Normal Hyperbolicity in Secondary Hopf Bifurcations

The paper’s Theorem 2 states that the torus born at a secondary Hopf (Neimark–Sacker) bifurcation is normally hyperbolic and has exactly ns stable directions in the subcritical case and ns+1 in the supercritical case, and proves this in two steps: (i) the invariant circle of the Poincaré map in a Neimark–Sacker bifurcation is normally hyperbolic (citing Chaperon), and (ii) its saturation by the flow is a normally hyperbolic torus (via Fenichel’s thin surface section) . The candidate solution reaches the same conclusion using a different route: it derives uniform normal expansion/contraction for the Poincaré map from the Neimark–Sacker normal form and then transports these rates to the flow using bounded return times and an adapted metric (Hirsch–Pugh–Shub). Both arguments are consistent with the bifurcation set-up and the stable-dimension count implied by Hypothesis (S) and Theorem 1 . Minor presentation gaps remain (e.g., explicit regularity needed for normal form and Lyapunov coefficient), but they do not affect the main correctness.