On phase-lock area parquet in a special slow-fast limit of model of Josephson junction

Abstract

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 the family of differential equations on the 2-torus, $\frac{d\theta}{d\tau}=\frac1{\omega}(\cos\theta+B+A\cos\tau)$, which is known as the RSJ model. It depends on three parameters: $B$ called the abscissa, $A$ called the ordinate, and a fixed frequency $\omega$. We study its rotation number $\rho(B,A;\omega)$ as a function of $(B,A)$ and the phase-lock areas: those its level subsets that have non-empty interiors. They exist only for integer values of the rotation number (Buchstaber, Karpov, Tertychnyi). In this paper we study asymptotics of the phase-lock area portrait in a special slow-fast limit, as $\omega\to0$ and $(B,A)\to(0,1)$ so that $(B,A)-(0,1)=O(\omega)$. We show that in the rescaled parameters $\ell:=\frac B{\omega}$ and $u:=\frac{A-1}{\omega}$ the phase-lock area portrait converges to a parquet with boundary lines being parallel to the lines $\{ u\pm\ell=0\}$. Namely, the limit of phase-lock area with rotation number $r$ is the union of an infinite chain of squares going up, with integer vertices and diagonals of length two lying on the line $\{\ell=r\}$, and an infinite strip going down (sector in the case, when $r=0$). We state and prove a generalization of this result to a wide class of slow-fast systems on 2-torus.