中国物理B ›› 2010, Vol. 19 ›› Issue (3): 30517-030517.doi: 10.1088/1674-1056/19/3/030517

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Shilnikov sense chaos in a simple three-dimensional system

田瑞兰1, 王炜2, 张琪昌2   

  1. (1)Centre for Nonlinear Dynamics Research, Shijiazhuang Railway Institute, Shijiazhuang 050043, China; (2)Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
  • 收稿日期:2009-01-02 修回日期:2009-09-21 出版日期:2010-03-15 发布日期:2010-03-15
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No.~10872141).

Shilnikov sense chaos in a simple three-dimensional system

Wang Wei(王炜)a), Zhang Qi-Chang(张琪昌) a)†, and Tian Rui-Lan(田瑞兰)b)   

  1. a Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China; b Centre for Nonlinear Dynamics Research, Shijiazhuang Railway Institute, Shijiazhuang 050043, China
  • Received:2009-01-02 Revised:2009-09-21 Online:2010-03-15 Published:2010-03-15
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No.~10872141).

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摘要： The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.

Abstract: The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.

Key words: chaos, Shilnikov theorem, homoclinic orbit, manifold

中图分类号:  (Solar wind plasma; sources of solar wind)

  • 96.50.Ci
94.30.-d (Physics of the magnetosphere) 94.20.-y (Physics of the ionosphere)