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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.