# Example for Carleman Linearization resulting in a linear system

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

• is preferably a well understood nonlinear system such as an multi fluid tank system, a single or double inverse pendulum, Dubins car, SIR model, Lotka-Volterra ... but that is not strictly necessary for some initial conditions of your choosing that show non linearity.
• $$\frac{dx}{dt} = f(x,t)$$ has a multidimensional $$x$$
• $$f(x,t)$$ is non linear in $$x$$

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 successfull in reproducing the dynamics?