### A Floquet-Liapunov theorem in Fréchet spaces

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We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equation$$x\frac{d\overrightarrow{y}}{dx}={\overrightarrow{G}}_{0}\left(x\right)+\left[\lambda \left(x\right)+{A}_{0}\right]\overrightarrow{y}+{x}^{\mu}\overrightarrow{G}(x,\overrightarrow{y}),$$the idea of our proof is to interpret a formal power series solution...

Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local...