On constrictions of phase-lock areas in model of overdamped Josephson effect and transition matrix of double confluent Heun equation

Abstract

We will discuss the model of the overdamped Josephson junction in superconductivity, which is given by a family of first order non-linear ordinary differential equations on two-torus depending on three parameters: a fixed parameter $\omega$ (the frequency); a pair of variable parameters $(B,A)$ (abscissa and ordinate). It is important to study the rotation number of the system as a function $\rho=\rho(B,A)$ and to describe the phase-lock areas: its level sets $L_r=\{\rho =r\}$ with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point called constriction (if it does not lie in the abscissa axis). There is a conjecture stating that all the constrictions of every phase-lock area $L_r$ lie in its axis $\Lambda_r=\{ B=r\omega\}$. Another conjecture states that for any two subsequent constrictions in $L_r$ with positive ordinates the interval between them also lies in $L_r$. In the present paper we give new results partially confirming both conjectures. The main result states that the intersection $L_r\cap\Lambda_r$ contains an explicit infinite interval of the axis $\Lambda_r$ (conjecturally a connected component of the latter intersection). The proof is done by studying an equivalent family of systems of second order linear differential equations on the Riemann sphere. We obtain new results on the Stokes multipliers and the transition matrix between appropriate canonical solution bases of the linear system and deduce the main result on non-linear systems on two-torus.