May 20 and June 10, 2022 – The Mezić Group is pleased to host Allan Avila from AIMdyn Inc.
Allan Avila from AIMdyn Inc., USA will give a Zoom seminar on May 20 and June 10, 2022 at 9AM PST.
Title: Spectral Properties of Pullback Operators on Vector Bundles of a Dynamical System
Abstract: The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this talk, we consider induced linear operators acting on the space of sections of the tangent, cotangent, and tensor bundles of the state space.
We begin by first demonstrating how these operators are indeed natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators' spectra and characterize the algebraic and differential topological properties of their spectrum. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an ``eigendistribution", provide conditions for an eigendistribution to be integrable and demonstrate how to recover the foliations arising from their integral manifolds via the level sets of certain Koopman eigenfunctions. Lastly, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle.
We then apply our results to generalize the well-known fact that the flows of commuting vector fields commute and recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold's Cat map via the spectrum of the induced operator on sections of the tangent bundle. If time permits, we will describe a numerical method for computing the spectrum of the induced operators on sections of the cotangent bundle.
Short Bio: Allan received his B.S. in Mechanical Engineering from the University of California Riverside in 2015, and an M.S. and PhD in Mechanical Engineering from the University of California Santa Barbara in 2020. He is a member of the Society of Industrial and Applied Mathematics, and a 2018 Hispanic Scholarship recipient. Allan is now a researcher at AIMdyn Inc. and his general research is on data-driven analysis and forecasting of dynamical systems via spectral operator methods.