# Dynamical systems via Koopman operators

## Composition operators, Dynamic Extended Mode decompositions… NB: Koopman B.O. Koopman not S.J. Koopman, who also works in dynamical systems.

I do not know how Koopman operators work, especially in the learning setting, but maybe this fragment of abstract will do for now :

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.

There is a brief literature review in Klus et al. (2020).

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 $$x(t)$$

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

## Recursive estimation

See recursive identification for generic theory of learning under the distribution shift induced by a moving parameter vector.

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