The approximation and portray of stochastic Poincaré maps for stochastic slow-fast systems in Hilbert spaces

This work is concerned with the stochastic slow-fast evolutionary systems with white noises in Hilbert spaces. We first establish the stochastic Poincaré maps of the stochastic slow-fast systems in the neighborhood of the periodic orbit for the approximate systems with colored noises in distribution. Further employing the random slow manifold theory, we prove that the stochastic Poincaré maps of the stochastic slow-fast systems converge to the same fixed point of the stochastic Poincaré maps for the approximate systems in distribution as the colored noise parameter tends to zero. Moreover, we apply the moving orthonormal system to construct the exact portray of the stochastic Poincaré maps for the stochastic slow-fast systems in distribution. A concrete example is provided to illustrate the stochastic Poincaré maps.