中国物理B ›› 2013, Vol. 22 ›› Issue (9): 90503-090503.doi: 10.1088/1674-1056/22/9/090503

• GENERAL • 上一篇    下一篇

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

冯晶晶a b c, 张琪昌a b, 王炜a b, 郝淑英c   

  1. a Tianjin Key Labortory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China;
    b State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China;
    c School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China
  • 收稿日期:2012-11-15 修回日期:2013-03-20 出版日期:2013-07-26 发布日期:2013-07-26
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11072168 and 11102127), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120006), and the Research Program of Application Foundation and Advanced Technology of Tianjin, China (Grant Nos. 12JCYBJC12500 and 11JCYBJC05800).

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

Feng Jing-Jing (冯晶晶)a b c, Zhang Qi-Chang (张琪昌)a b, Wang Wei (王炜)a b, Hao Shu-Ying (郝淑英)c   

  1. a Tianjin Key Labortory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China;
    b State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China;
    c School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China
  • Received:2012-11-15 Revised:2013-03-20 Online:2013-07-26 Published:2013-07-26
  • Contact: Wang Wei E-mail:wangweifrancis@tju.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11072168 and 11102127), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120006), and the Research Program of Application Foundation and Advanced Technology of Tianjin, China (Grant Nos. 12JCYBJC12500 and 11JCYBJC05800).

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摘要： In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

关键词: chaos, Shilnikov theorem, homoclinic orbit, Padé, approximation

Abstract: In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

Key words: chaos, Shilnikov theorem, homoclinic orbit, Padé, approximation

中图分类号:  (Control of chaos, applications of chaos)

  • 05.45.Gg
02.30.Oz (Bifurcation theory)