CIMMS Lunchtime Series: Spectral Theory for Nonlinear Dynamical Systems

Dr. Igor Mezic, Department of Mechanical Engineering, University of California, Santa Barbara

Nonlinear dynamical systems theory relies mainly on geometric tools of the type first employed by Poincare. In this talk we show how a lifting (representation) of the action of a nonlinear dynamical system to a linear, Koopman operator action on an infinite-dimesional Hilbert space of observables leads to methods of analysis of nonlinear dynamical systems in spectral terms. This theory, that relies on tools developed originally by Wiener, von Neumann and Koopman, is used to address questions of model validation and model reduction for high-dimensional, nonlinear systems. Much use is made of linking geometric concepts such as attractors and stable and unstable manifolds to spectral properties of the Koopman operator. When an adjoint (Perron-Frobenius) lifting is used, problems related to uncertainty propagation can be treated. For high-dimensional systems, graph theory methods are used in conjunction with the Perron-Frobenius lifting to address those.