An iterative method for the approximation of fibers in slow-fast systems

TL;DR

This paper introduces an iterative approach to improve the approximation of fibers in slow-fast systems, providing exponential estimates of tangent spaces and extending to curvature approximation, with applications to biological models.

Contribution

The paper extends existing methods to also approximate fiber directions and curvature in slow-fast systems, with proven exponential accuracy in finite-dimensional real analytic models.

Findings

Successfully applied to Michaelis-Menten-Henri model

Demonstrated on Lindemann mechanism in non-standard form

Extended method to curvature approximation

Abstract

In this paper we extend a method for iteratively improving slow manifolds so that it also can be used to approximate the fiber directions. The extended method is applied to general finite dimensional real analytic systems where we obtain exponential estimates of the tangent spaces to the fibers. The method is demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism. The latter example also serves to demonstrate the method on a slow-fast system in non-standard slow-fast form. Finally, we extend the method further so that it also approximates the curvature of the fibers.