利用雅可比椭圆函数计算强非线性杜芬振子的周期响应

Calculation for Periodic Responses of Strongly Nonlinear Duffing Oscillator Using Jacobian Elliptic Functions

应用雅可比椭圆函数及均值方法计算求解受到简谐外激扰的强非线性杜芬振荡系统＋ａυ＋γυ３＝ε（－βυ＋ＦｃｏｓΩＴ）的稳态周期响应。首先利用雅可比椭圆函数给出无扰动系统的周期解。然后，采用对无扰动系统周期解进行扰动的方法，求扰动系统的周期解。在这个过程中，采用均值方法对问题进行了简化。并通过对所得解的讨论与分析，最终得到原问题的稳态周期响应。实例验证的结果表明，我们所介绍的方法是成功的。并可应用于求解其它强非线性系统的周期响应。

In this paper, we consider a strongly nonlinear Duffing oscillator which has the following form v + αv +γv3 = ε( -βυ+ FcosΩT) . At first, we consider the corresponding unperturbed system with ε = 0 and use Jacobian elliptic functions to express its periodic solutions. We then find the periodic solutions of the perturbed system. In this process, average method is applied to simplify the related system. Finally, example calculations are presented to verify our approach. Results from our method are compared with those from numerical integration of the original equation, and quite good agreement between them suggests the effectiveness of our approach. Our method can be extended to the investigation of strongly nonlinear systems of other types.