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Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method

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Abstract

The hyperbolic perturbation method is applied to determining the homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators of the form \(\ddot{x}+c_{1}x+c_{3}x^{3}=\varepsilon f(\mu,x,\dot{x})\) , in which the hyperbolic functions are employed instead of the periodic functions in the usual perturbation method. The generalized Liénard oscillator with \(f(\mu,x,\dot{x})=(\mu -\mu_{1}x^{2}-\mu_{2}\dot{x}^{2})\dot{x}\) is studied in detail. Comparisons with the numerical simulations obtained by using R–K method are made to show the efficacy and accuracy of the present method.

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  1. Department of Applied Mechanics and Engineering, Sun Yat-Sen University, Guangzhou, 510275, China

    Y. Y. Chen & S. H. Chen

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  1. Y. Y. Chen
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  2. S. H. Chen
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Chen, Y.Y., Chen, S.H. Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method. Nonlinear Dyn 58, 417–429 (2009). https://doi.org/10.1007/s11071-009-9489-9

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