Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification

Abstract

While invariant measures are widely employed to analyze physical systems when a direct study of pointwise trajectories is intractable, e.g., due to chaos or noise, they cannot uniquely identify the underlying dynamics. Our first result shows that, in contrast to invariant measures in state coordinates, e.g., $[x(t), y(t), z(t)]$, the invariant measure expressed in time-delay coordinates, e.g., $[x(t), x(t-τ),\ldots, x(t-(m-1)τ)]$, can identify the dynamics up to a topological conjugacy. Our second result resolves the remaining ambiguity: by combining invariant measures constructed from multiple delay frames with distinct observables, the system is uniquely identifiable, provided that a suitable initial condition is satisfied. These guarantees require informative observables and appropriate delay parameters ($m,τ$), which can be limiting in certain settings. We support our theoretical contributions through a series of physical examples demonstrating how invariant measures expressed in delay-coordinates can be used to perform robust system identification in practice.