Synchronization Properties of Random Piecewise Isometries

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables—asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than 40 years ago.     

Lyapunov exponents and kolmogorov-sinai entropy for a high-dimensional convex billiard

We compute the Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a self-bound N-body system that is realized as a convex billiard. This system exhibits truly high-dimensional chaos, and 2N-4 Lyapunov exponents are found to be positive. The KS entropy increases linearly with the numbers of particles. We examine the chaos generating defocusing mechanism and investigate how high-dimensional chaos develops in this system with no dispersing elements.     

Nowhere Dispersing 3D Billiards with Non-vanishing Lyapunov Exponents

We describe a class of 3-dimensional regions with focusing components that generate a billiard system with non-vanishing Lyapunov exponents. To do this we answer affirmatively the long standing question whether or not the chaotic motion caused by defocusing can be produced in more than two dimensions. Received: 14 February 1996 / Accepted: 21 March 1997     

Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give a lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constant curvature. Using these conditions, we construct large classes of magnetic billiard tables with positive Lyapunov exponents on the plane, on the sphere and on the hyperbolic plane. Received: 7 April 2000 / Accepted: 19 September 2000     

Hyperbolic Billiards on Surfaces of Constant Curvature

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with positive Lyapunov exponents on the sphere and on the hyperbolic plane. Received: 26 January 1999 / Accepted: 17 May 1999     

The dynamical entropy of the quantum Arnold cat map

We present a rigorous computation of the dynamical entropyh of the quantum Arnold cat map. This map, which describes a flow on the noncommutative two-dimensional torus, is a simple example of a quantum dynamical system with optimal mixing properties, characterized by Lyapunov exponents ± 1n ⁺, ⁺ > 1. We show that, for all values of the quantum deformation parameter,h coincides with the positive Lyapunov exponent of the dynamics.     

Long-Time-Tail Effects on Lyapunov Exponents of a Random, Two-Dimensional Field-Driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, nonoverlapping hard-disk scatterers in a thermostatted electric field, . The low-density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz–Boltzmann equation. In this paper we develop a method to extend theses results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of the Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to nonanalytic, field-dependent contributions to both the positive and negative Lyapunov exponents which are of the form ~ ²ln~, where ~ is a dimensionless parameter proportional to the strength of the applied field. We show that these nonanalytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value due to the presence of the thermostatted field, and that the collision frequency also contains such nonanalytic terms.     

Numerical exploration of a family of strictly convex billiards with boundary of classC 2

We are interested in the possible existence of strictly convex ergodic billiards. Such billiards are searched for by means of numerical investigation. The boundary of a billiard is built with four arcs of classC . Adjacent arcs have equal curvatures at connecting points. The surface of section of the billiards is explored. It seems as if symmetric billiards always have invariant curves (islands). Asymmetric billiards have been found which look ergodic. They are built with an arc of an ellipse, two arcs of circles, and one-half of a Descartes oval.     

The Metric Entropy of Endomorphisms

We prove that, for a C ² non-invertible but non-degenerate map on a compact Riemannian manifold without boundary, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents. This generalizes Ledrappier-Young’s entropy formula [5] (for negative Lyapunov exponents of diffeomorphisms) to the case of endomorphisms. This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).     

Linearly stable orbits in 3 dimensional billiards

We construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart. It shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3 dimensions.Supported in part by the NSF Grant DMS-8807077 and the Sloan Foundation     

Polygonal Approaches to the Circular Billiard: the Lyapunov Exponent

The Lyapunov exponents for three polygonal approaches to the circular billiard quasiperiodic approach, random approach and equilateral approach are calculated, the chaotic behavior in polygonal billiards is discussed. The role of singularity presented by vertex angles is quantitatively discussed. It is found that for the equilateral approach, the Lyapunov exponents vary with the side number N according to a Poisson law, i.e., λ(N) = aN exp(-βN).     

Invariants for smooth conjugacy of hyperbolic dynamical systems. IV

We show that if twoC transitive Anosov flows in a three-dimensional manifold are topologically conjugate and the Lyapunov exponents on corresponding periodic orbits agree, then the conjugating homeomorphism isC .Partially supported by NSF grant # DMS 85-04984     

Coupling sensitivity of chaos and the Lyapunov dimension: The case of coupled two-dimensional maps

We numerically investigate the response of spectra of the Lyapunov exponents in chaotic two-dimensional (2-d) maps to perturbations generated by coupling two such maps. The results reveal the coupling sensitivity of chaos, which was discovered previously in coupled 1-d maps, with a number of features some of which are inherent in higher-dimensional systems. In particular, the Lyapunov dimension of a strange attractor is also found to be strongly sensitive to coupling perturbations. Our results suggest a new quantity characterizing chaos, χ_coup, which measures the strength of the coupling sensitivity.     

Ergodic properties of plane billiards with symmetric potentials

We study chaotic behaviour of the motion of a particle moving like in a billiard table outside some disks where a symmetric potential acts. Quadratic forms introduced in (Markarian, 1988) to study non-vanishing Lyapunov exponents are used.     

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of 3-dimensional billiards. Interestingly, we find that for some track billiards, the mechanism generating hyperbolicity is not the defocusing one, which requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.     

Smoothness of Invariant Manifolds for Nonautonomous Equations

For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C^1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.     

Local Lyapunov exponents for spatiotemporal chaos

Local Lyapunov exponents are proposed for characterization of perturbations in distributed dynamical systems with chaotic behavior. Their relation to usual and velocity-dependent exponents is discussed. Local Lyapunov exponents are analytically calculated for coupled map lattices using random field approximation. Boundary Lyapunov exponents describing reflection of perturbations at boundaries are also introduced and calculated.     

Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

We study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices without any particular structure the generalized Lyapunov exponents become equal in this limit and the value of one of the generalized Lyapunov exponents is obtained by simple arguments. On the contrary, for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized Lyapunov exponents remains different forD , indicating that high dimensionality cannot always destroy intermittency.     

Determining Lyapunov exponents from a time series

We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.     

A Functional Analytic Approach to Perturbations of the Lorentz Gas

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map of the classical Lorentz gas are flexible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic reflections with kicks and slips at the boundaries of the scatterers; and random perturbations comprised of these and possibly other classes of maps. The spectra and spectral projections of the transfer operators are shown to vary continuously with such perturbations so that the spectral gap enjoyed by the classical billiard persists and important limit theorems follow.