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Master-slave synchronization and invariant manifolds for coupled stochastic systems. (English) Zbl 1314.34116
Summary: We deal with abstract systems 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 and show that this system can be reduced to a single equation with modified nonlinearity. This result means that under some conditions, we observe (nonlinear) synchronization phenomena in the coupled system. Our applications include stochastic systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein-Gordon and Schrödinger equations. We also show that the random manifold constructed converges to its deterministic counterpart when the intensity of noise tends to zero.
©2010 American Institute of Physics

MSC:
34D06 Synchronization of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
34K50 Stochastic functional-differential equations
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34D45 Attractors of solutions to ordinary differential equations
60H40 White noise theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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