基于参数展开的同伦分析技术及其应用

提出了一种基于参数展开的新的同伦分析技术(PE-HAM)：结合参数展 开技术和同伦理论将一非线性动力系统(不要求系统内含有小参数)的求解问题转化为一组 线性微分方程的求解问题，并将之运用到强非线性振动领域. 用该方法研究了强非线性 Duffing系统的响应问题，得到了一阶近似解. 作为特例讨论了保守Duffing系统和受谐和 激励的耗散Duffing系统的稳态响应问题. 数值模拟的结果，说明了新方法的有效性.

A homotopy technique with the parameter expansion and its application

In this paper, a new homotopy technique based on the parameter expansion (PE-HAM) is proposed to study strongly nonlinear oscillation. By means of the technique of parameter expansion with the theory of homotopy, we transform the original strongly non-linear dynamical system into a set of linear differential equations which can be solved easily. As illustrative examples, strongly nonlinear conservative Duffing's oscillator and dissipated Duffing's oscillator with harmonic excitation are considered using PE-HAM method to obtain the zero-th and first order approximate solutions. Moreover, the numerical results by the Runge-Kutta of 4th order are carried out to compare with the theoretical ones and excellent agreement can be found, even in the case of strongly nonlinear magnitude.