Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.Comment: Final version for J. Stat. Phys., 18 pages, 4 figure

Trajectory versus probability density entropy

We study the problem of entropy increase of the Bernoulli-shift map without recourse to the concept of trajectory and we discuss whether, and under which conditions if it does, the distribution density entropy coincides with the Kolmogorov-Sinai entropy, namely, with the trajectory entropy.Comment: 24 page

Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.Comment: 30 pages, 18 figure

Chaos in Traveling Waves of Lattice Systems of Unbounded Media

. We describe coupled map lattices (CML) of unbounded media corresponding to some well-known evolution partial differential equations (including reaction-diffusion equation, KuramotoSivashinsky, Swift-Hohenberg and Ginzburg-Landau equation). Following Kaneko we view CML also as phenomenological models of the medium and present the dynamical system approach to study the global behavior of solutions of CML. In particular, we establish spatio-temporal chaos associated with the set of traveling wave solutions of CML as well as describe the dynamics of the evolution operator on this set. Several examples are given to illustrate the appearance of Smale horseshoes and the presence of the dynamics of Morse-Smale type. Introduction In this paper we deal with lattice dynamical systems of an unbounded medium. They are also called coupled map lattices (or briefly CML) and are described by the equation of the form u j (n + 1) = f(u j (n)) + "g j (fu i (n)g ji\Gammajjs ): (0.1) Here n 2 Zis the di..