Stochastic Hopf bifurcation: An example

Abstract

The stochastic Hopf bifurcation behavior of the noisy Duffing-van der Pol oscillator x ̈ = (α + σW)x + β x ̇ − x 2 x ̇ − x 3 , (A) is studied numerically. (α, β are bifurcation parameters, W is white noise, and σ is an intensity parameter.) When the qualitative change of the stationary solution of the Fokker-Planck equation is considered, the stochastic Hopf bifurcation appears as a change from a Dirac measure to a crater-like density. Unfortunately, this behavior is not related to the sample stability of the trivial solution zero. To capture all the stochastic dynamics of the equation, it is necessary to investigate the change of stability of invariant measures and the occurrence of new invariant measures for the generated random dynamical system.