Dynamical uncertainty geometry for nonlinear transport

Abstract

Uncertainty in nonlinear dynamical systems is often organized by transport structures that are not apparent from posterior geometry alone. We introduce Dynamical Uncertainty Geometry (DUG), a framework that separates three ingredients that are frequently conflated: posterior uncertainty, future observations, and model-induced transport. DUG indexes posterior credible sets and transport-conditioned subsets by a common enclosed-probability coordinate, allowing physically meaningful classes to be tracked across credibility levels rather than examined at a single threshold. The framework combines a mass-ranked credible filtration, a description of future experiments through their induced probability laws and Fisher geometry, and a labelled transport skeleton representing dynamically distinct outcomes. We establish stability results for the resulting mass-indexed persistence modules under simultaneous perturbations of posterior density and probability measure, including continuum, refinement, and finite atomic formulations. These results provide quantitative control of persistence computed from numerical approximations and weighted grids. Two benchmark problems illustrate the framework. In a short-arc orbit-determination setting, DUG identifies dynamically distinct return classes within a connected uncertainty region and highlights a difference between information-based and local Fisher-based observation-design criteria. In the Earth-Moon planar circular restricted three-body problem, DUG reveals multiple first-hit transport outcomes coexisting within a connected credible region and provides diagnostics for assessing their numerical resolution. Together, these examples show how topological summaries of posterior geometry, when coupled to transport labels and future experiments, yield a richer description of uncertainty than either posterior probabilities or dynamical classifications alone.