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

B. Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): 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: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as a function of parameters. The three-dimensional phase-lock areas are the level sets $L_r:=\{\rho=r\}\subset\mathbb{R}^3$ 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 $L_r\cap(\mathbb{R}^2_{B,A}\times\{\omega\})$ is a garland of domains going vertically to infinity and separated by points; those separating points for which $A\neq0$ are called constrictions. In a joint paper by Yu. Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa $\ell:=\frac B\omega$ is integer and $\ell=\rho$; 2) the family $\mathrm{Constr}_\ell$ of constrictions with given $\ell\in\mathbb{Z}$ is an analytic submanifold in $\mathbb{R}^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$. In the present paper we show that 1) the limit points of $\mathrm{Constr}_\ell$ are $\beta_{\ell,k}=(0,s_{\ell,k})$, where $s_{\ell,k}$ are the positive zeros of the $\ell$-th Bessel function $J_\ell(s)$; 2) to each $\beta_{\ell,k}$ accumulates exactly one its component $\mathcal{C}_{\ell,k}$ (constriction curve), and it lands at $\beta_{\ell,k}$ regularly. Known numerical phase-lock area pictures show that high components of interior of each phase-lock area $L_r$ 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 the rotation number function. We show that the Poincaré map is well defined on a neighborhood of the plane $\{ a=0\}\subset\mathbb{R}^2_{\ell,a}\times(\mathbb{R}_+)_s$, and it sends each constriction curve germ $(\mathcal{C}_{\ell,k},\beta_{\ell,k})$ to $(\mathcal{C}_{\ell,k+1},\beta_{\ell,k+1})$.