Complex Dynamics in a Class of Reversible Equations

Abstract

We consider the pendulum type equationwhere the functions Wand h satisfy (HI) WE C 2 (JR. x JR.; JR.) is I-periodic in t and T-periodic in u, (H2) hE C(JR.;JR.) is I-periodic and fol h(t)dt = 0, (H3) Vet, u) = Wet, u) -h(t)u is even in t.

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

“…Keeping this in mind, we now turn to the description of the variational approach used in [6] to produce evidence of chaotic dynamics in Eq. (1). In order to do this, let…”

“…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.…”

“…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.…”

“…We refer to [27] for comments on the validity of the above statements as indicators of the chaotic behavior of the dynamics of an equation of the form (1).…”

“…(1). Equation (1) generalizes the classical forced pendulum equation, one of the simplest but more studied problems in nonlinear analysis (we refer the reader to the interesting paper [16] and to its extensive references).…”