A coordinate-independent version of Hoppensteadt's convergence theorem

TL;DR

This paper presents a new, easier-to-verify convergence theorem for singular perturbation reduction in autonomous systems over unbounded time intervals, with applications to biochemical reaction equations.

Contribution

It introduces a coordinate-independent convergence theorem for autonomous systems on unbounded intervals, simplifying criteria especially for systems with one-dimensional slow manifolds.

Findings

The theorem applies to systems with one-dimensional slow manifolds.

Criteria for convergence are easier to verify than previous theorems.

Applications to biochemical reaction equations demonstrate practical utility.

Abstract

The classical theorems about singular perturbation reduction (due to Tikhonov and Fenichel) are concerned with convergence on a compact time interval (in slow time) as a small parameter approaches zero. For unbounded time intervals Hoppensteadt gave a convergence theorem, but his criteria are generally not easy to apply to concrete given systems. We state and prove a convergence result for autonomous systems on unbounded time intervals which relies on criteria that are relatively easy to verify, in particular for the case of a one-dimensional slow manifold. As for applications, we discuss several reaction equations from biochemistry.