A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

TL;DR

This paper investigates how non-Gaussian Levy noise influences stochastic bifurcations in dynamical systems, revealing complex bifurcation patterns that differ significantly from Gaussian noise and deterministic cases.

Contribution

It introduces the analysis of stochastic bifurcations under non-Gaussian Levy noise, highlighting novel phenomena in the most probable phase portraits.

Findings

Multiple pitchfork bifurcations occur depending on Levy noise parameters.

Non-Gaussian Levy noise induces more complex bifurcation scenarios.

Gaussian noise results in only one bifurcation, similar to deterministic systems.

Abstract

We study stochastic bifurcation for a system under multiplicative stable Levy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Levy noise varies, there is either one, two or none backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Levy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.