求解非线性动力系统周期解推广的打靶法

Generalized Shooting Method for Determining the Periodic Orbit of the Nonlinear Dynamics System

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度 ,将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中 ,然后对传统打靶法进行改造 ,将周期也作为一个参数一起参入打靶法的迭代过程 ,从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求 ,可以用于分析强非线性系统 ,而且对参数激励系统同样有效 ,对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维R¨ossler系统和八维非线性柔性转子 轴承系统的周期轨道和周期进行了求解 ,通过与四阶Runge Kutta数值积分结果比较 。

In this paper, a new generalized shooting method for determining the periodic orbit and its period of the nonlinear dynamic system is presented by rebuilding the traditional shooting method. At first, by changing the time scale the period of the periodic orbit of the nonlinear system is drawn into the governing equation of this system explicitly, then, the period as a parameter takes part into the iteration procedure of the shooting method together. The periodic orbit and its period of the system can be determined rapidly and precisely. The method needn't the rigor of the initial iteration conditions and it can be used for analyzing the forced nonlinear systems and also the parametric excited systems. As an illustrative example, the computation results of Rossler equation and an eight-dimensional nonlinear flex-rotor system are compared with those obtained via the Runge-Kutta integration algorithm. The validity of this method is verified by the results obtained in two examples.