Prediction of homoclinic bifurcation: the elliptic averaging method

A criterion to predict bifurcation of homoclinic orbits in strongly nonlinear autonomous oscillators is presented. The averaging method combined formally with the Jacobian elliptic functions is applied to determine an approximation of limit cycles near homoclinicity. We then introduce a criterion for predicting homoclinic orbits, based on the collision between the bifurcating limit cycle and the saddle equillibrium. In particular, we show that this criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic nonlinearities are included.