THE HOPF BIFURCATION OF THE VAN DER POL-DUFFING OSCILLATOR IN THE PRESENCE OF BOTH PARAMETRIC AND THE EXTERNAL INFLUENCE
随机干扰与随机参数激励联合作用下的Hopf分叉

In this paper, we study the Hopf bifurcations of the van der Pol-Duffing type nonlinear oscillator under the stochastic external influence and the internal parametric excitation. It is shown that the Hopf bifurcation point exhibited in the deterministic system is sensitive to the stochastic external influence and the parametric excitation. The addition of a small stochastic parametric excitation gives rise to a shift of the bifurcation point, which will be earlier than that of the ordinary system for a van der Pol-Duffing oscillator exhibiting the supercritical bifurcation. Also we have found that the non-zero response will appear in the system with the stochastic influence in the study of the steady movements, we have proved that the bifurcations of the movements are independent of the order of itself.