A generalized Padé–Lindstedt–Poincaré method for predicting homoclinic and heteroclinic bifurcations of strongly nonlinear autonomous oscillators

By combining the generalized Padé approximation method and the well-known Lindstedt–Poincaré method, a novel technique, referred to as the generalized Padé–Lindstedt–Poincaré method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Padé approximation method is generalized. According to this generalization, the numerator and denominator of the Padé approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Padé approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Padé approximation method is introduced into the Lindstedt–Poincaré method’s procedure for solving the perturbation equations. Via the proposed generalized Padé–Lindstedt–Poincaré method, the homo-/heteroclinic bifurcations of the generalized Helmholtz–Duffing–Van der Pol oscillator and Φ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^{6}$$\end{document}-Van der Pol oscillator are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge–Kutta method, which shows the method proposed in this paper is both effective and feasible. Furthermore, the proposed method can be also utilized to solve many other oscillators.