Global stability of perturbed complex-balanced systems

The paper proves that if a complex‑balanced mass‑action system (G, κ*) is robustly permanent, then on every stoichiometric compatibility class U there exists ε>0 such that for all κ in B(κ*,ε), (G,κ) has a unique globally attracting equilibrium in U. Its proof reduces the dynamics on U via a diffeomorphism to an s‑dimensional system (Theorem 3.4) and then applies Smith–Waltman’s perturbation theorem (Theorem 4.1) using the uniform compact attractor supplied by robust permanence; see the statement and proof of Theorem 4.2 and surrounding discussion. This checks all hypotheses (existence of a globally attracting steady state at κ*, spectral stability at that steady state, continuity of the Jacobian, and a common compact attractor K for all nearby κ) and yields the conclusion directly . The candidate solution proves the same result by a different route: it uses the Horn–Jackson free energy as a strict Lyapunov function for (G,κ*) on U to obtain a uniform margin of decrease on the common compact set K away from the equilibrium, extends this strict decrease to nearby κ by continuity, invokes hyperbolicity of the complex‑balanced equilibrium within U and the implicit function theorem to continue it uniquely as x̂(κ), and rules out any other equilibria by the strict Lyapunov inequality and robust permanence; convergence then follows from local exponential stability. Both arguments are sound and rest on the same core ingredients (robust permanence and linear stability of complex‑balanced equilibria). Minor presentation notes for the paper: the text asserts that the equilibrium at κ* is globally attracting without spelling out the standard Lyapunov/permanence argument, and there is a typographical error in the displayed Lyapunov formula in §3.1; neither affects correctness .