Synchronisation in coupled stochastic PDEs
Seminar Room 1, Newton Institute
AbstractWe first consider a system of semilinear parabolic stochastic partial differential equations with additive space-time noise on the union of thin bounded tubular domains with interaction via interface and give conditions which guarantee synchronized behaviour of solutions at the level of pullback attractors. In particular, we show that in some cases the limiting dynamics is described by a single stochastic parabolic equation with the averaged diffusion coefficient and a nonlinearity term, which essentially indicates synchronization of the dynamics on both sides of the interface. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case.
Then we deal with an abstract system of two coupled nonlinear stochastic (infinite dimensional) equations subjected to additive white noise type process. This kind of systems may describe various interaction phenomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invariant manifold for the coupled system. This result means that under some conditions we observe (nonlinear) master-slave synchronization phenomena in the coupled system. As applications we consider stochastic systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein-Gordon and Schroedinger equations.
Partially based on joint results with T. Caraballo, P. E. Kloeden, and B. Schmalfuss.