Analytics of homoclinic bifurcations in Three-Dimensional Systems

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.