约瑟夫森结模型在特殊快慢极限下的锁相区铺砌结构
On phase-lock area parquet in a special slow-fast limit of model of Josephson junction
浏览论文内容
中文总结 AI 辅助
本文研究过阻尼约瑟夫森结的RSJ模型，在$ω→0$的快慢极限下，其锁相区收敛到边界平行于$\text{ℓ}±u=0$的铺砌结构，并将结果推广到2-环面的一类快慢系统。
中文摘要 AI 辅助
该网址（1973年诺贝尔奖成果）预言了由窄电介质分隔的两个超导体系统（这类系统被称为约瑟夫森结）的隧穿效应：通过该结的超电流的存在及其控制方程。过阻尼约瑟夫森结由环面$T^2=R^2\slash2\pi Z^2$上的微分方程族建模，$\frac{d\theta}{d\tau}=\frac1{\omega}(\cos\theta+B+A\cos\tau)$，这被称为RSJ模型。它依赖三个参数：称为横坐标的$B$、称为纵坐标的$A$，以及固定频率$\omega$。我们研究其旋转数$\rho(B,A;\omega)$作为$(B,A)$的函数，以及锁相区：那些具有非空内部的水平子集。它们仅在旋转数取整数值时存在（Buchstaber、Karpov、Tertychnyi）。在本文中，我们研究特殊快慢极限下锁相区图像的渐近行为，当$\omega\to0$且$(B,A)\to(0,1)$，使得$(B,A)-(0,1)=O(\omega)$。我们证明，在重标参数$\ell:=\frac B{\omega}$和$u:=\frac{A-1}{\omega}$下，锁相区图像收敛到边界平行于直线$\{ u\pm\ell=0\}$的铺砌结构。具体而言，旋转数为$r$的锁相区的极限是一个向上延伸的无限链正方形的并集，这些正方形具有整数顶点，且长度为2的对角线位于直线$\{\ell=r\}$上，以及一个向下延伸的无限带（当$r=0$时为扇形）。我们陈述并证明了该结果对2-环面上一类广泛快慢系统的推广。
英文摘要
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, $\frac{dθ}{dτ}=\frac1ω(\cosθ+B+A\cosτ)$, which is known as the RSJ model. It depends on three parameters: $B$ called the abscissa, $A$ called the ordinate, and a fixed frequency $ω$. We study its rotation number $ρ(B,A;ω)$ 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 $ω\to0$ and $(B,A)\to(0,1)$ so that $(B,A)-(0,1)=O(ω)$. We show that in the rescaled parameters $\ell:=\frac Bω$ and $u:=\frac{A-1}ω$ 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.
