Limit cycles in mass-conserving deficiency-one mass-action systems

TL;DR

This paper constructs simple, mass-conserving mass-action systems with two-dimensional stoichiometric subspaces that exhibit multiple limit cycles, illustrating complex dynamics within the scope of the Deficiency-One Theorem.

Contribution

It provides explicit examples of mass-conserving systems with limit cycles, including networks with multiple limit cycles, and demonstrates the absence of limit cycles in certain bimolecular cases.

Findings

Mass-conserving systems can have multiple limit cycles.

Limit cycles are present in trimolecular and tetramolecular networks.

Bimolecular networks with two-dimensional stoichiometric subspace do not admit limit cycles.

Abstract

We present some simple mass-action systems with limit cycles that fall under the scope of the Deficiency-One Theorem. All the constructed examples are mass-conserving and their stoichiometric subspace is two-dimensional. Using the continuation software MATCONT, we depict the limit cycles in all stoichiometric classes at once. The networks are trimolecular and tetramolecular, and some exhibit two or even three limit cycles. Finally, we show that the associated mass-action system of a bimolecular reaction network with two-dimensional stoichiometric subspace does not admit a limit cycle.