*Submitted Paper*

**Inserted:** 27 may 2018

**Last Updated:** 8 jul 2018

**Year:** 2018

**Abstract:**

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

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