The theory of deterministic dynamical systems:

In the abstract, a dynamical system consists of a set of states and a rule for the evaluation of a system state. This general viewpoint may be applied to practically any deterministic dynamical system that evolves in time.

Continuous-time dynamics may be represented as:

\begin{equation} \dot{x} = f(x) \end{equation}

where \(x \in S \subset \mathbb{R}^n\) and \(f: S \rightarrow \mathbb{R}^n\) is a vector field on that state space.

We may also consider a dynamical system given by the discrete-time map:

\begin{equation} \forall k \in \mathbb{Z}, x_{k+1} = T \circ x_k \end{equation}

where \(x_k \in S \subset \mathbb{R}^n\), \(k\) is a discrete-time index and \(T: S \rightarrow S\) is a dynamic map.

While the discrete-time representation of dynamical systems usually doesn’t show up in the representation of physical systems, we may use it to represent discrete-time sampling of these systems.

This representation is also more practical because the data collected from dynamical systems always comes in discrete-time samples.

Data as evaluations of functions of the state:

Within the context of dynamical systems, we may interpret data as knowledge of variables related to the system’s state. In fact, we may think of data as evaluations of functions of the state, known as observables.

The collection of observables forms a vector space:

Let \(g: S \rightarrow \mathbb{R}\) be a real-valued observable of the dynamical system. As a corollary of the Buckingham-Pi theorem, the collection of all such observables forms a vector space.

The Koopman Operator \(U\) is a linear transform of this vector space given by:

\begin{equation} Ug(x) = g \circ T(x) \end{equation}

which in a discrete setting implies:

\begin{equation} Ug(x_k) = g \circ T(x_k) = g(x_{k+1}) \end{equation}

where the linearity of the Koopman Operator follows from the linearity of the composition operator:

\begin{equation} U \circ (g_1 + g_2)(x) = g_1 \circ T(x) + g_2 \circ T(x) = Ug_1(x) + Ug_2(x) \end{equation}

So we may think of the Koopman Operator as lifting dynamics of the state space to the space of observables.

References:

  1. Bernard Koopman. Hamiltonian systems and Transformations in Hilbert Space.

  2. Hassan Arbabi. Introduction to Koopman operator theory for dynamical systems. MIT. 2020.

  3. Steven L. Brunton. Notes on Koopman operator theory. 2019.