This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.

Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel-Laplace transformation. This article shows how to generalize the Borel-Laplace transformation in order to investigate bounded solutions of parameter dependent non-linear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniformly to the sectoral Borel sums. Our approach provides a unified treatment for all values of the complex parameter.Keywords: Ordinary differential equations irregular singularity unfolding confluence center manifold of a saddle-node singularity Borel summation

We describe the equivalence classes of germs of generic 2-parameter families of complex vector fieldsż = ω (z) on C unfolding a singular parabolic point of multiplicity k + 1:The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fieldṡ z = z k+1 + 1 z + 0 over CP 1 . This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.

The article presents a confluence approach to the study of the fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity. First the relation between the nonlinear monodromy of Painlevé VI and the nonlinear Stokes phenomenon in Painlevé V is explained in detail. Then the (wild) character variety of the isomonodromic problem for Painlevé V is constructed through confluence from the character variety of the corresponding isomonodromic problem for Painlevé VI and a birational transformation between them is provided. The known description of the action of the nonlinear monodromy of Painlevé VI on its character variety is then used to describe the action of the nonlinear wild monodromy of Painlevé V on its character variety, that is the action of the group generated by the nonlinear monodromy, the nonlinear Stokes operators and the nonlinear exponential torus.

S U M M A R YThe eikonal equation in an attenuating medium has the form of a complex-valued HamiltonJacobi equation and must be solved in terms of the complex-valued traveltime (complex-valued action function). A very suitable approximate method for calculating the complex-valued traveltime right in real space is represented by the perturbation from the reference traveltime calculated along real-valued reference rays to the complex-valued traveltime defined by the complex-valued Hamilton-Jacobi equation.The real-valued reference rays are calculated using the reference Hamiltonian function. The perturbation Hamiltonian function is parametrized by one or more perturbation parameters, and smoothly connects the reference Hamiltonian function with the Hamiltonian function corresponding to a given complex-valued Hamilton-Jacobi equation. Both the reference Hamiltonian function and the perturbation Hamiltonian function may be constructed in different ways, yielding differently accurate perturbation expansions of traveltime. All present perturbation methods use reference rays calculated in a reference anisotropic non-attenuating medium.In this paper, the reference Hamiltonian function is constructed directly using the Hamiltonian function corresponding to a given complex-valued Hamilton-Jacobi equation, and the perturbation Hamiltonian function is linear with respect to the perturbation parameter. The direct construction of the reference Hamiltonian function from the given complex-valued Hamilton-Jacobi equation is very general and accurate, especially for homogeneous Hamiltonian functions of degree N = −1 with respect to the slowness vector.