Existence of periodic orbits near heteroclinic connections

Abstract

We consider a potential $W:R^m\to R$ 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 $T_j\to +\infty$, converges locally to a heteroclinic solution that connects $a_{-}$ to $a_{+}$. We then focus on the elliptic system \begin{equation} Δu=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 ${\bar{u}}_{-},{\bar{u}}_{+}:R\to R^m$ 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 $u^L:R^2\to R^m$, which is $L$-periodic in $x$, converges to $a_{\pm }$ as $y\to \pm \infty$ and, along a sequence $L_j\to +\infty$, converges locally to a heteroclinic solution that connects ${\bar{u}}_{-}$ to ${\bar{u}}_{+}$.

36 pages, 4 figures