Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

doi:10.1088/1674-1056/21/2/020503

Abstract Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).

Keywords: nonlinear dynamical system Melnikov boundary irrational restoring force saddle-like singularity homoclinic-like orbit

Received: 30 August 2011
Revised: 02 October 2011
Accepted manuscript online:

PACS:	05.45.-a	(Nonlinear dynamics and chaos)
	05.45.Ac	(Low-dimensional chaos)
	82.40.Bj	(Oscillations, chaos, and bifurcations)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11002093, 11072065, and 10872136) and the Science Foundation of the Science and Technology Department of Hebei Province of China (Grant No. 11215643).

Corresponding Authors: Cao Qing-Jie,caoqingjie@hotmail.com
E-mail: caoqingjie@hotmail.com

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Tian Rui-Lan(田瑞兰), Yang Xin-Wei(杨新伟), Cao Qing-Jie(曹庆杰), and Wu Qi-Liang(吴启亮) Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force 2012 Chin. Phys. B 21 020503

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Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

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