Asymptotics of homoclinic bifurcation in a three-dimensional system

An analytical approach to predicting a critical parameter value of homoclin ic bifurcation in a three-dimensional system is reported. The multiple scal es method is first performed to construct a higher-order approximation of t he periodic solution. A criterion based on a collision between the periodic orbit and the fixed point involved in the bifurcation is applied. This cri terion developed initially to predict homoclinic bifurcations in planar aut onomous systems, is adapted here to derive a critical value of the homoclin ic bifurcation in a specific three-dimensional system. To support our analy tical predictions and to describe the dynamical behaviour of the system, a complete numerical study is provided.