基于近哈密顿系统的Hopf分岔

来源期刊：中南大学学报(自然科学版)2004年第2期

论文作者：王桥医 徐先懂 唐文评

文章页码：258 - 261

关键词：非线性动力学;小扰动;时变;哈密顿系统; Melnikov方法; Hopf分岔

Key words：nonlinear dynamics; perturbation; time-dependent; Hamiltonian system; Melnikov method; Hopf bifurcation

摘    要：针对三维时变小扰动哈密顿系统的Hopf分岔的理论仅仅适用于自治系统的情况,运用Melnikov方法研究了时变小扰动哈密顿系统周期轨道发生Hopf分岔的条件,并将这些条件应用于一类三维时变小扰动非自治系统,使之能用于非自治系统。研究结果表明,所研究的系统还存在复杂而有规律的环面分岔行为。

Abstract: This paper develops a newmethod to study the Hopf bifurcations of periodic orbits in three-dimen-sional, time dependent perturbation of planar Hamiltonian differential equations, and gets a series of concise formula to simplify the Hopf bifurcation conditions by some mathematical skill and subharmonic Melnikov method. TheHopf bifurcation existence parameter domains described in an equation is deduced, elliptic func-tions and elliptic integration are used to calculate these parameter domains. In order to verify the method, the system is integrated at the bifurcation existence parameter domainswith numerical method. The results indicatethat the method is well coincided with the numerical results.Further numerical integration indicates that more complicated torus bifurcation exist in the example system.