  Matching dynamical systems, through different forms of conjugacies and
equivalences, has long been a fundamental concept, and a powerful tool, in the
study and classification of nonlinear dynamic behavior (e.g. through normal
forms). In this paper we will argue that the use of the Koopman operator and
its spectrum is particularly well suited for this endeavor, both in theory, but
also especially in view of recent data-driven algorithm developments. We
believe, and document through illustrative examples, that this can nontrivially
extend the use and applicability of the Koopman spectral theoretical and
computational machinery beyond modeling and prediction, towards what can be
considered as a systematic discovery of "Cole-Hopf-type" transformations for
dynamics.
