Predicting homoclinic bifurcations in planar autonomous systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.