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一类非自治动力系统超次谐周期解的不存在性
引用本文:高经武,蔡中民,李庆士,武际可.一类非自治动力系统超次谐周期解的不存在性[J].力学学报,2004,36(5):629-633.
作者姓名:高经武  蔡中民  李庆士  武际可
作者单位:太原理工大学应用力学研究所
摘    要:在非线性动力系统的研究中, Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下,经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似,因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究,证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解,超次谐周期解不可能存在, 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用,文中考察了几个典型的算例,结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的,并提供了一个简单的几何上的解释.

关 键 词:动力系统  非自治  高阶Melnikov方法  次谐周期解  超次谐周期解  Poincare映射
修稿时间:2003年12月29

Nonexistence of ultra-subharmonic periodic solutions for a class of nonautonomous dynamic system
Gao Jingwu Cai Zhongmin Li Qingshi Wu Jike Institute of Applied Mechanics,Taiyuan University of Technology,Taiyuan ,China.Nonexistence of ultra-subharmonic periodic solutions for a class of nonautonomous dynamic system[J].chinese journal of theoretical and applied mechanics,2004,36(5):629-633.
Authors:Gao Jingwu Cai Zhongmin Li Qingshi Wu Jike Institute of Applied Mechanics  Taiyuan University of Technology  Taiyuan  China
Institution:Gao Jingwu Cai Zhongmin Li Qingshi Wu Jike Institute of Applied Mechanics,Taiyuan University of Technology,Taiyuan 030024,China Department of Mechanics and Engineering Science,Peking University,Beijing 100871,China
Abstract:In the study of the nonlinear dynamics, Melnikov function is widely used as a criterion to check whether subharmonic or ultra-subharmonic bifurcation even chaos will occur in a perturbed Hamilton system. However, for the most cases, the classical Melnikov method can merely show the existence of subharmonic periodic orbits. Such a result is attributed to that only first order approximation is adopted in the classical Melnikov method. So higher-order Melnikov method is developed to determine the existence of the ultra subharmonic periodic solution. In this paper, a class of non-autonomous differential dynamic system is studied. It is proved that if there exists a periodic solution in such a system, the solution can only be subharmonic, and the existence of ultra-subharmonic periodic solution is impossible. Moreover, the nonexistence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also confirmed. As an application of above conclusions, some typical examples are investigated. The results demonstrate that second-order Melnikov method used to justify the existence of ultra-subharmonic periodic orbits in a planar perturbation system may lead to a wrong conclusion. A simple geometric explanation is also provided.
Keywords:dynamic system  non-autonomous  higher-order Melnikov method  subharmonic periodic solution  ultra-subharmonic periodic solution  Poincare map
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