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 <italic>Pontryagin-Rodygin theorem</italic> , provides a leading-order approximation for this slow drift, the connection to the underlying geometry described by <italic>geometric singular perturbation theory</italic> (GSPT)—also known as <italic>Fenichel theory</italic> —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 constructs an averaged system that is guaranteed to govern the dynamics on the persisting normally hyperbolic manifold to any perturbative 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.