Invariant manifolds in singular perturbation problems for ordinary differential equations

Abstract

SynopsisBased on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles

“…Later on, the works by Knobloch and Aulbach [25], Nipp [30], and Sakamoto [33] also presented results similar to [12]. By now, the theory is fairly standard, and there have been enormous applications to traveling waves of partial differential equations, see [23] and the references there.…”

“…To apply geometric singular perturbation theorems to the vector field on R n × R m × [0, ε 0 ], we use the theorems stated in [33]. The following lemma is a restatement of the theorems in [33], and we refer to [33] for the proof.…”

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

“…This theorem is formulated in a specific setting: we assume that the invariant manifold M is (nearly) the zero section of a trivial vector bundle. This is a slightly more general formulation than in [Hen81;Sak90]. There, it is assumed that in a product X × Y of Euclidean (or Banach) spaces, the invariant manifold M is given as the graph of a function h : X → Y .…”

“…This is similar, but developed independently from Sakamoto's work [Sak90] in which he used the same ideas to study singular perturbation problems. We improve these results in a couple of ways.…”

Post-Coulombian Dynamics at Order c-3

“…The dynamics on I ε is governed byẋ = f 1 (x, h ε (x), ε), i.e., The C 1 b -estimate (5.34) is not given explicitly in [7], but may be validated along the lines of [16,15] …”