Global Convergence Regimes of the Return Dynamics in the Class ${\mathcal{O}}_C$

Abstract

This study investigates the global dynamics of discrete return maps $F:\partial C\to \partial C$ generated on the boundary of a convex core $C\subset {\mathbb{R}}^N$ via an outward normal displacement by a thickness function $d(c)$ followed by an inward normal return.By deriving a uniform global one-step asymptotic expansion preconditioned by the curvature operator $B_c=d(c)(I-d(c)S_c{)}^{-1}$, the authors establish a second-order expansion of the thickness increment governed by the preconditioned Hessian $K_c=B_c^{1/2}{H}_c{B}_c^{1/2}$. Under the spectral bound ${\lambda }_{\min }(K_c)\ge -2+\delta$, the map exhibits strict global thickness ascent, rendering $\mathcal{V}(c)=d_{\max }-d(c)$ a strict Lyapunov function. This guarantees finite dissipation, vanishing step sizes, and global convergence of orbits to unique critical points, with exponential rates at nondegenerate maxima and no nontrivial periodic behavior.Finally, numerical computations on the unit circle confirm that ${\lambda }_{\min }=-2$ marks a sharp stability threshold: crossing it leads to a period-doubling transition from curvature-preconditioned global convergence to stable period-two dynamics.