On Comparison of Dynamics of Dissipative and Finite-Time Systems Using Koopman Operator Methods*

Comparison of different dynamical systems or a dynamical system with data is one of the core issues in both dynamical systems and control theory. The fruitful approaches are particularly hard to come by for systems and data that show nonlinear behavior and non-Gaussian noise characteristics. We describe a theory that utilizes spectral theory of linear operators (composition, or Koopman) to provide the methodology, that was originally formulated for measure-preserving systems. Here we extend it to capture dissipative and finite-time dynamics. The approach combines a version of ergodic partition theory with Hardy space theory in observable space (rather than in time).