Mass action systems: two criteria for Hopf bifurcation without Hurwitz

The paper’s Theorem 5.2 proves the periodic-orbit claim via a global Hopf bifurcation argument: along any analytic path of positive diagonal scalings h=diag(1/x̄) connecting D1 and D2, the Jacobian remains invertible and a net change of inertia guarantees periodic orbits by Fiedler’s global Hopf theorem (stated as Theorem 5.1 in the paper) . The model instead asserts a local Hopf bifurcation by claiming (i) existence of a simple pair ±iω and (ii) transversality along a one-parameter diagonal curve, both obtained via generic perturbations inside the diagonal parameter space. However, the paper explicitly identifies precisely this step as an open linear algebra question Q* (existence of a path of positive diagonal matrices for which the crossing is simple and transverse) and avoids it by relying on global Hopf; it states “we do not know the answer to Q* yet” . The model’s core unproven steps therefore hinge on the still-open Q*, whereas the paper’s argument is complete. Both agree on the Jacobian factorization Jac = B(v̄) diag(1/x̄) with B(v̄)=N diag(v̄) Y^T , but the model’s claimed local-Hopf construction lacks the missing hypotheses needed to resolve Q*.