Phase diagram of the random frequency oscillator: The case of Ornstein-Uhlenbeck noise

TL;DR

This paper analyzes the stability of a stochastic oscillator with Ornstein-Uhlenbeck noise, deriving a phase diagram that characterizes noise-induced bifurcations and the transition between absorbing and steady states.

Contribution

It provides a quantitative phase diagram for a stochastic oscillator with finite memory noise, using multiple approximation methods to calculate the Lyapunov exponent.

Findings

Identification of the critical noise amplitude for bifurcation

Derivation of Lyapunov exponent expressions using various approximations

Quantitative phase diagram of the oscillator's stability regions

Abstract

We study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an Ornstein-Uhlenbeck noise. This system undergoes a noise-induced bifurcation when the amplitude of the noise grows. The critical curve, that separates the absorbing phase from an extended non-equilibrium steady state, corresponds to the vanishing of the Lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy. We derive various expressions for this Lyapunov exponent by using different approximation schemes. This allows us to determine quantitatively the phase diagram of the random parametric oscillator.