以单摆系统为例，将Wiggins提出的Hopf分岔条件进行了具体计算，从理论上获得了单摆系统发生次谐分岔的方式，并用数值模拟方法验证了结果的正确性.将Melnikov方法推广到二频驱动情况，由二频驱动单摆的Hopf分岔条件得出存在奇-奇阶次谐分岔和奇-偶阶次谐分岔，数值模拟结果与理论分析一致，表明Melnikov方法可以处理多频驱动系统的Hopf分岔问题.

Based on the necessary and sufficient conditions for existing of Hopf bifurcation, the ways to generate subharmonic bifurcation for single pendulum system is provided. Furthermore, this paper presents the application of Melnikov method in two-frequency drives single pendulum system. Therefore, the Hopf bifurcation and its conditions are discussed in detail and its application illustrates that this method solves many problems of Hopf bifurcation in multi-frequency drives single pendulum system.