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B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system(called Josephson junction) of two superconductors separated by a narrowdielectric: existence of a supercurrent through it and equations governing it.The overdamped Josephson junction is modeled by a family of differentialequations on 2-torus depending on 3 parameters: $B$, $A$, $\omega$. We studyits rotation number $\rho(B,A;\omega)$ as a function of parameters. Thethree-dimensional phase-lock areas are the level sets $L_r:=\{\rho=r\}$ withnon-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 goingvertically to infinity and separated by points; those separating points forwhich $A\neq0$ are called constrictions. In a joint paper by Yu.Bibilo and theauthor, it was shown that 1) at each constriction the rescaled abscissa$\ell:=\frac B\omega$ is equal to $\rho$; 2) the family of constrictions withgiven $\ell\in\mathbb Z$ is an analytic submanifold $Constr_\ell$ in $(\mathbbR^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$. Here we show that the limitpoints of $Constr_\ell$ are $\beta_{\ell,k}=(0,s_{\ell,k})$, where$s_{\ell,k}>0$ are zeros of the Bessel function $J_\ell(s)$, and it lands atthem regularly. Known numerical pictures show that high components of$Int(L_r)$ look similar. In his paper with Bibilo, the author introduced acandidate to the self-similarity map between neighbor components: thePoincar\'e map of the dynamical isomonodromic foliation governed by Painlev\'e3 equation. Whenever well-defined, it preserves $\rho$. We show that thePoincar\'e 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$\beta_{\ell,k}$ to $\beta_{\ell,k+1}$ for integer $\ell$.

On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlev\'e 3 equation