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

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 LindstedtPoincaré 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.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

“…This means that the traditional version of the generalized harmonic function perturbation method is unsuitable for deriving the analytical relationship between the system parameters and the amplitude of the limit cycle. Refer to [27] and [28]. To overcome this shortcoming, the compound Simpson integration formula was introduced to calculate Fourier coefficients p 2i and q 2i [33].…”

“…In addition, many other quantitative methods are available for solving the limit cycle of oscillators (1). Examples include the elliptic function perturbation method [25], elliptic Lindstedt-Poincaré method [26], generalized harmonic function perturbation method [27,28], generalized PadéLindstedtPoincaré method [29,30] and the perturbation incremental method [31,32]. However, these quantitative methods have certain limitations.…”