The smallest bimolecular mass-action systems admitting Andronov–Hopf bifurcation

We systematically address the question of which small bimolecular reaction networks endowed with mass-action kinetics are capable of Hopf bifurcation. It is easily shown that any such network must have at least $3$ species and at least $4$ reactions, and its rank is at least $3$. Terming the class of $n$-species, $m$-reaction, rank-$r$ networks as $(n,m,r)$ networks, we are able to fully classify bimolecular $(3,4,3)$ networks: with the extensive help of computer algebra, we divide these networks into those which forbid Hopf bifurcation and those which admit Hopf bifurcation.
Beginning with $14670$ bimolecular $(3,4,3)$ networks which admit positive equilibria, we show that the great majority of these are incapable of Hopf bifurcation. At the end of this process, we are left $138$ networks with the potential for Hopf bifurcation. These fall into $87$ distinct classes, up to a natural equivalence. Out of the $87$ classes we find that $86$ admit nondegenerate Hopf bifurcation (supercritical, subcritical, or both). The remaining exceptional class robustly admits a vertical Hopf bifurcation.
Finally, we can use the results on bimolecular $(3,4,3)$ networks, along with previously developed theory on inheritance, to predict the occurrence of Hopf bifurcation in networks with more species and/or reactions. Thus, in fact, finding all small networks with the capacity for Hopf bifurcation greatly expands our knowledge of which reaction networks, not necessarily small, admit Hopf bifurcation.
The talk is based on a recent joint paper with Murad Banaji:\\ \centerline{\url{https://iopscience.iop.org/article/10.1088/1361-6544/acb0a8/pdf}}