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

TL;DR

This paper introduces new dynamical systems on the torus related to general Heun equations, extending the RSJ model, and studies phase-lock areas, revealing that constrictions break down in the new family while the quantization effect persists.

Contribution

The paper constructs two new families of torus dynamical systems described by general and confluent Heun equations, extending the RSJ model and analyzing phase-lock areas and constriction behavior.

Findings

Quantization effect of phase-lock areas persists in the new family.

Constrictions break down in the dRSJ family.

Phase-lock areas are characterized by nonempty interior level sets of the rotation number.

Abstract

The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. 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 the torus are those level subsets of the rotation number function that have nonempty interiors. It is known that for the RSJ model, the rotation number quantization effect occurs: phase-lock areas exist only for integer rotation number values. Moreover, each phase-lock area is a chain of domains separated by points. Those separation points that do not lie on 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.