Prediction of homoclinic bifurcation: the elliptic averaging method. (English) Zbl 0953.34026

Summary: 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 to limit cycles near homoclinicity. The authors introduce a criterion for predicting homoclinic orbits, based on the collision between the bifurcating limit cycle and the saddle equilibrium. In particular, they show that this criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic nonlinearities are included.


34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C29 Averaging method for ordinary differential equations
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