Stability and constructions of the Poincaré maps for a class of stochastic partial differential equations

This work is concerned with a class of stochastic partial differential equations with a multiplicative noise. The Poincaré map, as a main tool to study the periodic orbits, plays a crucially important role in the invariant sets of the system, which captures the dominated dynamical behavior of the trajectories. It derives out the existence and the stability of the stochastic Poincaré maps of the system, which return to a small neighborhood of a fixed point of a deterministic Poincaré map one time or even infinite times with the small noise disturbing. It further expresses the constructions of the stochastic Poincaré maps through the random invariant manifold and the moving orthonormal system. •The existence of the stochastic Poincaré maps of the SPDE is established.•The stochastic Poincaré maps of the SPDE are stable.•The SPDE is reduced to a finite dimensional stochastic differential system.•The construction of the stochastic Poincaré maps for the SPDE is given.