A coordinate-independent Pontryagin-Rodygin theorem for slow-fast averaging

The slow drift along a manifold of periodic orbits is a key mathematical structure underlying bursting dynamics in many scientific applications. While classical averaging theory, as formalised by the Pontryagin-Rodygin theorem, provides a leading-order approximation for this slow drift, the connection to the underlying geometry described by Geometric Singular Perturbation Theory (GSPT)--also known as Fenichel theory--is often not explicit, particularly at higher orders. This paper makes that connection rigorous and constructive using the parametrisation method. We provide a detailed, self-contained exposition of this functional analytic technique, showing how it synthesizes the geometric insight of invariant manifold theory with a systematic, perturbative algorithm. By treating the manifold's embedding and the reduced flow as coupled unknowns, the method provides a constructive proof of an averaged system that is guaranteed to be geometrically consistent with the persistence of the normally hyperbolic manifold to any order. We translate the abstract theory into a concrete computational procedure using Floquet theory, spectral analysis, and the Fredholm alternative, yielding a practical guide for computing high-accuracy, higher-order averaged models, and we demonstrate its implementation, both analytically and numerically, through specific examples.