Some minimal bimolecular mass-action systems with limit cycles

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