中国物理B ›› 2005, Vol. 14 ›› Issue (7): 1365-1369.doi: 10.1088/1009-1963/14/7/017

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Nonexistence of ultra-subharmonic periodic orbits in periodically forced differential equation

李庆士1, 高经武2   

  1. (1)Institute of Applied Mechanics, Taiyuan University of　Technology, Taiyuan 030024, China; (2)Institute of Applied Mechanics, Taiyuan University of　Technology, Taiyuan 030024, China；Department of Physics, Shanxi Datong University, Datong 037009, China
  • 收稿日期:2003-12-22 修回日期:2004-12-15 出版日期:2005-06-20 发布日期:2005-06-20

Nonexistence of ultra-subharmonic periodic orbits in periodically forced differential equation

Gao Jing-Wu (高经武)ab, Li Qing-Shi(李庆士)a   

  1. a Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan 030024, China; b Department of Physics, Shanxi Datong University, Datong 037009, China
  • Received:2003-12-22 Revised:2004-12-15 Online:2005-06-20 Published:2005-06-20

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摘要： It is proved that if there exists a periodic solution for a class of non-autonomous differential dynamic systems, it can only be subharmonic, ultra-subharmonic periodic solution is impossible. Moreover, the existence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also denied. As an application of the above conclusions, through investigating some typical examples, it is pointed out that the existence of ultra-subharmonic periodic orbits in a planar perturbation system cannot be determined by second-order Melnikov method. An explanation is also provided.

关键词: nonlinear dynamic system, higher-order Melnikov method, ultra-subharmonic periodic solution, Poincar\'{e} map

Abstract: It is proved that if there exists a periodic solution for a class of non-autonomous differential dynamic systems, it can only be subharmonic, ultra-subharmonic periodic solution is impossible. Moreover, the existence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also denied. As an application of the above conclusions, through investigating some typical examples, it is pointed out that the existence of ultra-subharmonic periodic orbits in a planar perturbation system cannot be determined by second-order Melnikov method. An explanation is also provided.

Key words: nonlinear dynamic system, higher-order Melnikov method, ultra-subharmonic periodic solution, Poincar\'{e} map

中图分类号:  (Nonlinear dynamics and chaos)

  • 05.45.-a
04.20.Jb (Exact solutions) 02.30.Hq (Ordinary differential equations)