1. Why we need Koopman operator

Linearity is a desireable prpoperty in many problems. Linearization near a fixed point of a nonlinear system provides a local linearization expression, while Koopman operator provides a global linearization expression of the system using higher(infinite) number of dimansional nonlinear coordinates.

Under some constrains, Koopman decomposition is closely related with and approxiamted by DMD and DFT.

2. What is a Koopman operator

Koopman operator is an infinite-dimension linear system that advances ANY observation $g(x)$ one time step forward, e.g. $g(x) = x$ if use the measured observable directly. It’s a Markov operator.

\[\kappa_t g = g \circ F_t\]

Where $\kappa_t$ is our Koopman operator, and $F_t$ is the forward operator.

For a discrete-time system with timestep $\Delta t$, it can be write as

\[\kappa_t g(x_k) = g(F_t(x_k)) = g(x_{k+1})\]

For continous system, we have:

\[\frac{d}{dt}g=\kappa g\]

Operator $\kappa$ can be understand from the definition of differential, and linked with Koopman operator $\kappa_t$ by

\[\frac{d}{dt}g = \lim_{t \to 0} \frac{g(x_{k+1}-g{x_k})}{t} = \frac{\kappa_t g-g}{t} = \kappa g\]

The third term is a function of $g$ and can be denoted as $\kappa g$

3. How can I calculate/estimate Koopman eigenfunction?

We can calculate some Koopman eigenvectors if we have known governing equation of some simple forms. Mostly, we apply dynamical mode decomposition(DMD) to estimate it.