Local unstable manifolds of periodic orbits in thin domains

We compare the dynamics of a system of damped wave equations on a thin domain with the dynamics of the limit system when the thickness of the thin domain goes to 0. More precisely, under the assumption that the problem in the thin domain admits a periodic orbit γεγε close to a hyperbolic periodic orbit γ0γ0 of the problem in the limit domain, we show that γεγε is also hyperbolic and that the local unstable manifold of the periodic orbit γεγε is close to the local unstable manifold of γ0γ0.