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Estimating Lyapunov exponents in billiards.

George Datseris¹, Lukas Hupe¹, Ragnar Fleischmann¹

¹Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany and Faculty of Physics, Georg-August-Universität Göttingen, 37077 Göttingen, Germany.

Chaos (Woodbury, N.Y.)

|October 3, 2019

Summary

The Lyapunov exponent in chaotic dynamical systems is inversely proportional to the chaotic phase space volume. This study extends calculations to include magnetic fields, offering new software for dynamical billiards research.

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Area of Science:

Physics
Statistical Mechanics
Dynamical Systems

Background:

Dynamical billiards are fundamental models for chaotic Hamiltonian systems.
Understanding chaotic dynamics is crucial for various physics applications.
Lyapunov exponents quantify the rate of chaos in dynamical systems.

Purpose of the Study:

To investigate the relationship between Lyapunov exponents and phase space volume in dynamical billiards.
To explore the influence of external parameters, specifically magnetic fields, on chaotic dynamics.
To extend existing formalisms for calculating Lyapunov exponents.

Main Methods:

Analysis of phase space volume arguments for Lyapunov exponent estimation.
Extension of the Dellago, Posch, and Hoover formalism.
Implementation of a software tool for Lyapunov exponent calculation in billiards with magnetic fields.

Main Results:

A clear inverse proportionality was found between the leading Lyapunov exponent and the chaotic phase space volume across diverse billiards.
The study successfully extended the theoretical framework to incorporate external magnetic fields.
A functional software implementation for calculating Lyapunov exponents under magnetic fields was developed.

Conclusions:

The inverse relationship between Lyapunov exponent and chaotic phase space volume appears to be a general principle in dynamical billiards.
The extended formalism and software provide valuable tools for studying chaotic Hamiltonian systems, particularly under magnetic field influence.