arXiv subjects

Amaury Bittmann

Publications and source records attributed to Amaury Bittmann.

4 recordsLinked to original sources

arXiv2017

Doubly-resonant saddle-nodes in (C^3,0) and the fixed singularity at infinity in Painlev{é} equations: analytic classification

In this work, we consider germs of analytic singular vector elds in (C^3,0) with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector elds come from irregular two-dimensional dierential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at innity in Painlev{é} equations (P j) j=I,...,V for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in (C^2,0). We also prove that these maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in a previous paper. Finally we provide an analytic classication under the action of bered dieomorphisms, based on the study of the so-called Stokes dieomorphisms obtained by comparing consecutive sectorial normalizing maps {à} la Martinet-Ramis / Stolovitch for 1-resonant vector fields.

math.DS↗

arXiv2016

Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization

In this work, following [Bit15], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlev{é} equations (P\_j), j=I...V , for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic nor-malization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in $(\mathbb{C}^{2},0)$. We also prove that the normalizing map is essentially unique and weakly Gevrey-1 summable.

math.DS↗

arXiv2016

Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlevé equations (part III): local analytic classification

In this work, following [Bit15] and [Bit16a], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlev{é} equations (P\_j), j=I...V , for generic values of the parameters. Under suitable assumptions, we provide an analytic classification under the action of fibered diffeomorphisms, based on the study of the Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps {à} la Martinet-Ramis / Stolovitch. These normalizing maps over sectorial domains are obtained in the main theorem of [Bit16a], which is analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in $\mathbb{C}^{3}$. We also prove that these maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in [Bit15].

math.DS↗

arXiv2016

Doubly-resonant saddle-nodes in $C^3$ and the fixed singularity at infinity in the Painlev{é} equations: formal classification

In this work we consider formal singular vector fields in $ C^{3}$with an isolated and doubly-resonant singularity of saddle-node typeat the origin. Such vector fields come from irregular two-dimensionalsystems with two opposite non-zero eigenvalues, and appear for instancewhen studying the irregular singularity at infinity in Painlev{é} equations$(P\_{j})\_{j\in(I,II,III,IV,V)}$, for generic values of the parameters.Under generic assumptions we give a complete formal classificationfor the action of formal diffeomorphisms (by changes of coordinates)fixing the origin and fibered in the independent variable. Wealso identify all formal isotropies (self-conjugacies) of the normalforms. In the particular case where the flow preserves a transversesymplectic structure, e.g. for Painlev{é} equations, we provethat the normalizing map can be chosen to preserve the transversesymplectic form.

math.DS↗