P-Bifurcations in the Stochastic Version of the Duffing-van der Pol Equation

In this paper, we shall re-examine the stochastic version of the Duffing-van der Pol equation. As in [2], [9], [19], [22], we shall introduce a multiplicative and an additive stochastic excitation in our case, i.e.x¨=(α+σ1ξ1)x+βx˙+ax3+bx2x˙+σ2ξ2, where, α and β are the bifurcation parameters, ζ1 and ζ2 are white noise processes with intensities σ1 and σ2, respectively. The existence of the extrema of the probability density function is presented for the stochastic system. The method used in this paper is essentially the same as what has been used in [19]. We first reduce the above system to a weakly perturbed conservative system by introducing an appropriate scaling. The corresponding unperturbed system is then studied. Subsequently, by transforming the variables and performing stochastic averaging, we obtain a one-dimensional Itv equation of the Hamiltonian H. The probability density function is found by solving the Fokker-Planck equation. The extrema of the probability density function are then calculated in order to study the so-called P-Bifurcation. The bifurcation pictures for the stochastic version Duffing-van der Pol oscillator with a = −1.0, b = −1.0 over the whole (α, β)-plane are given. The related mean exit time problem has also been studied.