Second-order averaging and Melnikov analyses for forced non-linear oscillators

Abstract

An analysis is presented of a class of periodically forced non-linear oscillators. The systems have centers and families of periodic orbits and may have homoclinic and/or heteroclinic orbits when the forcing and damping terms are removed. First, bifurcation behavior is analyzed near the unperturbed centers when primary, subharmonic or superharmonic resonance occurs, by using the second-order averaging method. Second, Melnikov's method is applied and bifurcation behavior near the unperturbed homoclinic, heteroclinic and resonant periodic orbits is analyzed. The limits of saddle-node bifurcations of subharmonics near the unperturbed resonant periodic orbits as the resonant periodic orbits approach homoclinic and/or heteroclinic orbits or centers are obtained. The results of the second-order averaging and Melnikov analyses for saddle-node bifurcations of subharmonics near centers are compared and their relation is discussed. An example is given for the Duffing oscillator with double well potential. © 1996 Academic Press Limited.