The Poincaré maps of a slow-fast stochastic system

Abstract

This work is concerned with the Poincaré maps of a slow-fast stochastic system. Applying the random manifold reduced theory and the moving orthonormal system argument, it derives the effective construction of the stochastic Poincaré map. Moreover, employing the geometric distribution of the first exit time, it builds the stability of the stochastic Poincaré maps, that is, the stochastic Poincaré maps return to a small neighborhood of a fixed point of a deterministic Poincaré map with one time or even infinite times as the noise intensity tends to zero. It further investigates the limit distribution of the first exit location on the boundary of the domain via the harmonic measure.