# Mapping nonlinear systems into linear ones

This is a request for references. I'm not asking for the answer... Just give me a starting point! I'm interested in global linearization of linear equations. I guess the problem is well understood in math given its simplicity, but under "linearization" I only find local linearization (e.g. near equilibria), which I'm not interested in.

Consider a linear system of equations $$\dot{x} = L x$$ where $x$ is a vector. Now perform an invertible transformation $x \to y=f(x)$. In the new variables the system is in general nonlinear $$\dot{y} = f' L f^{-1} (y)$$ where $f'$ denotes the Jacobian matrix.

Now let's pose an inverse problem: Given a generic nonlinear system $$\dot{y}=F(y)$$ which general conditions should $F$ satisfy for there to exist a linear map $L$ and a transformation $f$ such that $F=f' L f^{-1}$? I would expect the answer to involve geometric concepts, like geodesic transport, parallelizability and the such...

Relaxing the hypothesis that $f$ is invertible, a sufficient condition to linearize globally the system with a change of variables $f$ is that $F$ is analytic. Then the Carleman linearization technique and other similar techniques (as can be seen in the book "Nonlinear dynamical systems and Carleman Linearization") can be applied to transform the nonlinear system in a infinite dimensional linear system.