求非线性动力系统周期解的切比雪夫多项式法

周期运动是一种在客观世界中普遍存在的运动形式，它与混沌运动之间存在十分密切的关系，因而具有很重要的研究价值。利用切比雪夫多项式的若干良好性质，对自治非线性动力系统进行分析，将状态矢量在主周期上展开为切比雪夫多项式的形式，从而将原问题转变为非线性代数方程组的求解问题，得出一种可以方便、迅速地获得周期轨道近似多项式表达式的方法。此方法不依赖于小参数假设，可以用于分析强非线性问题，而且对参数激励系统同样有效。在计算机条件允许时，对高维系统也能迅速、精确地得到其周期轨道的近似多项式表达式。以三维Rossler系统和五维非线性磁浮转子系统周期轨道的计算为例，通过与四阶Runge-Kutta数值积分结果比较，说明此方法的精确、高效性。

CHEBYSHEV POLYNOMIALS: A USEFUL METHOD TO GET THE PERIODIC SOLUTION OF NONLINEAR DYNAMICS

Except for the transient motion, the stable motion of object can be divided into periodic motion, quasi-periodic motion and Chaotic motion. Despite of the interesting of many researchers on chaos, the study of periodic motion is of great theoretical and practical importance in various fields of science and engineering, since periodic motion has a close relationship with chaos. Nevertheless, the theory of linear system can not be used directly to solve the nonlinear problems because of the speciality and complexity of it. When doing nonlinear analysis, the existing asymptotic methods have some of inherent shortcomings. For example, the perturbation method depends on the assumption of small parameters and the KBM method is very tedious in getting the high order asymptotic solution. In this paper, a new analysis technique in the study of periodical solution of nonlinear autonomous dynamical systems over one period is presented. The approach is based on the idea that the state vector and the parameters of the system can be expanded in terms of Chebyshev polynomials over the principal period. Such an expansion reduces the original problem to a set of nonlinear algebraic equations from which the solution in the interval of one period can be obtained. This new method does not need to be based on the assumption of small parameters. With the use of the good properties of Chebyshev polynomials, the method deals with linear and nonlinear systems in the same ways and is very convenience for the analysis of systems with periodical varying coefficients. The asymptotic polynomial solution of periodical orbit even for high dimension dynamical systems can be obtained very quickly. As illustration examples, the analytical results of Rossler equation and a five dimension magnetic levitation flex-rotator system are compared with those obtained via a Runge-Kutta integration algorithm. The results obtained in the two examples indicate that the suggested approach is extremely accurate and effective.