On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping

The dynamical system generated by a system describing nonlinear oscillations of two coupled Berger plates with nonlinear interior damping and clamped boundary is considered. The dependence of the long-time behavior of the system trajectories on the coupling parameter $\gamma$ is studied in the case of (i) same equations for both plates of the system and damping possibly degenerate at zero; and (ii) different equations and damping non-degenerate at any point. Ultimate synchronization at the level of attractors is proved for both cases, which means that the global attractor of the system approaches the diagonal of the phase space of the system as $\gamma\to\infty$. In case (ii) the structure of the upper limit of the attractor is studied. It coincides with the diagonal of the product of two samples of the attractor to the dynamical system generated by a single plate equation. If both the equations describing the plate dynamics are the same and the damping functions are non-degenerate at any point we prove the synchronization phenomenon for finite large $\gamma$. System synchronization rate is exponential in this case.