The Carleman linearization came to my attention due to this article. I tried to understand this method but so far i wasn't succesful i tried to understand page 39 of this presentation however the example didn't make sense for me.
Could someone demonstrate how to compute a Carleman linearization and demonstrate/argue that the resulting linear ODE behaves similar to the non linear system?
I would prefer a demonstration that
- $\frac{dx}{dt} = f(x,t)$ has a multidimensional $x$
- $f(x,t)$ is non linear in $x$
- It would be nice if the nonlinear system is well understood. Examples would be the single or double inverse pendulum, Dubins car, SIR model, Lotka-Volterra but that is not a requirement
- shows what role initial conditions play
If there is some visualization (for example a phase portrait) that shows how a finite approximation of the infinite dimensional linear equation breaks and how it gets better if a larger finite dimensional approximation is used please also add that.
Are there some well understood conditions when a Carleman linearization will be non successful in reproducing the dynamics?
