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

Abstract

In this work which follows directly [Bit16b, Bit16c], we consider analytic singular vector fields in C 3 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 Painlevé 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 [MR82, MR83, Sto96]. These normalizing maps over sectorial domains are obtained in the main theorem of [Bit16c], which is analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in C 2 . 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 [Bit16b].

Reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ and exceptional hyperbolic CR-singularities

“…Similar types of functional moduli spaces have since then been discovered in several other contexts (e.g. [Ily93,AhG05,Loh09,Bit18]...). The common thread through most of these works is that the divergent behavior is concentrated to a single variable or a single resonant monomial, and that there is a finite covering of a full neighborhood of the singularity by domains projecting to onto sectors in the divergent variable.…”

Wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence

“…Roughly speaking, the source of this nonlinear phenomenon is the existence of local normalizing transformations above certain sectors at the singularity and therefore of canonical 2-parameter families of solutions with well-behaved exponential asymptotics over the sectors. This was proved originally in the works of Takano [63,64] and Yoshida [65,66], and recently also by Bittmann [3].…”

“…Notation A.1. -The entries of 3×3 matrices will be indexed by (0, t, 1) rather than (1,2,3), in a correspondence to the eigenvalues of the matrix 0 0 0 0 t 0 0 0 1 . As before, the triple of indices (i, j, k) will always denote a permutation of (0, t, 1), and (i, j, k, l) will denote a permutation of (0, t, 1, ∞).…”

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

“…It seems likely that the global stable manifolds of Γ 1 (h) are also Gevrey-1 (as the stable manifold W s (q 1 ) of q 1 : (r 1 , v, l) = (1, 0, 0)), but this would require better normal forms. [3] considers related normal forms, but the condition in this paper regarding the trace, implies that q 1 is an attractor (or repeller). This is clearly violated in the present context, but it holds for a more general class of dissipation functions (linear damping is degenerate from this perspective) and in [28] we show that this condition relates directly to the property of circularization.…”

“…(5.16) after desingularization, corresponding to division of the right hand side by ρ 4 1 . Setting ρ 1 = l and dividing the right hand side by r 3 1 , we obtain (2.6). Consequently, within ρ 1 = 0, we have the Hamiltonian system with Hamiltonian function…”