On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation*

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated a narrow dielectric. It states existence of supercurrent through it and equations governing it. overdamped junction is modeled family differential on 2-torus depending 3 parameters: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as function $(B,A)$ with fixed $\omega$. phase-lock areas are level sets $L_r:=\{\rho=r\}$ non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ an infinite chain domains going vertically to infinity points called constrictions (expect those $A=0$). show that: 1) all in lie axis $\{ B=\omega r\}$ (confirming conjecture Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction positive: some punctured neighborhood vertical line lies $\operatorname{Int}(L_r)$ another conjecture). first prove deformability one, arbitrarily small $\omega$, same $\rho$, $\ell:=\frac B\omega$ type (positive or not), using equivalent description model linear systems $\bar{\mathbb C}$ Tertychnyi) studying their isomonodromic deformations described Painlev\'e equations. Then non-existence ghost (i.e., either $\rho\neq\ell$, non-positive type) given $\ell$ proved slow-fast methods. In Section 6 we present applications results elaborated methods open problems.