It has been a while since my last post. I have been busy! Today, I want to talk about some aspects of Koopman operator theory (I know this sounds spooky), but it is not that bad. It says that you can reformulate a nonlinear dynamical system into a (possibly) infinite-dimensional linear system. However, there are many cases where you don’t get an infinite-dimensional reformulation of the problem!
More or less, the trade-off there is that you boost into a higher-dimensional space, and in that higher-dimensional space, you can straighten out the nonlinearities.
Here is an example of the reformulation. This system is nonlinear!
\[\begin{pmatrix} \dot{x_1}\\ \dot{x_2} \end{pmatrix} = \begin{pmatrix} x_1\\ \lambda - x_1^2 \end{pmatrix}\]However, we can change this by adding a new dimension $x_3 = x_1^2$ and substitution back in to get a linear system with only one extra dimension!
\[\frac{dx_3(t)}{dt} = \frac{d}{dt}x_1^2(t) = 2x_1(t)\frac{dx_1(t)}{dt}= 2 x_1^2(t)\]With this we can rewrite $\dot{x_2} = \lambda - x_1^2(t) = \lambda - x_3$. And now we have a full blown linear system from the nonlinear one!
\[\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix} = \begin{pmatrix} 1&0&0\\ 0&0& -1\\ 0&0&2 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} + \begin{pmatrix} 0\\ \lambda\\ 0 \end{pmatrix}\]