Averaging and Chaotic Motions in Forced Oscillations

We review the averaging theorem of Krylov–Bogoliubov, which allows one to establish the local existence of periodic orbits in certain forced oscillation problems. We then consider some global features, in particular the existence of hetero- and homoclinic orbits, and describe a method, originally due to Melnikov, by which such orbits can be detected in the case that the averaged equation possesses a saddle connection. We give a number of examples and discuss the chaotic motions resulting from the presence of such orbits.