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)
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
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.
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