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On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B,A;ω) as a function of parameters. The three-dimensional phase-lock areas are the level sets Lr:={ρ=r} in the three-dimensional parameter space with non-empty interiors; they exist only for integer numbers r (Buchstaber, Karpov, Tertychnyi). For every fixed ω and integer r the corresponding planar ω-slice of the phase-lock area L_r is a garland of domains going vertically to infinity and separated by points; those separating points for which A is non-zero are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa l:=B/ω is integer and l=ρ; 2) the family Constr(l ) of constrictions with given integer l is an analytic submanifold in two-dimensional space of the rescaled parameters a:=1/ω, s:=A/ω. In the present paper we show that 1) the limit points of Constr(l ) are β(l,k)=(0,s(l,k)), where s(l,k) are the positive zeros of the l-th Bessel function Jl(s); 2) to each β(l,k) accumulates exactly one its component L(l,k) (constriction curve), and it lands at β(l,k) regularly. Known numerical phase-lock area pictures show that high components of interior of each phase-lock area Lr look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincaré map of the dynamical isomonodromic foliation governed by Painlevé 3 equation. Whenever well-defined, it preserves the rotation number function. We show that the Poincaré map is well-defined on a neighborhood of the plane {a=0} in the three-dimensional space of parameters l, a, s, and it sends each constriction curve germ (L(l,k),β(l,k)) to (L(l,k+1),β(l,k+1)).