We systematically address the question of which small, bimolecular, chemical reaction networks endowed with mass-action kinetics are capable of Hopf bifurcation. It is easily shown that any such network must have at least three species and at least four irreversible reactions. We are able to fully classify three-species, four-reaction, bimolecular networks: with the extensive help of computer algebra, we divide these networks into those which forbid Hopf bifurcation and those which admit Hopf bifurcation. We find that a previously known example due to Thomas Wilhelm is only one of many networks in this class which admit Hopf bifurcation.

The task of deciding which small networks admit Hopf bifurcation naturally breaks into two parts. First, we focus on ruling out Hopf bifurcation in the great majority of the networks; and second, we focus on confirming, where possible, that a nondegenerate bifurcation occurs in the remaining networks.

Part I. Beginning with 14670 three-species, four-reaction, bimolecular networks which admit positive equilibria, we show that the great majority of these are incapable of Hopf bifurcation. Often we can declare the absence of Hopf bifurcation in a given network by proving the positivity of an associated polynomial. This task can be approached using software, including semidefinite programming, to decompose the polynomials into sums of squares and positive terms. At the end of this process, we are left with 138 networks with the potential for Hopf bifurcation. These fall into 87 distinct classes, up to a natural equivalence.

Part II. Having shown that there are 87 distinct classes of three-species, four-reaction, bimolecular chemical reaction networks with the potential for Hopf bifurcation, the next question is how many of these networks actually admit a nondegenerate Hopf bifurcation. Out of the 87 classes, we find that 86 admit nondegenerate Hopf bifurcation. The remaining exceptional network robustly admits a degenerate Hopf bifurcation.

Amongst the 86 networks capable of nondegenerate Hopf bifurcation, we find that 57 admit a supercritical Hopf bifurcation, and 54 admit a subcritical Hopf bifurcation. At the intersection of these networks are 25 networks that admit both bifurcations and hence can have both stable and unstable periodic orbits. These claims involve extensive use of computer algebra to automate the process of checking nondegeneracy and transversality conditions. With the help of these computations, we are able to show that many of the networks admit the coexistence of stable equilibrium and a stable periodic orbit for some choices of rate constants. We also show the occurrence of bifurcations of higher codimension in these networks.

Finally, we can use the results on three-species, four-reaction, and bimolecular networks, along with previously developed theory, to predict the occurrence of Hopf bifurcation in networks with more species and/or reactions. Thus, finding all small networks with the capacity for Hopf bifurcation greatly expands our knowledge of which chemical reaction networks, not necessarily small, admit Hopf bifurcation.

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