# Tikhonov-Fenichel reduction for parameterized critical manifolds with applications to chemical reaction networks

## TL;DR

This paper develops a Tikhonov-Fenichel reduction method for singularly perturbed ODEs with known critical manifolds, applicable to chemical reaction networks, especially when the fast subsystem has complex balanced steady states.

## Contribution

It introduces a reduction formula that does not require separation of variables and applies to chemical networks with known stationary point parameterizations.

## Key findings

- Derived a reduction formula for singularly perturbed ODEs.
- Applied the theory to chemical reaction networks with mass action kinetics.
- Obtained a closed form for the reduced system in specific cases.

## Abstract

We derive a reduction formula for singularly perturbed ordinary differential equations (in the sense of Tikhonov and Fenichel) with a known parameterization of the critical manifold. No a priori assumptions concerning separation of slow and fast variables are made, or necessary.We apply the theoretical results to chemical reaction networks with mass action kinetics admitting slow and fast reactions. For some relevant classes of such systems there exist canonical parameterizations of the variety of stationary points, hence the theory is applicable in a natural manner. In particular we obtain a closed form expression for the reduced system when the fast subsystem admits complex balanced steady states.