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Calcul moulien et théorie des formes normales classiques et renormalisées

Abstract : This text is about normal forms of differential equations. The first part of this text deals with retarded (or delayed) differential equations; these equations appear for example in tide computations and also in physiological modeling, where they have a particular interest. The search of normal forms of retarded differential equations is made difficult as the initial conditions space is of infinite dimension. We present a computation by Faria which reduces this difficulty: first we make a projection on a finite dimensional central manifold, on which it is possible to make a classical (i.e. Poincaré-Dulac) normal form computation. Then we extend this result with the help of Gaeta's method of renormalization. By using these two methods, we prove a theorem giving existence of a renormalized normal form of a retarded differential equation. In the second part, we present the mould calculus by Jean Écalle. We use this formalism to compute normal forms of formal vector fields and apply it then to hamiltonian formal vector field in cartesian coordinates, then in action-angle coordinates. We obtain then a new proof of a formal version of Birkhoff's theorem on normal forms, and Kolmogorov's theorem. We present also a Maple worksheet, which shows how easily mould can be implemented in formal computations.
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https://tel.archives-ouvertes.fr/tel-00521709
Contributor : Guillaume Morin <>
Submitted on : Tuesday, September 28, 2010 - 2:12:35 PM
Last modification on : Monday, December 14, 2020 - 9:48:48 AM
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  • HAL Id : tel-00521709, version 1

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Guillaume Morin. Calcul moulien et théorie des formes normales classiques et renormalisées. Mathématiques [math]. Observatoire de Paris, 2010. Français. ⟨tel-00521709⟩

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