Estimating Lyapunov exponents in billiards

The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi et al. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955]. We introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi et al. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.

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“…The calculation scheme for Lyapunov exponents is based on the methods described in [53,54]. For given equations of motions˙ (t ) = F( (t )),…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Repeating the steps in [54], after a reflection of the ith particle the new deviations δx i and δ p i can be expressed by the deviations δx i and δ p i before the collision with the wall:…”

Section: Acknowledgmentsmentioning

confidence: 99%

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices

et al. 2021

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“…The speed of this exponential divergence is characterized by the (maximum) Lyapunov exponent λ . Here, we adopt the method of the “billiard map” in terms of the collisions of trajectories on the outer boundary, instead of the “billiard flow” with continuous time ( 55 ). In this way, the Lyapunov exponent is defined as

where

is the distance of the two chosen trajectories in phase space at the i th bounce.…”

Section: Resultsmentioning

confidence: 99%

Light chaotic dynamics in the transformation from curved to flat surfaces

Xu¹,

Dana²,

Wang³

et al. 2022

Proc. Natl. Acad. Sci. U.S.A.

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Significance Exploring light dynamics and chaos on curved surfaces is one of the new challenges met in modern cosmology. To address this question and investigate chaotic dynamics of light on three-dimensional curved surfaces, we consider their connection under a conformal transformation with flat billiards with spatially varying refractive index. Through this connection, we demonstrate that these two systems share the same dynamics; the complexity of the problem simplifies, and well-known tools originating from chaos analysis in planar billiards can be used to explore chaotic dynamics on curved space. We discover that the degree of chaos is fully controlled by a single curvature-related geometrical parameter, providing a degree of freedom in chaos engineering, as well as potentialities to design nonuniform table billiards/cavities/resonators.

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“…Repeating the steps in [47], after a reflection of the i-th particle the new deviations δx i and δp i can be expressed by the deviations δx i and δp i before the collision with the wall:…”

Section: Appendix A: Numerical Integrationmentioning

confidence: 99%

“…The calculation scheme for Lyapunov exponents is based on the methods described in [46,47]. For given equations of motions…”

Section: Appendix B: Calculation Of Lyapunov Exponentsmentioning

confidence: 99%

Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices

Hahn,

Urbina,

Richter

et al. 2020

Preprint

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The study of non-linear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi, Pasta, Ulam and Tsingou (FPUT). We introduce a new family of such systems which consist of chains of N harmonically coupled particles with the non-linearity introduced by confining the motion of each individual particle to a box/stadium with hard walls. The stadia are arranged on a one dimensional lattice but they individually do not have to be one dimensional thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N . Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh-Gordon theory, we do not see any evidence for the kind of non-ergodicity famously seen in the FPUT work. Finally, we examine the chain with particles confined to two dimensional stadia where the individual stadium is already chaotic, and find a much more chaotic phase space at small system sizes.

show abstract

Estimating Lyapunov exponents in billiards

Cited by 7 publications

References 36 publications

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices

Light chaotic dynamics in the transformation from curved to flat surfaces

Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices

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