The Smallest Bimolecular Mass-Action System with a Vertical Andronov–Hopf Bifurcation

The paper proves, for the mass-action system (2) at κ1 = κ2 + κ3, that V(x,y,z) = (κ2/2)(x−y+z)^2 + 2κ2xy − 2κ4 log(xy) is a first integral, identifies the stable manifold as {x−y+z=0, xy=κ4/κ2}, and constructs an analytic two-dimensional invariant surface M foliated by periodic orbits which, together with the equilibrium, contains every positive ω-limit set. It does so via an analytic change of variables decoupling a Hamiltonian (p,q)-subsystem and a contracting r-dynamics, yielding a unique periodic orbit on each level set (Theorem 1) . The candidate solution establishes the same three conclusions by closing the (s,p) subsystem, proving V̇≡0 directly, characterizing the stable manifold via convexity of V, and reducing the remaining equation to a nonautonomous logistic ODE with periodic coefficients that has a unique globally attracting periodic solution. The statements match the paper’s theorem for interior (x,y>0) initial data; the proofs are different but consistent. The candidate does not address the boundary analysis (paper’s Theorem 2), but that lies beyond the core claims of Theorem 1.