Some minimal bimolecular mass-action systems with limit cycles

TL;DR

This paper examines minimal bimolecular mass-action systems with three species that exhibit limit cycles, and constructs higher-dimensional systems with stable limit cycles, revealing complex dynamical behaviors.

Contribution

It introduces minimal bimolecular systems with limit cycles and extends these to four-species systems with coexistence of multiple attractors.

Findings

Existence of stable limit cycles in minimal three-species systems.

Coexistence of stable equilibrium, unstable, and stable limit cycles.

Construction of four-species systems with stable limit cycles.

Abstract

We discuss three examples of bimolecular mass-action systems with three species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants and then exhibits stable limit cycles, but no chaotic behaviour. For some rate constants in the Feinberg--Berner system, a stable equilibrium, an unstabe limit cycle, and a stable limit cycle coexist. All three networks are minimal in some sense. By way of homogenising the above three examples, we construct bimolecular mass-conserving mass-action systems with four species that admit a stable limit cycle. The homogenised Feinberg--Berner system and the homogenised Wilhelm--Heinrich system admit the coexistence of a stable equilibrium, an unstable limit cycle, and a stable limit cycle.