typically infinite-dimensional operator, which is capable of describing the full dynamics of an underlying system. For this paper we focus on the practical implementation of Koopman operator theory. An in-depth overview about theoretical concepts and applications can be found in [@Budisic.2012].

Each dynamic system of the form with state $x \in \mathbb{R}^n$ and time $t$ has a linear representation of the form $$\frac{d}{dt}g(x) = \mathcal{K} g(x),$$

where $g = [h_1(x), ...]^T$ is a function of possibly infinite observables $h_i(x)$ of the state. The Koopman operator $\mathcal{K}$ evolves observable functions $h_i(x)$ instead of direct traces of the states, which can be arbitrary functions. An example is shown in Eq. . $$\label{eq:observables_example}
g(x) = [h_1(x), h_2(x)...]^T = [1, x, x^2, x^3, \sin(x), \sqrt{x}, e^x, ...]^T$$

These observable functions span a subspace of the Hilbert space[@BruntonKoopmanInvariant.2016b]. In general, the resulting system is infinite dimensional but linear. This enables tools like spectral analysis without any inaccuracy caused by linearization[@Budisic.2012]. In 2005, Igor Mezic presented the application of the Koopman operator for spectral analysis of high-dimensional, nonlinear systems[@Mezic.2005]. The Dynamic Mode Decomposition (DMD) extracts eigenvalues and modes of the Koopman operator from measurement data[@Schmid2010], i.e. it finds the most probable system matrix $A$ for The 