标题： 环面上与一般Heun方程相关的动力系统：相位锁定区域和收缩破坏 显示英文标题

标题： Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking

摘要： 超阻尼约瑟夫森结在超导性中由二维环面上的动力系统族来建模：即所谓的RSJ模型。 正如V.M. 布赫斯泰因和S.I. 特尔奇赫斯基所发现的，这个族可以通过一个二阶微分方程族：特殊的双合流赫恩方程来等价描述。 在本文中，我们构建了环面上的两个新的动力系统族，这些系统可以由一个具有四个奇点的一般赫恩方程族（GHE）和一个具有三个奇点的合流赫恩方程族来等价描述。 第一个族与GHE相关，是RSJ模型的变形，将被记为dRSJ。 环面上动力系统族的相位锁区是旋转数函数的那些具有非空内部的水平子集。 对于RSJ模型，V.M. 布赫斯泰因、O.V. 卡波夫和S.I. 特尔奇赫斯基发现了旋转数量化效应：相位锁区仅存在于整数旋转数值时存在。 正如A.V. 克里门科和O.L. 罗马斯凯维奇所显示的，每个相位锁区是一个由点分隔的区域链。 那些不位于横轴上的分离点称为收缩点。 在本文中，我们研究新族dRSJ中的相位锁区。 量化效应在这个族中仍然有效。 另一方面，我们表明在新族dRSJ中，收缩点会破裂。 显示英文摘要

摘要： The overdamped Josephson junction in superconductivity is modeled by a family of dynamical systems on two-dimensional torus: the so-called RSJ model. As was discovered by V.M. Buchstaber and S.I. Tertychnyi, this family admits an equivalent description by a family of second order differential equations: special double confluent Heun equations. In the present paper we construct two new families of dynamical systems on torus that can be equivalently described by a family of general Heun equations (GHE), with four singular points, and confluent Heun equations, with three singular points. The first family, related to GHE, is a deformation of the RSJ model, which will be denoted by dRSJ. The phase-lock areas of a family of dynamical systems on torus are those level subsets of the rotation number function that have non-empty interiors. For the RSJ model, V.M. Buchstaber, O.V. Karpov and S.I. Tertychnyi discovered the Rotation Number Quantization Effect: phase-lock areas exist only for integer rotation number values. As was shown by A.V. Klimenko and O.L. Romaskevich, each phase-lock area is a chain of domains separated by points. Those separation points that do not lie in the abscissa axis are called constrictions. In the present paper we study phase-lock areas in the new family dRSJ. The Quantization Effect remains valid in this family. On the other hand, we show that in the new family dRSJ the constrictions break down.