Prediction of homoclinic bifurcation: the elliptic averaging method

A criterion to predict bifurcation of homoclinic orbits in strongly nonline ar autonomous oscillators is presented. The averaging method combined forma lly with the Jacobian elliptic functions is applied to determine an approxi mation of limit cycles near homoclinicity. We then introduce a criterion fo r predicting homoclinic orbits, based on the collision between the bifurcat ing limit cycle and the saddle equillibrium. In particular, we show that th is criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic: nonlinearities are in cluded. (C) 2000 Elsevier Science Ltd. All rights reserved.