STOCHASTIC HOPF-BIFURCATION - AN EXAMPLE

The stochastic Hopf bifurcation behavior of the noisy Duffing-van der Pol oscillator x = (alpha + sigma W)x + beta x - x(2)x - x(3), (A) is studied numerically. (alpha, beta are bifurcation parameters, W is whi te noise, and sigma is an intensity parameter.) When the qualitative c hange of the stationary solution of the Fokker-Planck equation is cons idered, the stochastic Hopf bifurcation appears as a change from a Dir ac measure to a crater-like density. Unfortunately, this behavior is n ot related to the sample stability of the trivial solution zero. To ca pture all the stochastic dynamics of the equation, it is necessary to investigate the change of stability of invariant measures and the occu rrence of new invariant measures for the generated random dynamical sy stem. Copyright (C) 1996 Elsevier Science Ltd.