Multibump homoclinic solutions to periodic orbits in a center manifold

It is well known that a periodically forced pendulum equation exhibits a chaotic behavior provided some nondegeneracy conditions are satisfied. In this paper we review and describe how critical point theory can be used to prove, for a wider class of systems, and with weaker nondegeneracy conditions, results on the existence of solutions asymptotic to stationary point or to invariant manifold, and also results showing that the system behaves chaotically.