随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。

THE HOPF BIFURCATION OF THE VAN DER POL-DUFFING OSCILLATOR IN THE PRESENCE OF BOTH PARAMETRIC AND THE EXTERNAL INFLUENCE

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.