基于Hopf分岔的周期激励van der Pol-Duffing振荡器出现混沌的解析预测

周期激励ｖａｎｄｅｒＰｏｌ－Ｄｕｆｉｎｇ振荡器是能够呈现混沌行为的简单二阶非自治动态系统之一。本文利用谐波平衡技术和分岔理论获得了振荡器近似基谐波幅度发Ｈｏｐｆ分岔的曲线；探讨了Ｈｏｐｆ分岔与混沌出现的关系，首次剖析了ｖａｎｄｅｒＰｏｌ－Ｄｕｆｉｎｇ振荡器的Ｈｏｐｆ分岔结果是拟周期解，且拟周期解的崩溃出现混沌；数值摸拟计算出了ｖａｎｄｅｒＰｏｌ－Ｄｕｆｉｎｇ振荡器的混沌参数区域，结果表Ｈｏｐｆ分岔比Ｍｅｌｎｉｋｏｖ方法有更高的预测精度，可有效地用于预测混沌可能出现的参数区域。

Analytical Prediction of Chaotic Behaviors in the van der Pol-Duffing Oscillator Using Hopf Bifurcations

The periodically excited van der Pol-Duffing oscillator is one of the simplest secondorder nonautonomous dynamical systems having chaotic behaviors.In this paper,we first derive Hopfbifurcation condition of the amplitude of approximate fundamental harmonics and then discuss the relations between the Hopf bifurcation and chaos appearing in the oscillator.It is analysed in the first time that the Hoprbifurcation results in the quasiperiodic solutions and the breakdown of the solutions generate the chaotic solutions for the van der Pol-Duffing oscillator.Finally,we depict the parametet ranges of the oscillator having chaotic behaviors.The results show that the Hopf bifurcation has much higher prediction accuracy than Melnikov method does and it can effectively be used to predict the parameter ranges of occurrence of the chaos in the van der Pol-Duffing oscillator.