Slow foliation of a slow–fast stochastic evolutionary system

Abstract

This work is concerned with the dynamics of a slow-fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore, the geometric structure of the slow foliation is investigated: every fiber of the slow foliation parallels each other, with the slow manifold as a special fiber. In fact, when an arbitrarily chosen point of a fiber falls in the slow manifold, the fiber must be the slow manifold itself.

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

“…It is well known that a random invariant manifold is an invariant set in the state space, carries essential dynamical behavior and characteristics, and provides a geometric structure to describe random dynamics of stochastic partial differential equations. More works about random invariant manifolds of SPDE see [4,5,6,8,10,11,26,28].…”

Slow manifold and parameter estimation for a nonlocal fast-slow stochastic evolutionary system

“…Let Z ǫ (t, ω, Z 0 ) = (U ǫ (t, ω, (U 0 , V 0 ) T ), V ǫ (t, ω, ( be the solution of ( 8)-( 9) with initial value Z 0 := (U ǫ (0), V ǫ (0)) T = (U 0 , V 0 ) T . By the classical theory for evolutionary equations [19], under the hypothesis (H1-H3) the system (8)- (9) with initial data has unique global solution for every ω = (ω 1 , ω 2 ) T ∈ Ω = Ω 1 × Ω 2 just like to [9].…”

“…Proof. Suppose that there are two solutions for random system (8)- (9). Say, Z ǫ (t) = (U ǫ (t), V ǫ (t)) T and Zǫ (t) = ( Ũ ǫ (t), Ṽ ǫ (t)) T .…”

The role of slow manifolds in parameter estimation for a multiscale stochastic system with $α$-stable Lévy noise

“…To make progress in understanding these complex dynamics, it is of a great importance to have a suitable tool for the reduction of such systems and their models to only their slow components, which is often essential for scientific computation and further analysis. The reduction method based on the random slow manifold is one of such effective tool [9][10][11] .…”