributionsofBirkhoffsumsS n ofasequenceofrandomvariablesofdynamicalsystemswithzeroentropyandLebesguespectrumtype.Adynamicalsystemofthisfamilyisaskewproductoveratranslationbyananglea.Thesequencehaslongmemoryeffects.Itcomesthatwhena=p isirrationaltheasymptoticbehaviorofthemomentsofthenormalizedsumsS n =f n dependsonthepropertiesofthecontinuousfractionexpansionofa.Inparticular,themomentsoforderk,EððS n =ffiffiffinpÞ k Þ,arefiniteandboundedwithrespectton whena=p hasboundedcontinuousfractionexpansion.Theconsequencesofthesepropertiesonthevalidityornotofthecentrallimittheoremarediscussed. 2003ElsevierB.V.Allrightsreserved. PACS:05.40;05.45Keywords:Weakchaos;Centrallimittheorem;Diffusioncoefficients 1. IntroductionKinetictheoryofgaseshaspermittedtoderivetransportprocesses,liketheBrownianmotion,fromtheLiouvilleequation.Manyworkshavebeendevotedtoderivediffusionprocessesfromchaoticdeterministicdynamicalsystems.OneofthepioneeringworkshasbeendonebySina€iifortheLorentzgasintwodimensionswhichisasystemofnon-interactingparticlesmovingwithconstantvelocityandbeingelasticallyreflectedfromperiodicallydistributedscattererswith

It is shown that a \({\mathbb{Z}}^2 \)-action on a Lebesgue space is intrinsically random (IR) iff it is a Kolmogorov action (K-action). As a consequence we obtain the fact that the \({\mathbb{Z}}^2 \)-action defined by the Lorentz gas is an IR-action and the \({\mathbb{Z}}^2 \)-action defined by the ideal gas is not an IR-action.

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.

In this paper, we propose a simple quantum model of the kaons decay providing an estimate of the CP symmetry violation parameter. We use the two-level Friedrich's Hamiltonian model to obtain a good quantitative agreement with the experimental estimate of the violation parameter for neutral kaons. A temporal wave-function approach, based on an analogy with spatial wave-functions, plays a crucial role in our model.

We show that the stationary processes associated to some two-cell partitions induced by the irrational rotation of the circle and by Kronecker Gaussian processes are uniquely determined by their two-dimensional marginals.

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.

Travelling waves (TW) solutions under the dynamics of one-dimensional infinite cellular automata (CA) exist abundantly in many cases. We show that for any permutative CA, unstable TW are dense in the space of configurations. Then, we consider the cases where the number of states is a prime number, so that the state space is a finite field K , and the automata rules are linear on K . We give an algorithm for the computation of the TW for any integer velocity of propagation larger than the interaction range. Then, we show that their wavelengths are characterized in terms of zeros of an associated family of polynomials over K and we describe the mathematical complexity of wavelengths distributions in various linear CA laws. We also obtain some exponential lower bound for the growth of the number of waves in terms of the velocity in rule 90.

Traveling wave solutions of cellular automata (CA) with two states and nearest neighbors interaction on one-dimensional (1-D) infinite lattice are computed. Space and time periods and the number of distinct waves have been computed for all representative rules, and each velocity ranging from 2 to 22. This computation shows a difference between spatially extended systems, generating only temporal chaos and those producing as well spatial complexity. In the first case wavelengths are simply related to the velocity of propagation and the dispersivity is an affine function, while in the second case (which coincides with Wolfram class 3), the dispersivity is multiform and its dependence on the velocities is highly random and discontinuous. This property is typical of space-time chaos in CA. (c) 1999 American Institute of Physics.

For any ergodic Markov chain ( X n ) on a finite state space K , we construct a class of ergodic stationary processes on K having the same two-dimensional marginal laws as ( X n ) but with distinct laws. The same construction also allows us to exhibit projections of some ergodic Markov chains that are not determined by their two-dimensional marginal laws.

The waves we study are the analog of traveling waves u(x,t) = H(x + vt), where x represents the space variable, t the time and v the velocity. Here we consider the simplest waves generated by the configurations (ui): u(i,t) = (θtu)i, i ∈ Z, ui ∈ {0, 1…,k − 1}, where θ is the deterministic dynamics of cellular automata on the configuration space. We study intial configurations which evolve as spatially periodic waves under rule 90. The wavelength distribution for different velocities of propagation is reduced to an eigenvalue problem of a class of matrices in finite fields. We show that the set of the wavelengths is unbounded.

