2023/08/08 by Alexey Glutsyuk, Glutsyuk, Alexey · 1 citation
Materials Science · Physics and Astronomy · #34A26 #34E15 #34M03 #Dynamical Systems (math.DS) #FOS: Mathematics #Magnetism in coordination complexes #Organic and Molecular Conductors Research #Physics of Superconductivity and Magnetism
paper · pdf · doi:10.48550/arxiv.2308.04310
openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: 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 3 parameters: B, A, ω. 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\ with non-empty interiors; they exist for r∈\mathbb Z (Buchstaber, Karpov, Tertychnyi). For every fixed ω>0 and r∈\mathbb Z the planar slice Lr∩(\mathbb R2B,A×\ω\) is a garland of domains going vertically to infinity and separated by points; those separating points for which A≠0 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:=\frac Bω is equal to ρ; 2) the family of constrictions with given ℓ∈\mathbb Z is an analytic submanifold Constr_ℓ in (\mathbb R2+)a,s, a=ω-1, s=\frac Aω. Here we show that the limit points of Constr_ℓ are βℓ,k=(0,sℓ,k), where sℓ,k>0 are zeros of the Bessel function J_ℓ(s), and it lands at them regularly. Known numerical pictures show that high components of Int(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 ρ. We show that the Poincaré map is well-defined on a neighborhood of the plane \ a=0\⊂\mathbb R2ℓ,a×(\mathbb R+)s, and it sends βℓ,k to βℓ,k+1 for integer ℓ.