Koopmanoperator

The Koopman operator is a concept in dynamical systems theory that provides a linear perspective on the evolution of nonlinear systems. Instead of tracking the state of a system directly, the Koopman operator describes how scalar observables (functions of the state) evolve over time. This operator acts on the space of these observables, rather than the state space itself.

A key advantage of the Koopman operator is that its eigenvalues and eigenfunctions can reveal fundamental properties of the underlying nonlinear dynamics, such as periodicities and growth rates. For finite-dimensional systems, the Koopman operator is an infinite-dimensional operator. However, for certain classes of systems, it is possible to find finite-dimensional approximations or to construct a finite basis of "Koopman eigenfunctions" that are, in essence, coordinates in which the original nonlinear system behaves linearly.

The study of the Koopman operator has found applications in various fields, including fluid dynamics, robotics, and control theory. Techniques like Dynamic Mode Decomposition (DMD) are closely related to approximating the Koopman operator from data. By decomposing the system's dynamics into a set of modes characterized by frequencies and growth rates, DMD and similar methods offer a powerful tool for analyzing and predicting the behavior of complex systems, even when their underlying equations are unknown or too difficult to solve analytically. The spectral properties of the Koopman operator offer a bridge between the often complex, nonlinear state-space description and a simpler, linear spectral decomposition.