
Minimax Theory and its Applications 04 (2019), No. 1, 113149 Copyright Heldermann Verlag 2019 Existence of Periodic Orbits Near Heteroclinic Connections Giorgio Fusco Dip. di Matematica Pura ed Applicata, Università dell'Aquila, Via Vetoio, 67010 Coppito  L’Aquila, Italy fusco@univaq.it Giovanni F. Gronchi Dip. di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy giovanni.federico.gronchi@unipi.it Matteo Novaga Dip. di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy matteo.novaga@unipi.it [Abstractpdf] \newcommand{\R}{\mathbb{R}} We consider a potential $W\colon \R^m\rightarrow\R$ with two different global minima $a_, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system\\[2mm] \centerline{\hfill $\ddot{u}=W_u(u)$, \hfill(*)}\par has a family of $T$periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_$ to $a_+$. We then focus on the elliptic system\\[2mm] \centerline{\hfill $\Delta u=W_u(u),\;\; u\colon \R^2\rightarrow\R^m$, \hfill(**)}\par that we interpret as an infinite dimensional analogous of (*), where $x$ plays the role of time and $W$ is replaced by the action functional $J_\R(u)=\int_\R(\frac{1}{2}\vert u_y\vert^2+W(u))dy$. We assume that $J_\R$ has two different global minimizers $\bar{u}_, \bar{u}_+\colon \R\rightarrow\R^m$ in the set of maps that connect $a_$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (**) has a family of solutions $u^L\colon \R^2\rightarrow\R^m$, which is $L$periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_$ to $\bar{u}_+$. Keywords: Actionminimizing solutions, periodic orbits, homoclinic orbits, heteroclinic orbits, variational methods. MSC: 37J50, 37J45 [ Fulltextpdf (240 KB)] for subscribers only. 