Existence of periodic orbits near heteroclinic connections

TL;DR

This paper proves the existence of periodic solutions near heteroclinic connections in Hamiltonian and elliptic systems, showing convergence to heteroclinic orbits as periods tend to infinity, using variational methods and symmetry assumptions.

Contribution

It introduces a variational approach to establish periodic solutions near heteroclinic connections in both finite and infinite-dimensional systems, under symmetry conditions.

Findings

Existence of families of periodic solutions converging to heteroclinic orbits.

Construction of solutions in Hamiltonian systems with symmetry.

Extension of results to elliptic PDE systems with variational structure.

Abstract

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} 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 \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional \[J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.\] We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, 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 $\bar{u}_+$.