Breathing is maintained and controlled by a network of automatic neurons in the brainstem that generate respiratory rhythm and receive regulatory inputs. Breathing complexity therefore arises from respiratory central pattern generators modulated by peripheral and supra-spinal inputs. Very little is known on the brainstem neural substrates underlying breathing complexity in humans. We used both experimental and theoretical approaches to decipher these mechanisms in healthy humans and patients with chronic obstructive pulmonary disease (COPD). COPD is the most frequent chronic lung disease in the general population mainly due to tobacco smoke. In patients, airflow obstruction associated with hyperinflation and respiratory muscles weakness are key factors contributing to load-capacity imbalance and hence increased respiratory drive. Unexpectedly, we found that the patients breathed with a higher level of complexity during inspiration and expiration than controls. Using functional magnetic resonance imaging (fMRI), we scanned the brain of the participants to analyze the activity of two small regions involved in respiratory rhythmogenesis, the rostral ventro-lateral (VL) medulla (pre-Botzinger complex) and the caudal VL pons (parafacial group). fMRI revealed in controls higher activity of the VL medulla suggesting active inspiration, while in patients higher activity of the VL pons suggesting active expiration. COPD patients reactivate the parafacial to sustain ventilation. These findings may be involved in the onset of respiratory failure when the neural network becomes overwhelmed by respiratory overload We show that central neural activity correlates with airflow complexity in healthy subjects and COPD patients, at rest and during inspiratory loading. We finally used a theoretical approach of respiratory rhythmogenesis that reproduces the kernel activity of neurons involved in the automatic breathing. The model reveals how a chaotic activity in neurons can contribute to chaos in airflow and reproduces key experimental fMRI findings.

We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy, 19p. 3 fig, Discrete and Continuous Dynamical Systems-B (Vol. 20, No. 10) December 2015

International audience; In this paper, we use the time-super-operator formalism and the two-level Friedrichs model to obtain a phenomenological model of mesons decay. Our approach provides a fairly good estimation of the CP-symmetry violation parameter in the case of K, B and D mesons.

In this paper, we study the one-level Friedrichs model by using the quantum time super-operator that predicts the excited state decay inside the continuum. Its survival probability decays exponentially in time.

We prove a generalization of the individual ergodic theorem of the information theory and we apply it to give a new proof of the Shereshevsky inequality connecting the metric entropy and Lyapunov exponents of dynamical systems generated by cellular automata.

Recent progress of the theory of dynamical systems and billiards sheds new light on the nonequilibrium statistical mechanics. Mixing, weak mixing and continuous spectrum are associated to relaxation to equilibrium via entropy increase. The properties of the relaxation time are reflected in the transport properties, which could be anomalous both in Sinai’ billiard with infinite horizon and in the barrier billiard. Numerical simulations are presented to corroborate these properties.

In order to illustrate the class of conservative dynamical systems for which a Boltzmann entropy can be obtained under finite coarse-graining [2], we consider dynamical systems defined by the shift transformation on K ℤ, where K is any finite set of integers. We give a class of non-Markovian invariant measures that verify the Chapman-Kolmogorov equation (equivalent to a Boltzmann entropy) for any positive stochastic matrix and that are ergodic but not weakly mixing.

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.

We consider particles transport in the Sinai billiard with infinite horizon. The simulation shows that the transport is superdiffusive in both continuous and discrete time. Also, it is shown that the moments do not converge to the Gaussian moments even in the logarithmically renormalized time scale, at least for a fairly long computational time. These results are discussed with respect to the existent rigorous theorems. Similar results are obtained for the stadium billiard.

The Hamiltonian Friedrichs model [1] describing the evolution of a two- level system coupled to a continuum is used in order to modelize the decay of the kaon states K-1, K-2. Using different cut- off functions of the continuous degrees of freedom, we show that this model leads to a CP violation that qualitatively fits with experimental data improving previous numerical estimates. We also discuss the relation of our model to other models of open systems.

In the present Letter, we study in the framework of the Friedrichs model the evolution of a two-level system coupled to a continuum. This unitary evolution possesses a non-unitary component with a non-Hermitian effective Hamiltonian. We show that this model is well adapted in order to describe kaon phenomenology (oscillation, regeneration) and leads to a CP violation, although in this case the prediction is not quantitatively quite satisfying. (c) 2006 Elsevier B.V. All rights reserved.

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.

Space-time directional Lyapunov exponents are introduced. They describe the maximal velocity of propagation to the right or to the left of fronts of perturbations in a frame moving with a given velocity. The continuity of these exponents as function of the velocity and an inequality relating them to the directional entropy is proved.

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

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration parameter $\alpha$. We show that when the parameter $\alpha /\pi$ is rational then all the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are unbounded with respect to n, while for irrational $\alpha /\pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n., Comment: To be published in J. Phys.A

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking., To be published in CHAOS

We use spectral projections of time operator in the Liouville space for simple quantum scattering systems in order to define a space of unstable particle states evolving under a contractive semi-group. This space includes purely exponentially decaying states that correspond to complex eigenvalues of this semi-group. The construction provides a probabilistic interpretation of the resonant states characterized in terms of the Hardy class.

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.