Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking (2507.07282v1)

Abstract: 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 {\it 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 {\it 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.