# Coordinate-independent singular perturbation reduction for systems with three time scales

## TL;DR

This paper introduces a coordinate-independent reduction method for systems with three time scales, extending existing theories to arbitrary parameter-dependent systems and applying the approach to biochemical models.

## Contribution

It develops a novel coordinate-independent reduction technique for three time-scale systems and extends Tikhonov-Fenichel parameter analysis to this setting.

## Key findings

- Applicable to systems with three time scales.
- Extended Tikhonov-Fenichel parameter analysis.
- Demonstrated on biochemical systems.

## Abstract

On the basis of recent work by Cardin and Teixeira on ordinary differential equations with more than two time scales, we devise a coordinate-independent reduction for systems with three time scales; thus no a priori separation of variables into fast, slow etc. is required. Moreover we consider arbitrary parameter dependent systems and extend earlier work on Tikhonov-Fenichel parameter values -- i.e. parameter values from which singularly perturbed systems emanate upon small perturbations -- to the three time-scale setting. We apply our results to two standard systems from biochemistry.