具有多个极限环非线性动力系统的解析近似

应用一种新的解析方法------同伦分析法，研究了一种具有多个 极限环的Rayleigh振子问题. 与所有其他传统方法不同，该方法不依赖于小参数， 且提供了一个简便的途径以确保级数解的收敛, 因此，特别适用于强非线性问题. 将同伦分析法与平均法以及四阶的龙格库塔方法(数值解)做了比较. 结果 表明，平均法在强非线性情况失效, 四阶的龙格库塔法不能找到非稳定的极限环，而同伦分析法不仅适用于强非线性情 况，而且给出了非稳定的极限环.

ANALYTICAL APPROXIMATIONS FOR NONLINEAR DYNAMIC SYSTEM WITH MULTIPLE LIMIT CYCLES

A modified Rayleigh oscillator with multiple limit cycles is investigated by means of a new analytical method for nonlinear problems, namely, the homotopy analysis method (HAM). The HAM is independent upon small parameters. More importantly, unlike other traditional techniques, the HAM provides us with a simple way to ensure the convergence of solution series. Thus, the HAM can be used for strongly nonlinear problems. Comparisons of the solutions given by the HAM, the method of averaging, and Runge-Kutta method show that the method of averaging is not valid for strongly nonlinear cases, and the Runge-Kutta numerical technique does not work for the instable limit cycles, however, the HAM not only works for strongly nonlinear cases, but also can give good approximations for the instable limit cycles.