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Abstract Résumé Dans cette thèse on s'intéressè a deuxprobì emes faisant intervenir des formes normales de champs de vecteurs et des phénomènes exponentiellement petits ,
´ equilibre d'une famille de systèmes hamiltoniens au voisinage d'une résonance 0 2 i?. On démontre l'existence d'une famille d'orbites périodiques entourant l'´ equilibre puis l'existence d'orbites homoclinesàhomoclines`homoclinesà plusieurs bouclesàbouclesà chacune de ces orbites périodiques, aussi proche de cetéquilibrecetéquilibre que l'on veutàveut`veutà l'exception de l'´ equilibre lui-même ,
On obtient ensuite le résultat grâcè a des arguments géométriques liésliésà la petite dimension etàetà un théorème KAM qui permet de confiner les boucles. Pour le mêmeprobì eme dans le cadre d'un champ de vecteurs réversible non hamiltonien , l'apparition d'exponentiellement petits lors de la perturbation de l'orbite homocline de la forme normale empêche la démonstration de l'existence d'orbites homoclinesàhomoclines`homoclinesà des orbites périodiques de taille exponentiellement petite ,