中文摘要：

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.

英文摘要：

　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.