Geometric singular perturbation theory with real noise

Abstract

We prove the existence of families of random invariant manifolds for singularly perturbed systems of ordinary differential equations with sufficiently small real noise. We use these invariant manifolds to prove a random version of the inclination theorem or exchange lemma. Published by Elsevier Inc.

Singular fold with real noise

“…Under the above conditions, the main theorem in [5] implies that outside an arbitrary small neighborhood V of (0, 0), the critical manifolds S a and S r perturb smoothly to locally invariant random slow manifolds S a, ,ω and S r, ,ω for sufficiently small = 0. Under the two-dimensional set-up, S a, ,θ t ω and S r, ,θ t ω are actually trajectories of (1.10).…”

“…In [5], we extended the classical geometric singular perturbation theory of Fenichel [7]: System (1.1) has a critical manifold of equilibria when = 0 given by f (x, y, 0) = 0. The system may also be written in 'slow' time τ = t:…”

“…In this paper, we study the random system (1.8) with the method of random dynamical systems. The noise in our case is assumed to be uniformly C 1 small, allowing us to use the random invariant manifold and foliation theory developed in [3,4,5].…”

“…(Though the -direction is nonhyperbolic, its triviality enables dealing with the above parts as hyperbolic.) By the main theorem of [5], there exist attracting and repelling center-like locally invariant random manifolds M a,ω and M r,ω , -sections of which give S a, ,ω and S r, ,ω .…”

“…This paper, following our recent work [5], is devoted to the geometrical theory for slow-fast systems of ordinary differential equations driven by real noise near a fold point. By real noise, we mean the kind that can be studied in the pathwise sense (see page 57 in [1] for more details).…”