# 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...

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Isn't that asking for the Abel-function/Abel-equation for F? see: en.wikipedia.org/wiki/Abel_equation –  Gottfried Helms Jun 7 '13 at 21:09

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.

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If the non-wandering set of Your system consists only of one equilibrium, Your local linearization will be global. You could also find a kind of global linearization (stable and unstable manifolds) with respect to a hyperbolic non-wandering set resp. hyperbolic measures, see for instance Katok, Hasselblatt: Introduction to the Modern Theory of Dynamical Systems. But your condition on a global linearization seems to be to strong in this setting.

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I have to think about this... Does this mean that a dynamical system can be linearized all over a neighborhood of an equilibrium point that is locally asymptotically stable, via a variable transformation? –  tomate Jun 9 '13 at 9:34