  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.
