Chapter 6 Analytic ordinary differential equations and their local classification

Abstract

The article is devoted to the local theory of analytic differential equations. We describe classification of meromorphic systems ẋ = A ( t ) x near regular and irregular singular point. We present a local theory of non-linear holomorphic equations ẋ = V ( x ), with the proof of the resolution theorem and its application to the center–focus problem, with Ecalle–Voronin–Martinet–Ramis moduli and with Briuno–Yoccoz classification of non-resonant saddles. Finally, we present formal classification of nilpotent singularities and prove analyticity of the Takens prenormal form.