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Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique

Pascal Remy (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

Etant donné un système différentiel linéaire de niveau unique quelconque, nous explicitons des formules donnant les multiplicateurs de Stokes en fonction de constantes de connexion dans le plan de Borel, généralisant ainsi les formules obtenues dans l’article Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems (M. Loday-Richaud, P. Remy). Pour ce faire, nous nous ramenons à un système de niveaux ≤ 1 par la méthode classique de réduction du rang ; puis, nous...

Moduli spaces for linear differential equations and the Painlevé equations

Marius van der Put, Masa-Hiko Saito (2009)

Annales de l’institut Fourier

A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map R H : ℳ → ℛ , where ℳ is a moduli space of connections and ℛ , the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of R H (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces ℳ . The induced Painlevé equations are computed explicitly....

Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle

Christiane Rousseau, Colin Christopher (2007)

Annales de l’institut Fourier

We consider germs of one-parameter generic families of resonant analytic diffeomorphims and we give a complete modulus of analytic classification by means of the unfolding of the Écalle modulus. We describe the parametric resurgence phenomenon. We apply this to give a complete modulus of orbital analytic classification for the unfolding of a generic resonant saddle of a 2-dimensional vector field by means of the unfolding of its holonomy map. Here again the modulus is an unfolding of the Martinet-Ramis...

Monodromy and topological classification of germs of holomorphic foliations

David Marín, Jean-François Mattei (2012)

Annales scientifiques de l'École Normale Supérieure

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representations whose topological invariance was conjectured in the eighties by Cerveau and Sad and is proved here under mild hypotheses.

Morales-Ramis Theorems via Malgrange pseudogroup

Guy Casale (2009)

Annales de l’institut Fourier

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

Multiplicity of a foliation on projective spaces along an integral curve.

Julio García (1993)

Revista Matemática de la Universidad Complutense de Madrid

We compute the global multiplicity of a 1-dimensional foliation along an integral curve in projective spaces. We give a bound in the way of Poincaré problem for a complete intersection curves. In the projective plane, this bound give us a bound of the degree of non irreducible integral curves in function of the degree of the foliation.

Multisummability and ordinary meromorphic differential equations

Boele Braaksma (2012)

Banach Center Publications

In this expository paper we consider various approaches to multisummability. We apply it to nonlinear ODE's and give a somewhat modified proof of multisummability of formal solutions of ODE's with levels 1 and 2 via Écalle's method involving convolution equations.

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