Oscillations in planar deficiency-one mass-action systems

TL;DR

This paper explores oscillatory behaviors in deficiency-one mass-action systems, providing examples with centers and multiple limit cycles, highlighting the complex dynamics beyond deficiency-zero systems.

Contribution

It introduces new examples of oscillations in deficiency-one systems, expanding understanding of their dynamic behaviors compared to previously studied deficiency-zero cases.

Findings

Existence of centers in deficiency-one systems

Presence of multiple limit cycles

Oscillatory behaviors are common in deficiency-one networks

Abstract

Whereas the positive equilibrium of a mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, with centers or multiple limit cycles.