A Minimal Set of Koopman Eigenfunctions -- Analysis and Numerics

Abstract

This work provides the analytic answer to the question of how many Koopman eigenfunctions are necessary to generate the whole spectrum of the Koopman operator, this set is termed as a \emph{minimal set}. For an $N$ dimensional dynamical system, the cardinality of a minimal set is $N$. In addition, a numeric method is presented to find such a minimal set. The concept of time mappings, functions from the state space to the time axis, is the cornerstone of this work. It yields a convenient representation that splits the dynamic into $N$ independent systems. From them, a minimal set emerges which reveals governing and conservation laws. Thus, equivalency between a minimal set, flowbox representation, and conservation laws is made precise. In the numeric part, the curse of dimensionality in samples is discussed in the context of system recovery. The suggested method yields the most reduced representation from samples justifying the term \emph{minimal set}