Homoclinic Solution and Bifurcation of Self-Excited System with Quintic Strong Nonlinearity

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Journal of Vibration and Shock ›› 2013, Vol. 32 ›› Issue (11) : 117-125.

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Homoclinic Solution and Bifurcation of Self-Excited System with Quintic Strong Nonlinearity

Chen Yangyang 1 Yan Lewei 2 Chen Shuhui 3

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Abstract

The hyperbolic Lindstedt-Poincaré (L-P) perturbation procedure is extended for homoclinic solution and homoclinic bifurcation analysis of self-excited system, in which the generating system is with quintic strong nonlinearity. In the procedure, the homoclinic bifurcation value for limit cycle is expanded in power of perturbation parameter, the definition of secular terms for homoclinic perturbation solutions is given. And then the homoclinic bifurcation values can be determined by eliminating secular terms. The explicit homoclinic solutions by which the homoclinic conditions can strictly satisfy are obtained. The general solution formula up to arbitrary perturbation order can also be derived. By the present method, the homoclinic bifurcation of a general Liénard oscillator is studied in detail, in which the advantage and problems to be solved are discussed. Phase portraits and bifurcation values of typical examples are presented. Comparisons of results between the present method and the Runge-Kutta numerical method are made to illustrate the accuracy and efficiency of the present method. Base on the procedure and idea, the present method can be extended to deal with homoclinic (heteroclinic) solution and homoclinic (heteroclinic) bifurcation problems for more general systems.

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hyperbolic Lindstedt-Poincaré method / self-excited oscillator / quintic nonlinearity / homoclinic bifurcation / homoclinic solution

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Chen Yangyang Yan Lewei Chen Shuhui . Homoclinic Solution and Bifurcation of Self-Excited System with Quintic Strong Nonlinearity[J]. Journal of Vibration and Shock, 2013, 32(11): 117-125