Melnikov method for homoclinic bifurcation in nonlinear impact oscillators
Melnikov method for homoclinic bifurcation in nonlinear impact oscillators
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DOI：
10.1016/j.camwa.2005.03.007
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发表时间：
2005-08
影响因子：
2.9
通讯作者：
Zhengdong Du;Weinian Zhang
中科院分区：
文献类型：
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作者：
Zhengdong Du;Weinian Zhang
基于一个倒立摆在外部周期激励下对刚性壁面的冲击，讨论了一类非线性冲击振子的同斜分岔问题。将建立于光滑动力系统的Melnikov方法推广到非光滑动力系统。对于非线性冲击系统，冲击间的封闭解通常是不可用的。由于缺乏闭形式解，使得估计稳定流形与不稳定流形之间的间隙变得困难。本文给出了一种计算n阶Melnikov函数的方法，从而得到了由冲击准则所给出的辨识所形成的同斜环的参数持续存在的条件。
Based on an inverted pendulum impacting on rigid walls under external periodic excitation, a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation. The Melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one. For nonlinear impact systems, closed form solutions between impacts are generally unavailable. The absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold. In this paper, we give a method to compute the Melnikov functions up to the nth-order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule.