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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?

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