Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

We consider a nonnegative potential $W$ that vanishes on a finite set and study the existence of periodic orbits of the equation $$\ddot{u}=W_u(u),\;\;t\in\R,$$ that have the property of visiting neighborhoods of zeros of $W$ in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable $x=\epsilon t$, $\epsilon>0$ small, these orbits correspond to stationary solutions of the parabolic equation $$u_t=u_{xx}-W_u(u),\;\;x\in(0,1),\;t>0,$$ with periodic boundary conditions. In the second paper of the paper we study solutions of this equation that, as the stationary solutions, have a layered structure. We derive a system of ODE that describes the dynamics of the layers and show that their motion is extremely slow.