On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation

Yulia Bibilo , Alexey Glutsyuk (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We study family of dynamical systems on 2-torus modeling overdamped Josephson junction in superconductivity. It depends on three parameters (B, A; ω): B (abscissa), A (ordinate), ω (a fixed frequency). We study the rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. A phase-lock area is the level set L r := {ρ = r}, if it has a non-empty interior. This holds for r ∈ Z (a result by V.M.Buchstaber, O.V.Karpov and S.I.Tertychnyi). It is known that each phase-lock area is an infinite garland of domains going to infinity in the vertical direction and separated by points called constrictions (expect for the separation points with A = 0). We show that all the constrictions in L r lie in its axis {B = ωr}, confirming an experimental fact (conjecture) observed numerically by S.I.Tertychnyi, V.A.Kleptsyn, D.A.Filimonov, I.V.Schurov. We prove that each constriction is positive: the phaselock area germ contains the vertical line germ (confirming another conjecture). To do this, we study family of linear systems on the Riemann sphere equivalently describing the model: the Josephson type systems. We study their Jimbo isomonodromic deformations described by solutions of Painlevé 3 equations. Using results of this study and a Riemann-Hilbert approach, we show that each constriction can be analytically deformed to constrictions with the same := B ω and arbitrarily small ω 1. Then non-existence of "ghost" constrictions (nonpositive or with ρ =) with a given for small ω is proved by slow-fast methods.

Domaines: Systèmes dynamiques [math.DS]

Yulia Bibilo, Alexey Glutsyuk. On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation. 2020. ⟨hal-03026462v2⟩