Homoclinic solutions to periodic motions in a class of reversible equations

Generic-Type Results for Chaotic Dynamics in Equations with Periodic Forcing Terms

“…As we pointed out in the Introduction, the existence of consecutive minimizers is a weaker condition (first introduced in [8]) than the isolatedness or nondegeneracy of minimizers in the variational sense. Moreover, it can be shown [12] that when this assumption is violated, then all nonperiodic solutions of (1) are unbounded; hence the existence of consecutive minimizers is a necessary condition for the existence of chaotic dynamics.…”

“…The renormalized functional J was introduced for the first time by Rabinowitz in [20] (and later used in [1,8,17,21,22]) as the right tool to build a global approach to the study of homoclinic solutions to periodic motions in certain classes of equations. The analysis of [6] established that the structure of the sets of heteroclinics between u 0 and u 1 is closely related to the existence of chaotic dynamics in Eq.…”

“…If (11) is violated, then (see [6,8] For simplicity we will always drop e=0 when we refer to solutions of the unperturbed equation (1).…”

“…The main result of [6] (see Theorem 2.2 below) states the sufficient conditions to guarantee that the above described construction be successful. Roughly speaking, to obtain the starting class of heteroclinic solutions one needs a nondegeneracy assumption on the periodic asymptotic states (see Definition 2.1 at the beginning of Section 2) which was first introduced in [8] to replace the stronger hypotheses of isolatedness or nondegeneracy of minimizers in the variational sense. Then, if the family of heteroclinic orbits is discrete in a suitable sense (see condition (f) in Theorem 2.2), one can proceed to find multibump type dynamics which, in turn, implies the structure described in Definition 1.1.…”

Dynamics in the neighbourhood. of a continuum of fixed points

“…It can be proved that a degenerate equation always has a continuum of 2π-periodic solutions and so the corresponding Poincaré map is a diffeomorphism of the plane having a continuum of fixed points. There have been several papers on heteroclinic solutions of (1), see [10,3]. The next result shows that degeneracy is an obstruction for the existence of such solutions.…”