Mass action systems: two criteria for Hopf bifurcation without Hurwitz

We state two sufficient criteria for periodic oscillations in mass action systems. Neither criterion requires a computation of the Hurwitz determinants. The first criterion also applies to general kinetics and it concerns fully-open systems: in essence, it guarantees periodic oscillations whenever the system admits a steady state with a simple pair of complex conjugate eigenvalues with positive real part. The second criterion states an algebraic condition based on the steady-state Jacobian matrix expressed in convex coordinates via Stoichiometric Network Analysis. Both criteria have the potential for extensions and generalizations.