# Dynamical systems via Koopman operators

(Koopman here is B. O. Koopman (Koopman 1931) not SJ Koopman who also works in dynamical systems.)

I do not know how this works, but maybe this fragment of abstract will do for now (Budišić, Mohr, and Mezić 2012):

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on “dynamics of states.” While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of “big data” measurements. This overview article presents an alternative framework for dynamical systems, based on the “dynamics of observables” picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics.

The Chaos book folks describe it so:

the Koopman operator action on a state space function $$a(x)$$ is to replace it by its downstream value time $$t$$ later, $$a(x) \rightarrow a(x(t)),$$ evaluated at the trajectory point $$x(t)$$

$\left[\mathcal{K}^{t} a\right](x)=a\left(f^{t}(x)\right)=\int_{\mathcal{M}} d y \mathcal{K}^{t}(x, y) a(y),$ $$\mathcal{K}^{t}(x, y)=\delta\left(y-f^{t}(x)\right)$$

Arbabi, Hassan. 2020. “Introduction to Koopman Operator Theory of Dynamical Systems,” 32. https://www.mit.edu/~arbabi/research/KoopmanIntro.pdf.

Budišić, Marko, Ryan Mohr, and Igor Mezić. 2012. “Applied Koopmanism.” Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (4): 047510. https://doi.org/10.1063/1.4772195.

Koopman, B. O. 1931. “Hamiltonian Systems and Transformation in Hilbert Space.” Proceedings of the National Academy of Sciences 17 (5): 315–18. https://doi.org/10.1073/pnas.17.5.315.

Mauroy, Alexandre, and Jorge Goncalves. 2020. “Koopman-Based Lifting Techniques for Nonlinear Systems Identification.” IEEE Transactions on Automatic Control 65 (6): 2550–65. https://doi.org/10.1109/TAC.2019.2941433.

Williams, Matthew O., Ioannis G. Kevrekidis, and Clarence W. Rowley. 2015. “A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition.” Journal of Nonlinear Science 25 (6): 1307–46. https://doi.org/10.1007/s00332-015-9258-5.