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Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

Guillaume Duval, Andrzej J. Maciejewski (2009)

Annales de l’institut Fourier

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential V ( q ) , q ∈ ℂ n , of degree k ∈ ℤ ☆ . The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix V ′ ′ ( c ) calculated at a non-zero point c ∈ ℂ n , such that V ′ ( c ) = c . The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix V ′ ′ ( c ) is not diagonalizable....

Kovalevska vs. Kovacic-two different notions of integrability and their connections

Paweł Goldstein (2002)

Banach Center Publications

Ordinary differential equations all share the same common root-real physical problems. But, although the physical motivation remains the most important one, the way the subject develops does depend highly on the methods available. In the exposition I would like to show some connections between two methods of checking the ODE for integrability (whatever it should mean), with distant motivations and techniques. These are the so-called Painlevé tests and the methods originating in Ziglin's theory and...

Liouvillian first integrals of differential equations

Guy Casale (2011)

Banach Center Publications

In this paper we generalize to any dimension and codimension some theorems about existence of Liouvillian solutions or first integrals proved by M. Singer in Liouvillian first integrals of differential equations (1992) for first order differential equations.

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.

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