Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems

A family of polynomial differential systems describing the behavior of a chemical reaction network with generalized mass action kinetics is investigated. The coefficients and monomials are given by graphs. The aim of this investigation is to clarify the algebraic-discrete aspects of a Hopf bifurcation in these special differential equations. We apply concepts from toric geometry and convex geometry. As usual in stoichiometric network analysis we consider the solution set as a convex polyhedral cone and we intersect it with the deformed toric variety of the monomials. Using Gröbner bases the polynomial entries of the Jacobian are expressed in different coordinate systems. Then the Hurwitz criterion is applied in order to determine parameter regions where a Hopf bifurcation occurs. Examples from chemistry illustrate the theoretical results.