Synchronization in coupled second order in time infinite-dimensional models

TL;DR

This paper investigates the conditions under which coupled second order in time infinite-dimensional models synchronize, focusing on dissipative wave and elastic structures, with applications to various nonlinear PDEs and physical models.

Contribution

It establishes synchronization results in the infinite coupling limit and for finite intensities in identical subsystems, using novel compactness and stability methods.

Findings

Synchronization occurs at infinite coupling strength.

Finite coupling induces synchronization in identical subsystems.

Results apply to nonlinear wave and elastic models in multiple dimensions.

Abstract

We study asymptotic synchronization at the level of global attractors in a class of coupled second order in time models which arises in dissipative wave and elastic structure dynamics. Under some conditions we prove that this synchronization arises in the infinite coupling intensity limit and show that for identical subsystems this phenomenon appears for finite intensities. Our argument involves a method based on "compensated" compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension. We consider synchronization in sine-Gordon type models which describes distributed Josephson junctions.