Bifurcations leading to oscillation in small chemical reaction networks

Which systems of chemical reactions permit oscillation? The study of chemical oscillations has a long history, but we are still far from any structural characterisation of oscillatory reaction networks. I'll summarise recent work on identifying Andronov-Hopf and Bautin bifurcations, leading to oscillation and even multiple nondegenerate periodic orbits, in bimolecular mass action networks. In fact, a mixture of theory and computational algebra allows us to fully enumerate networks of minimal size admitting these bifurcations, including checking nondegeneracy and transversality conditions across the whole parameter space. We find that bifurcations leading to (stable) oscillation in small networks are not quite as rare as often believed. The results can be used alongside previous theory on "inheritance" to identify larger oscillatory networks via examination of their subnetworks. This is joint work with Balázs Boros.