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## On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

Journal of Dynamical and Control Systems. 2019. Vol. 25. No. 3. P. 323-349.

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamentaleffect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B,A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B,A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber,

O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper 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, which is called constriction
(provided that it does not lie in the abscissa axis). Those points of intersection of the boundary
∂Lr of the phase-lock area Lr with the line  {B = rω} (which is called its axis) that
are not constrictions are called simple intersections. It is known that our family of dynamical
systems is related to appropriate family of double–confluent Heun equations with the
same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to
some of those parameter values for which the corresponding “conjugate” double-confluent
Heun equation has a polynomial solution (follows from results of a joint paper of V.M.Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock area Lr lie in its axis. This conjecture was studied and partially proved in a joint paper of the author with
V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two
subsequent constrictions in Lr with positive ordinates, the interval between them also lies
in Lr . In this paper, we present new results partially confirming both conjectures. The main
result states that for every r ∈ Z \ {0} the phase-lock area Lr contains the infinite interval
of its axis  issued upwards from the simple intersection in ∂Lr  with the biggest
possible ordinate. The proof is done by studying the complexification of the system under
question, which is the projectivization of a family of systems of second-order linear equations
with two irregular non-resonant singular points at zero and at infinity. We obtain new
results on the transition matrix between appropriate canonical solution bases of the linear
system; on its behavior as a function of parameters. A key result, which implies the main
result of the paper, states that the off-diagonal terms of the transition matrix are both nonzero
at each constriction.We also show that their ratio is real at the constrictions. We reduce
the above conjectures on constrictions to the conjecture on negativity of the ratio of the
latter off-diagonal terms at each constriction.