Generalized Lyapunov exponents of the random harmonic oscillator: cumulant expansion approach

TL;DR

This paper employs cumulant expansion and importance-sampling Monte Carlo methods to estimate generalized Lyapunov exponents in random harmonic oscillators with various stochastic processes, revealing connections to many-particle systems.

Contribution

It introduces a cumulant expansion approach combined with importance sampling to compute Lyapunov exponents for different stochastic processes in harmonic oscillators.

Findings

Successfully estimates Lyapunov exponents for Gaussian, Ornstein-Uhlenbeck, and Poisson noise.

Addresses numerical challenges with importance-sampling Monte Carlo.

Links random oscillator behavior to many-particle interaction systems.

Abstract

The cumulant expansion is used to estimate generalized Lyapunov exponents of the random-frequency harmonic oscillator. Three stochastic processes are considered: Gaussian white noise, Ornstein-Uhlenbeck, and Poisson shot noise. In some cases, nontrivial numerical difficulties arise. These are mostly solved by implementing an appropriate importance-sampling Montecarlo scheme. We analyze the relation between random-frequency oscillators and many-particle systems with pairwise interactions like the Lennard-Jones gas.