A generalized Pad,-Lindstedt-Poincar, method for predicting homoclinic and heteroclinic bifurcations of strongly nonlinear autonomous oscillators

摘要

By combining the generalized Pad, approximation method and the well-known Lindstedt-Poincar, method, a novel technique, referred to as the generalized Pad,-Lindstedt-Poincar, method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Pad, approximation method is generalized. According to this generalization, the numerator and denominator of the Pad, approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Pad, approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Pad, approximation method is introduced into the Lindstedt-Poincar, method's procedure for solving the perturbation equations. Via the proposed generalized Pad,aEuro"Lindstedt-Poincar, method, the homo-/heteroclinic bifurcations of the generalized Helmholtz-Duffing-Van der Pol oscillator and -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.