Existence of Periodic Orbits Near Heteroclinic Connections

Giorgio Fusco ; Giovanni F. Gronchi,Matteo Novaga

Minimax theory and its applications, Tome 4 (2019) no. 1

Résumé

We consider a potential Rm Rwith two different global minima − + and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system u , has a family of-periodic solutions T which, along a sequence j (*) , converges locally to a heteroclinic solution that connects − to +. We then focus on the elliptic system ∆ u R2 Rm, ∫ (**) that we interpret as an infinite dimensional analogous of (*), where plays the role of time and is replaced by the action functional R 1 R 2 y 2 . We assume that R has two different global minimizers − + R Rm in the set of maps that connect − to +. We work in a symmetric context and prove, via a minimization procedure, that (**) has a family of solutions L R2 Rm, which is-periodic in , converges to ± as and, along a sequence j , converges locally to a heteroclinic solution that connects − to +.

Détail

Mots-clés : Action-minimizing solutions, periodic orbits, homoclinic orbits, heteroclinic orbits, variational methods