中国物理B ›› 2010, Vol. 19 ›› Issue (6): 60510-060510.doi: 10.1088/1674-1056/19/6/060510

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Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion

李伟义, 张琪昌, 王炜   

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China and State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China
  • 收稿日期:2009-07-28 出版日期:2010-06-15 发布日期:2010-06-15
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No.~10872141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.~20060056005).

Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion

Li Wei-Yi(李伟义), Zhang Qi-Chang(张琪昌)†, and Wang Wei(王炜)   

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China and State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China
  • Received:2009-07-28 Online:2010-06-15 Published:2010-06-15
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No.~10872141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.~20060056005).

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摘要： Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.

Abstract: Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.

Key words: Silnikov criterion, chaos, homoclinic orbit, period-doubling bifurcation

中图分类号:  (Numerical simulations of chaotic systems)

  • 05.45.Pq
02.60.Lj (Ordinary and partial differential equations; boundary value problems) 02.30.Oz (Bifurcation theory)