Invariant Manifold Reduction and Bifurcation for Stochastic Partial Differential Equations

Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences. Invariant manifolds often provide geometric structure for understanding stochastic dynamics. In this paper, a random invariant manifold reduction principle is proved for a class of stochastic partial differential equations. The dynamical behavior is shown to be described by a stochastic ordinary differential equation on an invariant manifold, under suitable conditions. The change of dynamical structures for the stochastic partial differential equations is thus obtained by investigating the stochastic ordinary differential equation. The random cone invariant property is used in the approach. Moreover, the invariant manifold reduction principle is applied to detect bifurcation phenomena and stationary states in stochastic parabolic and hyperbolic partial differential equations.