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Remarks on the uniqueness of second order ODEs

Dalibor Pražák (2011)

Applications of Mathematics

We are concerned with the uniqueness problem for solutions to the second order ODE of the form x ' ' + f ( x , t ) = 0 , subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x , for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit...

Resurgence in a Hamilton-Jacobi equation

Carme Olivé, David Sauzin, Tere M. Seara (2003)

Annales de l’institut Fourier

We study the resurgent structure associated with a Hamilton-Jacobi equation. This equation is obtained as the inner equation when studying the separatrix splitting problem for a perturbed pendulum via complex matching. We derive the Bridge equation, which encompasses infinitely many resurgent relations satisfied by the formal solution and the other components of the formal integral.

Resurgence of the Euler-MacLaurin summation formula

Ovidiu Costin, Stavros Garoufalidis (2008)

Annales de l’institut Fourier

The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series,...

Résurgence-sommabilité de séries formelles ramifiées dépendant d’un paramètre et solutions d’équations différentielles linéaires

Jean-Marc Rasoamanana (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cet article, nous établissons le caractère résurgent-sommable de séries formelles ramifiées solutions d’une classe d’équations différentielles linéaires. Nous analysons d’une part le problème de la dépendance analytique des sommes de Borel de telles séries par rapport aux paramètres de cette classe d’équations différentielles linéaires d’ordre deux, et d’autre part, nous analysons la structure résurgente complète associée à ces séries formelles via l’outil des singularités générales (ou microfonctions)....

Semicompleteness of homogeneous quadratic vector fields

Adolfo Guillot (2006)

Annales de l’institut Fourier

We investigate the quadratic homogeneous holomorphic vector fields on  C n that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...

Semi-formal theory and Stokes' phenomenon of non-linear meromorphic systems of ordinary differential equations

Werner Balser (2012)

Banach Center Publications

This article continues earlier work of the author on non-linear systems of ordinary differential equations, published in Asymptotic Analysis 15 (1997), MR no. 98g:34015b. There, a completely formal theory was presented, while here we are concerned with a semi-formal approach: Solutions of non-linear systems of ordinary meromorphic differential equations are represented as, in general divergent, power series in several free parameters. The coefficients, aside from an exponential polynomial, a general...

Singular sets of holonomy maps for algebraic foliations

Gabriel Calsamiglia, Bertrand Deroin, Sidney Frankel, Adolfo Guillot (2013)

Journal of the European Mathematical Society

In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set...

Singularités des flots holomorphes

Julio C. Rebelo (1996)

Annales de l'institut Fourier

Nous mettons en évidence une obstruction au prolongement d’un germe de champ de vecteurs holomorphe en un champ holomorphe complet. En particulier, on démontre que toute singularité isolée d’un champ holomorphe complet sur une surface complexe possède un deuxième jet non nul.

Small divisors and large multipliers

Boele Braaksma, Laurent Stolovitch (2007)

Annales de l’institut Fourier

We study germs of singular holomorphic vector fields at the origin of ℂ n of which the linear part is 1 -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

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