Koopman operators in continuous dynamical systems

Abstract:

Mathematical models provide fundamental tools for understanding the evolution of real-world systems. Among them, dynamical systems offer a powerful framework for describing how a system evolves over time according to deterministic laws. While linear systems can be characterized using well-established techniques, nonlinear systems can exhibit complex phenomena such as multiple attractors, oscillations, and chaos.

This seminar explores how linear structures can be uncovered within nonlinear dynamical systems through the Koopman operator framework. Rather than studying the evolution of the state variables directly, Koopman theory considers the evolution of observables, such as energy or velocity. Although the underlying dynamics are nonlinear, the Koopman operator governing the evolution of observables is linear, at the cost of acting on an infinite-dimensional function space.

The talk will focus on the spectral analysis of the Koopman operator and its infinitesimal generator for continuous-time systems. Particular attention will be given to the choice of functional space, with an emphasis on the Segal–Bargmann space, a reproducing kernel Hilbert space particularly suited to systems with stable equilibria. Numerical and data-driven approaches will also be discussed, including the Adaptive Koopman Spectral Method and Dynamic Mode Decomposition (DMD), which allows spectral information to be extracted directly from time-series data.

The seminar will highlight how functional analysis and operator theory can provide a linear perspective on nonlinear dynamics, combining rigorous theoretical tools with practical computational methods.