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 Duffi ng-Van der Pol equation. As in [3], [4] [5], [6], we shall introduce a multiplicative and an additive stochastic excitation in our case, i.e . (1) x = (alpha + sigma(1) xi(1)) x + beta x + ax(3) + bx(2)x +sigma( 2) xi(2) where, alpha and beta are the bifurcation parameters, xi(1) a nd xi(2) are white noise processes with intensities sigma(1) and sigma (2) respectively. The method used in this paper is essentially the sam e as what has been used in [4]. We first reduce system (1) to a weakly perturbed conservative system by intruducing an appropriate rescaling . The corresponding unperturbed system is then studied. The problem of the existence of the extrema of the probability density function is p resented for the stochastic system. Second, by transforming the variab les and performing stochastic averaging, we obtain a one-dimensional I to equation. The probability density function is found by solving the Fokker-Planck equation. The extrema of the probability density functio n are then culculated so we can study the so called P-bifurcation for the Duffing-van der Pol oscillator with a = -1.0, b = -1.0 over the wh ole (alpha, beta)-plane by making use of the system Hamiltonian.