Abstract
This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis, and the global operator theoretic Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory.
Highlights
SignificanceTwo approaches to analyzing dynamic systems are the geometric approach and operator theoretic approach, exemplified in recent years by invariant manifolds and Koopman operators
Electrical & Computer Engineering & C3 S2, The Clarkson Center for Complex Systems Science, Clarkson University, Potsdam, NY 13699, USA
The geometric invariant manifold approach is closely related to the instantaneous version of the finite-time Lyapunov exponent (FTLE) field
Summary
Two approaches to analyzing dynamic systems are the geometric approach and operator theoretic approach, exemplified in recent years by invariant manifolds and Koopman operators. The geometric invariant manifold approach is closely related to the instantaneous version of the finite-time Lyapunov exponent (FTLE) field. The very different, spectral and measure-based operator theoretic approach of evolution operators, known as “Koopmanism” involves Koopman eigenfunctions (KEIGs). We ask a simple question, “Is the FTLE field a KEIG?” The answer is: in general, no. This motivates the explicit construction of vector fields where the answer is yes, in the sense that the FTLE field in the infinitesimal time limit, i.e., the instantaneous. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure
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