# Inheritance of oscillation in chemical reaction networks

## TL;DR

This paper investigates how oscillatory behavior in chemical reaction networks (CRNs) can be inherited when networks are enlarged, providing conditions under which oscillations are preserved, especially in fully open CRNs with mass action kinetics.

## Contribution

It establishes four key methods for enlarging CRNs while maintaining oscillation capacity and extends results to various kinetics, highlighting the importance of small oscillatory motifs.

## Key findings

- Enlargement methods preserve oscillation in CRNs.
- Nondegenerate and stable oscillations are inherited in fully open CRNs.
- Small oscillatory motifs are significant sources of oscillation.

## Abstract

Some results are presented on how oscillation is inherited by chemical reaction networks (CRNs) when they are built in natural ways from smaller oscillatory networks. The main results describe four important ways in which a CRN can be enlarged while preserving its capacity for oscillation. The results are for general CRNs, not necessarily fully open, but lead to an important corollary for fully open networks: if a fully open CRN R with mass action kinetics admits a nondegenerate (resp., linearly stable) periodic orbit, then so do all such CRNs which include R as an induced subnetwork. This claim holds for other classes of kinetics, but fails, in general, for CRNs which are not fully open. Where analogous results for multistationarity can be proved using the implicit function theorem alone, the results here call on regular and singular perturbation theory. Equipped with these results and with the help of some analysis and numerical simulation, lower bounds are put on the proportion of small fully open CRNs capable of stable oscillation under various assumptions on the kinetics. This exploration suggests that small oscillatory motifs are an important source of oscillation in CRNs.