Hopf bifurcation with additive noise

TL;DR

This paper investigates how additive noise affects the dynamics of a system near a Hopf bifurcation, revealing three distinct phases of attractor behavior and the conditions for synchronization loss.

Contribution

It identifies three dynamical phases under noise, analyzes the effects of shear and stability, and conjectures a critical shear strength for loss of synchronization.

Findings

Three dynamical phases with different attractor types.

Synchronization depends on shear and stability conditions.

Critical shear strength leads to loss of synchronization and emergence of strange attractors.

Abstract

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent. We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (ampitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).