Multiple timescales and the parametrisation method in geometric singular perturbation theory

TL;DR

This paper introduces a coordinate-independent, iterative method inspired by the parametrisation approach to accurately compute slow manifolds and fast fibre bundles in systems with multiple timescales, revealing hidden dynamics.

Contribution

It develops a novel, high-accuracy, coordinate-independent method for analyzing multiple timescale systems using the parametrisation approach, applicable to reaction networks.

Findings

Successfully computes slow manifolds and fast fibre bundles with high accuracy.

Uncovers hidden timescales in complex systems.

Demonstrates applicability to reaction network problems.

Abstract

We present a novel method for computing slow manifolds and their fast fibre bundles in geometric singular perturbation problems. This coordinate-independent method is inspired by the parametrisation method introduced by Cabr\'e, Fontich and de la Llave. By iteratively solving a so-called conjugacy equation, our method simultaneously computes parametrisations of slow manifolds and fast fibre bundles, as well as the dynamics on these objects, to arbitrarily high degrees of accuracy. We show the power of this top-down method for the study of systems with multiple (i.e., three or more) timescales. In particular, we highlight the emergence of hidden timescales and show how our method can uncover these surprising multiple timescale structures. We also apply our parametrisation method to several reaction network problems.