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Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

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Abstract

In this paper, a novel construction of solutions of nonlinear oscillators are proposed which can be called as the quadratic generalized harmonic function. Based on this novel solution, a modified generalized harmonic function Lindstedt–Poincaré method is presented which call the quadratic generalized harmonic function perturbation method. Via this method, the homoclinic and heteroclinic bifurcations of Duffing-harmonic-van de Pol oscillator are investigated. The critical value of the homoclinic and heteroclinic bifurcation parameters are predicted. Meanwhile, the analytical solutions of homoclinic and heteroclinic orbits of this oscillator are also attained. To illustrate the accuracy of the present method, all the above-mentioned results are compared with those of Runge–Kutta method, which shows that the proposed method is effective and feasible. In addition, the present method can be utilized in study many other oscillators.

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Acknowledgments

The authors acknowledge support by the National Natural Science Foundation of China (Grant No. 11172093, 11372102) and the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2012B159).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, China

    Zhenbo Li, Jiashi Tang & Ping Cai

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  1. Zhenbo Li
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  2. Jiashi Tang
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  3. Ping Cai
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Correspondence to Zhenbo Li.

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Li, Z., Tang, J. & Cai, P. Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator. Qual. Theory Dyn. Syst. 15, 19–37 (2016). https://doi.org/10.1007/s12346-015-0138-z

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  • DOI: https://doi.org/10.1007/s12346-015-0138-z

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