On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation

B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$, $A$, $\omega$. We study its rotation number $\rho (B,A;\omega )$ as a function of parameters. The three-dimensional phase-lock areas are the level sets $L_r:=\{\rho =r\}$ with non-empty interiors; they exist for $r\in \mathbb{Z}$ (Buchstaber, Karpov, Tertychnyi). For every fixed $\omega >0$ and $r\in \mathbb{Z}$ the planar slice $L_r\cap ({\mathbb{R}}_{B,A}^2\times \{\omega \})$ is a garland of domains going vertically to infinity and separated by points; those separating points for which $A\mathrm{\ne }0$ are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa $\mathrm{\ell }:=\frac{B}{\omega }$ is equal to $\rho$; 2) the family of constrictions with given $\mathrm{\ell }\in \mathbb{Z}$ is an analytic submanifold $Const{r}_{\mathrm{\ell }}$ in $({\mathbb{R}}_{+}^2{)}_{a,s}$, $a={\omega }^{-1}$, $s=\frac{A}{\omega }$. Here we show that the limit points of $Const{r}_{\mathrm{\ell }}$ are ${\beta }_{\mathrm{\ell },k}=(0,s_{\mathrm{\ell },k})$, where $s_{\mathrm{\ell },k}>0$ are zeros of the Bessel function $J_{\mathrm{\ell }}(s)$, and it lands at them regularly. Known numerical pictures show that high components of $Int(L_r)$ look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincaré map of the dynamical isomonodromic foliation governed by Painlevé 3 equation. Whenever well-defined, it preserves $\rho$. We show that the Poincaré map is well-defined on a neighborhood of the plane $\{a=0\}\subset {\mathbb{R}}_{\mathrm{\ell },a}^2\times ({\mathbb{R}}_{+}{)}_s$, and it sends ${\beta }_{\mathrm{\ell },k}$ to ${\beta }_{\mathrm{\ell },k+1}$ for integer $\mathrm{\ell }$.