Chaos and Noise

TL;DR

This paper explores how chaos and noise influence simple dynamical systems, proposing methods to define Lyapunov exponents in noise-driven models derived from Hamiltonian systems.

Contribution

It introduces a framework for defining Lyapunov exponents in noise-affected dynamical systems based on Hamiltonian models, bridging chaos theory and stochastic analysis.

Findings

Lyapunov exponents can be properly defined in noise-driven systems.

Reduced equations from coupled systems can have stochastic interpretations.

The approach applies to a class of Hamiltonian-derived models.

Abstract

Simple dynamical systems -- with a small number of degrees of freedom -- can behave in a complex manner due to the presence of chaos. Such systems are most often (idealized) limiting cases of more realistic situations. Isolating a small number of dynamical degrees of freedom in a realistically coupled system generically yields reduced equations with terms that can have a stochastic interpretation. In situations where both noise and chaos can potentially exist, it is not immediately obvious how Lyapunov exponents, key to characterizing chaos, should be properly defined. In this paper, we show how to do this in a class of well-defined noise-driven dynamical systems, derived from an underlying Hamiltonian model.