Existence of periodic orbits near heteroclinic connections

We consider a potential W: ℝ m → ℝ with two different global minima a - , a + and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system ü = W u (u), has a family of T-periodic solutions u T which, along a sequence T j → +∞, converges locally to a heteroclinic solution that connects a - to a + . We then focus on the elliptic system Δu = W u (u); u: ℝ 2 → ℝ m , 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 =∫ ℝ(1/2|u y | 2 +W(u)dy. We assume that J ℝ has two different global minimizers ū-ū+: ℝ → 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 : ℝ 2 →ℝ m , which is L-periodic in x, converges to a ± as y →± ∞ and, along a sequence L j → +∞ converges locally to a heteroclinic solution that connects ū - to u +