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

作　　者：李伟义 张琪昌 王炜 

机构地区：[1]Department of Mechanics,School of Mechanical Engineering,Tianjin University [2]State Key Laboratory of Engines,Tianjin University

出　　处：《Chinese Physics B》2010年第6期139-147,共9页中国物理B（英文版）

基　　金：Project supported by the National Natural Science Foundation of China (Grant No.10872141);the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20060056005)

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.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.

关 键 词：Silnikov criterion CHAOS homoclinic orbit period-doubling bifurcation 

分 类 号：O415.5[理学—理论物理]