Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Abstract

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.