  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.
