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Homoclinic orbits in three-dimensional shilnikov-type chaotic systems

Chinese Physics B 2013年第9期22卷 312-323页

作者：冯晶晶张琪昌王炜郝淑英 Tianjin Key Labortory of Nonlinear Dynamics and Chaos Control School of Mechanical EngineeringTianjin University State Key Laboratory of Engines Tianjin University School of Mechanical Engineering Tianjin University of Technology

In this paper, the Pad6 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 P1D 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 Pad6 approximation

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shilnikov sense chaos in a simple three-dimensional system

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Chinese Physics B 2010年第3期19卷 207-216页

作者：王炜张琪昌田瑞兰 Department of Mechanics School of Mechanical EngineeringTianjin University Centre for Nonlinear Dynamics Research Shijiazhuang Railway Institute

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 equa- tions. 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.

关键词： chaos shilnikov theorem homoclinic orbit manifold

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