Predictability of Observables of Dynamical Systems

We study the evolution of observables of dynamical systems. For linear systems, we show that observables satisfy a closed differential equation whose minimal order is determined by the dynamical system and observation operator. This yields a minimal order closure and an equivalent discrete delay representation of the observable dynamics. For nonlinear systems we introduce the notion of diminishing ambiguity, which provides a framework under which the instantaneous observable dynamics can be approximately determined from sufficiently long output history, resulting in delay differential equation representation. These results clarify when observable dynamics can be inferred from past history without knowledge of the dynamical system and its full state.