Fr\'echet Means of Periodic Orbits in Dynamical Systems: Geometry, Dynamics, and Diagnostics

Abstract

In many dynamical systems applications, one seeks representative trajectories summarizing families of periodic orbits arising from parameter variation or uncertainty. While the Fr\'echet mean provides a natural notion of averaging in nonlinear metric spaces, its application to periodic trajectories is complicated by the interplay between geometric shape and temporal dynamics. Inspired by ideas from shape analysis, we develop a framework for computing Fr\'echet means of periodic orbits by representing trajectories as closed curves and introducing a metric structure on a quotient space that accounts for circular phase shifts. Within this framework, we establish the existence of empirical Fr\'echet means in the resulting infinite-dimensional quotient space. A central finding is that the resulting Fr\'echet mean depends strongly on the chosen parametrization: time parametrization preserves dynamical information, whereas arc length parametrization emphasizes geometric structure. To reconcile these viewpoints, we propose a decoupled approach that computes a geometric Fr\'echet mean using arc length parametrization and subsequently reconstructs representative dynamics through harmonic averaging of aligned speed profiles. We further introduce curvature- and medoid-based diagnostic measures that quantify the representativeness of the resulting mean and identify situations in which averaging produces geometric artifacts or fails to capture heterogeneous ensembles. Numerical experiments for the Van der Pol oscillator, the Rosenzweig-MacArthur predator-prey model, and the Morris-Lecar neuronal model demonstrate that the proposed methodology yields robust geometric summaries together with consistent averaged dynamics. The framework provides a principled approach for constructing representative periodic trajectories and assessing their validity in uncertainty quantification and dynamical systems.