Lyapunov exponent of the random frequency oscillator: cumulant expansion approach

TL;DR

This paper analytically computes the generalized Lyapunov exponent of a one-dimensional harmonic oscillator with random frequency using cumulant expansion, addressing numerical challenges in strong intermittency regimes.

Contribution

It introduces an analytical method to calculate the generalized Lyapunov exponent for a stochastic oscillator, extending understanding of intermittency effects.

Findings

Analytical expression for $oldsymbol{ ext{lambda}^ ext{star}}$ using cumulant expansion up to fourth order.

Identification of numerical difficulties in computing $oldsymbol{ ext{lambda}^ ext{star}}$ under strong intermittency.

Connections made between the oscillator problem and many-body systems with smooth interactions.

Abstract

We consider a one-dimensional harmonic oscillator with a random frequency, focusing on both the standard and the generalized Lyapunov exponents, $\lambda$ and $\lambda^\star$ respectively. We discuss the numerical difficulties that arise in the numerical calculation of $\lambda^\star$ in the case of strong intermittency. When the frequency corresponds to a Ornstein-Uhlenbeck process, we compute analytically $\lambda^\star$ by using a cumulant expansion including up to the fourth order. Connections with the problem of finding an analytical estimate for the largest Lyapunov exponent of a many-body system with smooth interactions are discussed.