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In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.
The singularly perturbed Riccati equation is the first-order nonlinear ODE $hbar partial_x f = af^2 + bf + c$ in the complex domain where $hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with presc
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The soluti
We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to
New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x
We consider the Cauchy problem $(mathbb D_{(k)} u)(t)=lambda u(t)$, $u(0)=1$, where $mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {bf 71} (2011), 583--600), $lambda >