Invariant manifolds and nonlinear master-slave synchronization in coupled systems

Abstract

We consider an abstract system of two coupled nonlinear (infinite dimensional) equations. This kind of systems may describe various interaction phenomena in a continuum medium. Under some conditions we prove the existence of an exponentially attracting 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. As applications we consider coupled systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein–Gordon and Schrödinger equations.

Keywords:

• Infinite dimensional coupled systems
• Invariant manifolds
• Synchronization

Keywords:

• 2000 Mathematics Subject Classifications:
• MSC2000
• 37L15
• Secondary: 37L25
• 35B40