Jean-Pierre Ramis

Problèmes De Modules Pour Des Équations Différentielles Non Linéaires Du Premier Ordre

© Publications mathématiques de l’I.H.É.S., 1982, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Complexe dualisant et théorèmes de dualité en géométrie analytique complexe

© Publications mathématiques de l’I.H.É.S., 1970, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Analytic Classification of Resonant Saddles and Foci

Publisher Summary This chapter describes analytical classification of resonant saddles and foci. The chapter applies some earlier results to the study of resonant analytic saddles or foci in the real plane. The chapter classifies exhaustively, up to local analytic transformations of R , the phase portraits near zero of the differential systems. The resonance means that, at first order, the phase portraits are defined by polynomials. But, in general, phase portraits for saddles and foci are not linearizable; in case Foci, linearizability means that one has a center instead of a focus; in this case, a finite number of tests on the Taylor series provides an order k at which the system “essentially” departs from the linear one. This chapter deals only with such systems, which are often called “weak” saddles or foci or order k; in this case, the integral curves spiral to the singular point very slowly, in contrast with “strong” foci (non resonant ones), for which they approach it exponentially fast; A similar interpretation can be given for weak saddles, by considering the complexification of the system. The chapter elaborates in details the formal classification of resonant differential forms. The chapter also explains the nature of the analytic invariants of resonant complex systems.