2ND-ORDER AVERAGING AND MELNIKOV ANALYSES FOR FORCED NONLINEAR OSCILLATORS

An analysis is presented of a class of periodically forced cion-linear oscillators. The systems have centers and families of periodic orbits and may have homoclinic and/or heteroclinic orbits when the forcing a nd damping terms are removed. First, bifurcation behavior is analyzed near the unperturbed centers when primary, subharmonic or superharmoni c resonance occurs, by using the second-order averaging method. Second , Melnikov's method is applied and bifurcation behavior near the unper turbed homoclinic, heteroclinic and resonant periodic orbits is analyz ed. The limits of saddle-node bifurcations of subharmonics near the un perturbed resonant periodic orbits as the resonant periodic orbits app roach homoclinic and/or heteroclinic orbits or centers are obtained. T he results of the second-order averaging and Melnikov analyses for sad dle-node bifurcations of subharmonics near centers are compared and th eir relation is discussed. An example is given for the Duffing oscilla tor with double well potential. (C) 1996 Academic Press Limited