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69
The Lyapunov Characteristic Exponents and their
 Computation, Lect. Notes Phys
, 2010
"... For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present ..."
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Cited by 29 (2 self)
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For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. 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 [99], 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. [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. 1
2008] “Improved estimates for correlations in billiards
 Comm. Math. Phys
"... Abstract We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich's stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were suboptim ..."
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Cited by 15 (1 self)
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Abstract We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich's stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were suboptimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.
Geometrical structure of Laplacian eigenfunctions
, 2013
"... We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and com ..."
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Cited by 11 (3 self)
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We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries
"... We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor λ, of the incident angle. These pinball billiards interpolate between a onedimensional map when λ = 0 and the classical Hamiltonian case of elastic coll ..."
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Cited by 10 (1 self)
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We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor λ, of the incident angle. These pinball billiards interpolate between a onedimensional map when λ = 0 and the classical Hamiltonian case of elastic collisions when λ = 1. Forall λ < 1, the dynamics is dissipative, and thus gives rise to attractors, which may be periodic or chaotic. Motivated by recent rigorous results of Markarian,Pujals and Sambarino [MPS08], we numerically investigate and characterise the bifurcations of the resulting attractors as the contraction parameter is varied. Some billiards exhibit only periodic attractors, some only chaotic attractors, and others have coexistence of the two types.
Spectral analysis of the transfer operator for the Lorentz gas.
 J. Mod. Dyn.,
, 2011
"... Abstract We study the billiard map associated with both the finite and infinite horizon Lorentz gases having smooth scatterers with strictly positive curvature. We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasicompact. The mixing prop ..."
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Cited by 9 (3 self)
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Abstract We study the billiard map associated with both the finite and infinite horizon Lorentz gases having smooth scatterers with strictly positive curvature. We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasicompact. The mixing properties of the billiard map then imply the existence of a spectral gap and related statistical properties such as exponential decay of correlations and the central limit theorem. Finer statistical properties of the map such as the identification of Ruelle resonances, large deviation estimates and an almostsure invariance principle follow immediately once the spectral picture is established.
New horizons in multidimensional diffusion: the Lorentz gas and the Riemann hypothesis
 J. Stat. Phys
"... The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor t are ..."
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Cited by 8 (1 self)
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The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor t are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by t ln t, with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d = 6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions. 1
Hyperbolic billiards with nearly flat focusing boundaries
, 2007
"... The standard Wojtkowski–Markarian–Donnay–Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is ..."
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Cited by 6 (1 self)
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The standard Wojtkowski–Markarian–Donnay–Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the socalled defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstardard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms. Mathematics Subject Classification: 37D50, 37D25, 37A25. 1
Explaining ThermodynamicLike Behavior in Terms of Epsilon Ergodicity*
"... Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections in particular arguments based on the KolmogorovArnoldMos ..."
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Cited by 6 (3 self)
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Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections in particular arguments based on the KolmogorovArnoldMoser theorem and the MarkusMeyer theorem are beside the point. We then argue that something close to Boltzmann’s proposal is true: gases behave thermodynamiclike if they are epsilonergodic, that is, ergodic on the phase space except for a small region of measure epsilon. This answer is promising because there is evidence that relevant systems are epsilonergodic. 1. Introduction. Consider