Stochastic perturbation of integrable systems: a window to weakly chaotic systems

TL;DR

This paper investigates how additive noise induces chaos in integrable Hamiltonian systems, providing analytic expressions for Lyapunov exponents and demonstrating how stochastic models can approximate weakly chaotic deterministic systems.

Contribution

It derives Lyapunov exponent formulas for noisy integrable systems and shows how stochastic perturbations can model weak chaos in near-integrable systems.

Findings

Additive noise causes chaos in integrable systems at any amplitude.

Derived explicit formulas for Lyapunov exponents under white and colored noise.

Stochastic models can qualitatively replicate weak chaos in deterministic systems.

Abstract

Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor's diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being `multi-resonance' -- by nature well beyond the Kolmogorov-Arnold-Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.