On Germs of Constriction Curves in Model of Overdamped Josephson Junction, Dynamical Isomonodromic Foliation and Painlevé 3 Equation

Abstract

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 $L_r\cap(\mathbb R^2_{B,A}\times\{\omega\})$ 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 equal to $\rho$; 2) the family of constrictions with given $\ell\in\mathbb Z$ is an analytic submanifold $Constr_\ell$ in $(\mathbb R^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$. Here we show that the limit points 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 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 Poincar\'e map of the dynamical isomonodromic foliation governed by Painlev\'e 3 equation. Whenever well-defined, it preserves $\rho$. We show that the Poincar\'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 $\ell$.