Equilibria and Their Stability in Networks with Steep Sigmoidal Nonlinearities

The paper’s Theorem 3.11 states exactly the target result: for a regular switching parameter Z, a zero-dimensional cell τ is an equilibrium cell iff τ is loop characteristic and Φj(τ)=0 for all j, and moreover every sigmoidal perturbation S(Z,ε) has a unique equilibrium xε with xε→τ as ε→0; see Theorem 3.11 and its proof in Section 6 . The paper proves necessity by contradiction using the regularity condition and sufficiency via sign changes on opposite faces of a small box τη around τ, applying the Intermediate Value Theorem and Brouwer’s fixed point theorem for existence, and then the Implicit Function Theorem plus a Jacobian estimate (eq. (6.9)) for uniqueness; cf. the construction (6.3)–(6.4), the existence step, and the non-singularity/IFT step . The candidate solution follows a different but sound route: it restates the local decoupling (paper’s Lemma 5.2) to show that, in a neighborhood of τ, each Λi depends only on xρ−1(i) among singular directions, then invokes Poincaré–Miranda for existence and a ρ-cyclewise one-dimensional monotone return-map argument for uniqueness. This is consistent with the paper’s local decomposition (Sec. 5.1) and the sign pattern encoded by Φ; cf. Lemma 5.2 and the cyclic-feedback decomposition . Minor phrasing in the model about constants (Ai,Ci) can be interpreted as ε-dependent across the box, which does not affect the steps used. Hence both are correct and logically consistent; they use different proof tools (Brouwer/IFT vs. Poincaré–Miranda/monotone composition).