Some minimal bimolecular mass-action systems with 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.

References in corpus (3)

• The smallest bimolecular mass action reaction networks admitting Andronov-Hopf bifurcation
• Adding species to chemical reaction networks: preserving rank preserves nondegenerate behaviours
• Limit cycles in mass-conserving deficiency-one mass-action systems

Cited by in corpus (5)

• The smallest bimolecular mass action reaction networks admitting Andronov-Hopf bifurcation
• Oscillations in three-reaction quadratic mass-action systems
• Limit cycles in mass-conserving deficiency-one mass-action systems
• Adding species to chemical reaction networks: preserving rank preserves nondegenerate behaviours
• The inheritance of local bifurcations in mass action networks