Predictability of Observables of Dynamical Systems

TL;DR

This paper investigates how the evolution of observables in dynamical systems can be characterized, providing minimal order representations for linear systems and a new framework for nonlinear systems to infer observable dynamics from output history.

Contribution

It introduces a minimal order closure for linear systems and the concept of diminishing ambiguity for nonlinear systems, enabling the approximation of observable dynamics from output history.

Findings

Linear observables satisfy a closed differential equation with minimal order.

Nonlinear systems can be approximated using delay differential equations.

Framework clarifies when observable dynamics can be inferred from output history.

Abstract

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.