A variational approach to chaotic dynamics in periodically forced nonlinear oscillators

Abstract

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Mixed states for an Allen‐Cahn type equation

“…The proof of the existence of U is by a minimization argument using elements of the proof of Theorem 2.1 combined with ideas from a related situation of Bosetto and Serra [4] in a simpler ODE setting. Much more care is needed in the current setting than is shown in Section 2, partly due to the fact that the natural functional for the current problem,…”

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

“…Following this approach, in a recent paper, [6] (see also [7,24]), the authors investigated the existence of a certain class of multibump-type heteroclinic solutions to periodic orbits of (1), whose presence implies all the requirements of Definition 1.1. The search for this kind of chaotic dynamics is carried out in [6] in two successive stages. First one establishes the existence of a basic type of heteroclinic solutions (1-bump solutions) connecting two periodic minimizers of the natural action functional associated to (1).…”

“…Next one uses these solutions as building blocks to obtain, in the spirit of the shadowing lemma, complex orbits which are asymptotic to the given periodic states at infinity and which oscillate a prescribed (finite or infinite) number of times between them. 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. It is proved in [6] that the nondegeneracy condition (f) is weaker than the standard transversality assumption in the perturbative approach.…”

Spatially heteroclinic solutions for a semilinear elliptic P.D.E.

“…Thus another approach is needed here. For such an approach, we were motivated by some recent work [7] on an Allen-Cahn model equation which in turn has antecedents in work of Moser [5] and Bangert [1] on minimal laminations of a torus and of Bosetto and Serra [2] on an ODE problem related to [1]. Γ will be replaced by a class of functions asymptotic from v to w and having an additional monotonicity property and J by a related function J * .…”