The hyperbolic perturbation method is applied to determining the homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators of the formẍ + c 1 x + c 3 x 3 = εf (μ, x,ẋ), in which the hyperbolic functions are employed instead of the periodic functions in the usual perturbation method. The generalized Liénard oscillator with f (μ, x,ẋ) = (μ − μ 1 x 2 − μ 2ẋ 2 )ẋ 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.
A hyperbolic Lindstedt-Poincaré method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Liénard oscillator is studied in detail, and the present method's predictions are compared with those of RungeKutta method to illustrate its accuracy.
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