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. ẍ = (α+σ₁ξ₁)x+βẋ+ax ³+bx₂ẋ+ σ₂ξ₂, where, α and β are the bifurcation parameters, ξ₁and ξ₂ are white noise processes with intensities σ₁and σ₂, respectively. The existence of the extrem a 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 Itô 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.