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

doi:10.1088/1674-1056/19/6/060510

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.

Keywords: Silnikov criterion chaos homoclinic orbit period-doubling bifurcation

Received: 28 July 2009
Accepted manuscript online:

PACS:	05.45.Pq	(Numerical simulations of chaotic systems)
	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)
	02.30.Oz	(Bifurcation theory)

Fund: 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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Li Wei-Yi(李伟义), Zhang Qi-Chang(张琪昌), and Wang Wei(王炜) Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion 2010 Chin. Phys. B 19 060510

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

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