Noise induced Hopf bifurcation

TL;DR

This paper investigates how stochastic noise influences Hopf bifurcations, showing that noise can both destroy and induce resonance in limit cycles within systems like the Lorenz model.

Contribution

It provides a theoretical analysis of noise effects on Hopf bifurcations and applies these results to the Lorenz system, revealing conditions for noise-induced resonance.

Findings

Noise can destroy limit cycles depending on variable scales.

Resonance occurs when fast variations involve coupled variables.

Departure from equilibrium steady state can be noise-dependent.

Abstract

We consider effect of stochastic sources upon self-organization process being initiated with creation of the limit cycle induced by the Hopf bifurcation. General relations obtained are applied to the stochastic Lorenz system to show that departure from equilibrium steady state can destroy the limit cycle in dependence of relation between characteristic scales of temporal variation of principle variables. Noise induced resonance related to the limit cycle is found to appear if the fastest variations displays a principle variable, which is coupled with two different degrees of freedom or more.