On partial synchronization of nonlinear oscillations of two Berger plates coupled by internal subdomains

The problem of nonlinear oscillations of two Berger plates occupying bounded domains ΩΩ in different parallel planes and coupled by internal subdomains Ω1⊂ΩΩ1⊂Ω is considered. A dynamical system generated by the problem in the space H=[H02(Ω)]2×[L2(Ω)]2 is studied. The long-time behavior of the trajectories of the system and its dependence on the value of the coupling parameter γγ is described in terms of the system global attractor. In particular, we prove a synchronization phenomenon at the level of attractor for the system. Namely, we consider a (limiting) dynamical system generated by a suitable second order in time evolution equation in the space H̃ consisting of the elements from HH with coordinates equal for the values of the spatial variable xx from the closed set Ω1¯: H̃={y=(y1,y2,y3,y4)∈H:y1(x)=y2(x),y3(x)=y4(x),x∈Ω1¯}, and prove that the attractor of the system describing oscillations of two partially coupled Berger plates approaches the attractor of the limiting system as γγ tends to the infinity.