%0 Journal Article
%T Nonexistence of ultra-subharmonic periodic solutions for a class of nonautonomous dynamic system<br>一类非自治动力系统超次谐周期解的不存在性
%A Gao Jingwu Cai Zhongmin Li Qingshi Wu Jike Institute of Applied Mechanics
%A Taiyuan University of Technology
%A Taiyuan
%A China
%A <br>高经武
%A 蔡中民
%A 李庆士
%A 武际可
%J 力学学报
%D 2004
%I 
%X 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.
%K dynamic system
%K non-autonomous
%K higher-order Melnikov method
%K subharmonic periodic solution
%K ultra-subharmonic periodic solution
%K Poincare map<br>动力系统
%K 非自治
%K 高阶Melnikov方法
%K 次谐周期解
%K 超次谐周期解
%K Poincare映射
%U http://www.alljournals.cn/get_abstract_url.aspx?pcid=6E709DC38FA1D09A4B578DD0906875B5B44D4D294832BB8E&cid=5D344E2AD54D14F8&jid=4100DA4A1A3BA1B0CE5AD99AE1DFB420&aid=84F8A97C68D74BCD&yid=D0E58B75BFD8E51C&vid=933658645952ED9F&iid=94C357A881DFC066&sid=E4EC39E73004B593&eid=06F643376BC2509E&journal_id=0459-1879&journal_name=力学学报&referenced_num=0&reference_num=10