# Principles for the design of billiards with nonvanishing Lyapunov exponents

TL;DR: In this paper, a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents was introduced, where the exponents have non-vanishing exponents.

Abstract: We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

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01 Jan 2005TL;DR: In this article, the authors discuss the existence and non-existence of caustics and periodic trajectories of billiards inside conics and quadrics, as well as in polygons.

Abstract: Motivation: Mechanics and optics Billiard in the circle and the square Billiard ball map and integral geometry Billiards inside conics and quadrics Existence and non-existence of caustics Periodic trajectories Billiards in polygons Chaotic billiards Dual billiards Bibliography Index.

403 citations

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TL;DR: A survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as numerical techniques developed for the computation of the maximal, of few and of all of them, can be found in this article.

Abstract: We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec (102), which pro- vides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been exten- sively used as an indicator of chaos, and the algorithm of the so-called standard method, developed by Benettin et al. (14), for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although we are mainly interested in finite-dimensional conservative systems, i.e., autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

259 citations

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TL;DR: In the semiclassical limit, quantum mechanics shows differences between classically integrable abd chaotic systems as discussed by the authors, and a review of recent developments in this field is presented in this paper.

259 citations

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TL;DR: A survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them, is presented.

Abstract: We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68}, which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

248 citations

### Cites background or methods from "Principles for the design of billia..."

...Finding an invariant family of cones (83) in TxS, which are mapped strictly into themselves by dxΦ t , guarantees that the values of the n− k largest LCEs are positive [142, 143]....

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...The method was suggested by Wojtkowski [142] and has been extensively applied for the study of chaotic billiards [142, 143, 43, 97] and geodesic flows [41, 42, 19]....

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01 Jan 1995TL;DR: In this paper, the dynamics of systems with hyperbolic properties are studied, and the relationship between the expanding properties of a map and its invariant measures in the Lebesgue measure class is examined.

Abstract: These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, µ), where f is a diffeomorphism of a Riemannian manifold and µ is an f-invariant Borel probability measure. After a brief review of abstract ergodic theory, Lyapunov exponents are introduced, and families of stable and unstable manifolds are constructed. Some relations between metric entropy, Lyapunov exponents and Hausdorff dimension are discussed. In the second half we address the following question: given a differentiable mapping, what are its natural invariant measures? We examine the relationship between the expanding properties of a map and its invariant measures in the Lebesgue measure class. These ideas are then applied to the construction of Sinai-Ruelle-Bowen measures for Axiom A attractors. The nonuniform case is discussed briefly, but its details are beyond the scope of these notes.

220 citations

##### References

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01 Jan 1974

TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

11,008 citations

01 Jan 1968

1,753 citations

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TL;DR: In this article, the authors define the ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure and the Bernoullian property of geodesic flows on closed Riemannian manifolds.

Abstract: CONTENTSPart I § 1. Introduction § 2. Prerequisites from ergodic theory § 3. Basic properties of the characteristic exponents of dynamical systems § 4. Properties of local stable manifoldsPart II § 5. The entropy of smooth dynamical systems § 6. "Measurable foliations". Description of the π-partition § 7. Ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure. The K-property § 8. The Bernoullian property § 9. Flows § 10. Geodesic flows on closed Riemannian manifolds without focal pointsReferences

1,393 citations

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TL;DR: In this paper, the authors consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals.

Abstract: In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. We prove that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.

914 citations