Geometric Considerations of a Good Dictionary for Koopman Analysis of Dynamical Systems

Representation of a dynamical system in terms of simplifying modes is a central premise of reduced order modelling and a primary concern of the increasingly popular DMD (dynamic mode decomposition) empirical interpretation of Koopman operator analysis of complex systems. In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants are to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. However, as far as we know there has not been an in depth study regarding the underlying geometry as related to an efficient representation. To this end we present that a good dictionary, that quite different from other's constructions, we need only to construct optimal initial data functions on a transverse co-dimension one set. Then the eigenfunctions on a subdomain follows the method of characteristics. The underlying geometry of Koopman eigenfunctions involves an extreme multiplicity whereby infinitely many eigenfunctions correspond to each eigenvalue that we resolved by our new concept as a quotient set of functions, in terms of matched level sets. We call this equivalence class of functions a ``primary eigenfunction" to further help us to resolve the relationship between the large number of eigenfunctions in perhaps an otherwise low dimensional phase space. This construction allows us to understand the geometric relationships between the numerous eigenfunctions in a useful way. Aspects are discussed how the underlying spectral decomposition as the point spectrum and continuous spectrum fundamentally rely on the domain.