Exact solutions describing very slow layer oscillations in a shadow reaction-diffusion system

We show in a rigorous way that a stable internal single-layer stationary solution is destabilized by the Hopf bifurcation as the time constant exceeds a certain critical value. Moreover, the exact critical value and the exact period of oscillatory solutions can be obtained. The exact period indicates that the oscillation is very slow, i.e., the period is of order $O(e^{C/\varepsilon})$. We also rigorously prove that Hopf bifurcations from multi-layer stationary solutions occur. In this case anti-phase horizontal oscillations of layers are shown by formal calculations. Numerical experiments show that the exact period agrees with the numerical period of a nearly periodic solution near the Hopf bifurcation point. Anti-phase (out of phase) horizontal oscillations of layers are numerically observed.