Estimating Lyapunov exponents in billiards.

George Datseris,Ragnar Fleischmann,Lukas Hupe

doi:10.1063/1.5099446

Abstract

Dynamical billiards are paradigmatic examples of chaotic Hamiltonian dynamical systems with widespread applications in physics. We study how well their Lyapunov exponent, characterizing the chaotic dynamics, and its dependence on external parameters can be estimated from phase space volume arguments, with emphasis on billiards with mixed regular and chaotic phase spaces. We show that in the very diverse billiards considered here, the leading contribution to the Lyapunov exponent is inversely proportional to the chaotic phase space volume and subsequently discuss the generality of this relationship. We also extend the well established formalism by Dellago, Posch, and Hoover to calculate the Lyapunov exponents of billiards to include external magnetic fields and provide a software on its implementation.

Highlights

Dynamical billiards are a well-studied class of dynamical systems, having applications in many di erent elds of physics
We study how well their Lyapunov exponent, characterizing the chaotic dynamics, and its dependence on external parameters can be estimated from phase space volume arguments, with emphasis on billiards with mixed regular and chaotic phase spaces
We have examined the value of the Lyapunov exponent λ in chaotic billiards

Summary

Introduction

Dynamical billiards are a well-studied class of dynamical systems, having applications in many di erent elds of physics. Besides playing a prominent role in ergodic theory,[1,2,3] billiards are important example systems for understanding quantum chaos,[4,5] with practical applications, e.g., in modeling optical microresonators for lasers[6,7] and room acoustics.[8] Billiard models have been successful in helping to understand transport properties of electronic nanostructures such as quantum dots and antidot superlattices.[9–18]. A billiard consists of a nite (or periodic) domain in which a point particle performs free ight with a unit velocity.

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