Symbolic extended dynamic mode decomposition

In this paper, we present a new method of performing extended dynamic mode decomposition (EDMD) on systems, which admit a symbolic representation. EDMD generates estimates of the Koopman operator, K, for a dynamical system by defining a dictionary of observables on the space and producing an estimate, Km, which is restricted to be invariant on the span of this dictionary. A central question for the EDMD is what should the dictionary be? We consider a class of chaotic dynamical systems with a known or estimable generating partition. For these systems, we construct an effective dictionary from indicators of the “cylinder sets,” which arise in defining the “symbolic system” from the generating partition. We prove strong operator topology convergence for both the projection onto the span of our dictionary and for Km. We also prove practical finite-step estimation bounds for the projection and Km as well. Finally, we demonstrate some numerical results on eigenspectrum estimation and forecasting applied to the dyadic map and the logistic map.