参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明，除参数激励的幅值和频率外，零解的稳定性只与原方程中线性项的系数和线性反馈有关，但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞，可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性，但选择合适的反馈增益和时滞，可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．

Response of a Parametrically Excited Duffing-Van der Pol Oscillator With Delayed Feedback

The dynamical behaviour of a parametrically excited Duffing_Van der Pol oscillator under linear_plus_nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow_flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciation choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original systems. It is demonstrated that nontrivial steady state responses may lose their stability by saddle_node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.