\(
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bz}{\mathbf{z}}
\newcommand{\bL}{\mathbf{L}}
\)
In this review, we will consider dynamical systems of the form
%
\begin{align}
\frac{d}{dt}\bx(t) = \mathbf{f}(\bx(t)),\label{Eq:ContinuousDynamicsGeneral}
\end{align}
%
where $\bx\in\mathcal{X}\subseteq\mathbb{R}^n$ is the state of the system and $\mathbf{f}$ is a vector field describing the dynamics.
In general, the dynamics may also depend on time $t$, parameters $\boldsymbol{\beta}$, and external actuation or control $\bu(t)$.
Although we omit these here for simplicity, they will be considered in later sections.
A major goal of modern dynamical systems is to find a new vector of coordinates $\bz$ such that either
\begin{equation}
\bx=\boldsymbol{\varphi}(\bz) \quad\text{ or }\quad \bz=\boldsymbol{\varphi}(\bx) \label{eq:linearizing-coordinates}
\end{equation}
where the dynamics are simplified, or ideally, linearized:
\begin{align}
\frac{d}{dt}\bz = \bL \bz.
\end{align}
While in geometric dynamics, one asks for homeomorphic
Koopman's 1931 paper was central to the celebrated proofs of the ergodic theorem by von Neumann~\cite{neumann1932pnas} and Birkhoff~\cite{birkhoff1931pnas,birkhoff1932pnas}.
The history of these developments is fraught with intrigue, as discussed by Moore~\cite{moore2015pnas}.