The smallest bimolecular mass action reaction networks admitting Andronov–Hopf bifurcation


Banaji, M. and Boros, B. 2023. The smallest bimolecular mass action reaction networks admitting Andronov–Hopf bifurcation. Nonlinearity. 36 (2), pp. 1398-1433.
TitleThe smallest bimolecular mass action reaction networks admitting Andronov–Hopf bifurcation
AuthorsBanaji, M. and Boros, B.

We address the question of which small, bimolecular, mass action chemical reaction networks (CRNs) are capable of Andronov–Hopf bifurcation (from here on abbreviated to ‘Hopf bifurcation’). It is easily shown that any such network must have at least three species and at least four irreversible reactions, and one example of such a network with exactly three species and four reactions was previously known due to Wilhelm. In this paper, we develop both theory and computational tools to fully classify three-species, four-reaction, bimolecular CRNs, according to whether they admit or forbid Hopf bifurcation. We show that there are, up to a natural equivalence, 86 minimal networks which admit nondegenerate Hopf bifurcation. Amongst these, we are able to decide which admit supercritical and subcritical bifurcations. Indeed, there are 25 networks which admit both supercritical and subcritical bifurcations, and we can confirm that all 25 admit a nondegenerate Bautin bifurcation. A total of 31 networks can admit more than one nondegenerate periodic orbit. Moreover, 29 of these networks admit the coexistence of a stable equilibrium with a stable periodic orbit. Thus, fairly complex behaviours are not very rare in these small, bimolecular networks. Finally, we can use previously developed theory on the inheritance of dynamical behaviours in CRNs to predict the occurrence of Hopf bifurcation in larger networks which include the networks we find here as subnetworks in a natural sense.

KeywordsPaper, chemical reaction networks, bimolecular networks, Hopf bifurcation, Bautin bifurcation, 34C23, 34C25, 37G10, 80A30
Sustainable Development Goals9 Industry, innovation and infrastructure
Middlesex University ThemeCreativity, Culture & Enterprise
PublisherIOP Publishing
Publication dates
Online23 Jan 2023
Print23 Jan 2023
Publication process dates
Deposited23 Jan 2023
Submitted07 Aug 2022
Accepted05 Jan 2023
Output statusPublished
Publisher's version
Digital Object Identifier (DOI)
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