On the effect of multiplicative noise in a supercritical pitchfork bifurcation

TL;DR

This paper investigates how multiplicative noise influences the stability of a supercritical pitchfork bifurcation, showing that noise can cause the system to tend to zero even when deterministic conditions suggest stability.

Contribution

It provides a detailed analysis of the effects of multiplicative noise on a classical bifurcation, revealing conditions under which stability is compromised.

Findings

Multiplicative noise can destabilize the positive branch of the bifurcation.

The system tends to zero almost surely under certain noise intensities.

Small noise levels lead to a meta-stable state before eventual extinction.

Abstract

The most important characteristic of {\em multiplicative noise} is that its effects of system's dynamics depends on the recent system's state. Consideration of multiplicative noise on self-referential systems including biological and economical systems therefore is of importance. In this note we study an elementary example. While in a deterministic super critical pitchfork bifurcation with positive bifurcation parameter $\lambda$ the positive branch $\sqrt{\lambda}$ is stable, multiplicative white noise $\lambda_t ={\lambda} + \sigma \zeta_t$ on the unique parameter reduces stability in that the system's state tends to 0 almost surely, even for ${\lambda}>0$, while for 'small' noise $\sigma < \sqrt{2 \lambda}$ the point $\sqrt{\lambda-\sigma^2/2}$ is a meta-stable state. In this case, correspondingly, the system will 'die out', i.e. $X_t \to 0$ within finite time.