On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation
Résumé
The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. The phase-lock areas are those level sets Lr := {ρ = r} that have non-empty interiors; they exist for r ∈ Z (Buchstaber, Karpov, Tertychnyi). Each Lr is an infinite chain of domains going vertically to infinity and separated by points. Those separating points that have non-zero ordinate A are called constrictions. We show that (1) all the constrictions in Lr lie on the axis {B = ωr}; (2) each constriction is positive: this means that some its punctured neighbourhood on the axis {B = ωr} lies in Int(Lr). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same ratio l=B/ω, using equivalent description of model by linear systems of differential equations on the Riemann sphere (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of "ghost" constrictions (i.e., constrictions either with ρ different from the ratio l=B/ω, or of non-positive type) with a given l for small ω is proved by slow-fast methods.
Domaines
Systèmes dynamiques [math.DS]
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