P-Bifurcations in the Noisy Duffing-van der Pol Equation

In this paper, we examine the stochastic version of the Duffing-van der Pol equation. As in [2], [8], [19], [16], we introduce both multiplicative and additive stochastic excitations to the original Duffingvan der Pol equation, i.e. $$ \ddot x = \left( {\alpha + \sigma _1 \xi _1 } \right)x + \beta x + a\dot x^3 + bx^2 \dot x + \sigma _2 \xi _2 , $$ where, α and α are the bifurcation parameters, ξ 1 and ξ 2 are white noise processes with intensities σ 1 and σ 2, 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 that which was 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 for 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 diagrams for the stochastic version of the 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 is also studied.