Noise-induced reentrant transition of the stochastic Duffing oscillator

TL;DR

This paper analytically derives the bifurcation diagram of a stochastic Duffing oscillator, revealing a noise-induced reentrant transition where noise stabilizes or destabilizes the fixed point depending on its amplitude.

Contribution

It provides the exact bifurcation diagram of the stochastic Duffing oscillator using Lyapunov exponents, highlighting a novel noise-induced reentrant transition phenomenon.

Findings

Identification of a noise amplitude interval that stabilizes the fixed point

Discovery of noise-induced reentrant transition in the oscillator

Analytical characterization of bifurcation behavior under stochastic influence

Abstract

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise belongs to a well-defined interval. Noisy oscillations are found outside that range, i.e., for both weaker and stronger noise.