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

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