Maurice Courbage

On Intrinsic Randomness of Dynamical Systems

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 beintrinsically 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-calledK systems andK 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 tononisomorphic K flows are necessarily non-isomorphic.

On the equivalence between Bernoulli dynamical systems and stochastic Markov processes

We extend to Bernoulli systems the explicit construction (elaborated previously for the baker transformation) of non-unitary, invertible transformations Λ, which associate Markovian processes admitting an H-theorem with the unitary dynamical group, through a similarity relation. We characterize the symmetries of the Bernoulli system as well as those of the associated Markov processes and provide examples of symmetry breaking under the passage, through a Λ transformation, from Bernoulli systems to stochastic Markov processes.

Chaotic oscillations in a map-based model of neural activity

We propose a discrete time dynamical system (a map) as a phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find conditions under which this map has an invariant region on the phase plane, containing a chaotic attractor. This attractor creates chaotic spiking-bursting oscillations of the model. We also show various regimes of other neural activities (subthreshold oscillations, phasic spiking, etc.) derived from the proposed model.

On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics

It is shown that time and entropy operators may exist as superoperators in the framework of the Liouville space provided that the Hamiltonian has an unbounded absolutely continuous spectrum. In this case the Liouville operator has uniform infinite multiplicity and thus the time operator may exist. A general proof of the Heisenberg uncertainty relation between time and energy is derived from the existence of this time operator.

MAP BASED MODELS IN NEURODYNAMICS

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.

Space-Time Directional Lyapunov Exponents for Cellular Automata

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.

Neural Mechanisms Underlying Breathing Complexity

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-Bötzinger 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.

Quantum-mechanical decay laws in the neutral kaons

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 K1, K2. 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.

Markov Evolution and H-Theorem under Finite Coarse Graining in Conservative Dynamical Systems

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.

Chaotic dynamics and transport in classical and quantum systems

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