Duffing-Van der pol系统的Hopf分岔

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振动与冲击 ›› 2010, Vol. 29 ›› Issue (7) : 204-209.

论文

Duffing-Van der pol系统的Hopf分岔

符五久

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Hopf Bifurcation of Duffing-Van der pol System

FU Wu-jiu

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摘要

本文将保守Duffing系统作为未扰系统,并对它分四种情形进行了严格求解.用Melnikov函数方法研究了Duffing-Van der pol系统的次谐分岔,获得了Duffing-Van der pol系统的Hopf分岔条件.根据这些条件,在参数空间中确定了Hopf分岔曲线.在分岔曲线上取参数进行了数值模拟,所获得的奇、偶阶Hopf分岔与理论分析的结果完全一致.

Abstract

Take Duffing equation as a non-perturbed Hamiltonian system in Duffing-van der pol equation ,its four conditions is solved rigorously.Subharmonic bifurcation of the Duffing-van der pol system is studied by sub-Melnikov method,Hopf bifurcation conditions are obtained for the Duffing-van der pol system. Hopf bifurcation curves are geted to base of the bifurcation conditions in the parameter space. on the bifurcation curve we take the parameters that do numerical simulate, to obtain odd number order and even number order Hopf bifurcations in correspondence with our theoretical analysis.