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Journal Articles Russian Mathematical Surveys Year : 2020

On families of constrictions in model of overdamped Josephson junction.

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Alexey Glutsyuk
Yulia Bibilo
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Abstract

We study family of dynamical systems on 2-torus modeling overdamped Josephson junction in superconductivity. It depends on two variable parameters B (abscissa), A (ordinate), and a fixed frequency. We study the rotation number as a function of (B,A) with fixed frequency. A phase-lock area is a level set of the rotation number function having a non-empty interior. This holds for integer values of the rotation number (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 asymptotically vertical direction and separated by points called constrictions (expect for the separation points on the abscissa axis A=0). We show that all the constrictions in each indivudual phase-lock area lie on the same vertical line (called the axis of the phase-lock area) with abscissa equal to the rotation number times the frequency. This confirms 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 phase-lock 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 ratio of the abscissa and the frequency (which should be an integer number) and arbitrarily small frequency. Then non-existence of "ghost" constrictions with a given above ratio (nonpositive constriction or a constriction with the latter ratio being different from the rotation number) for small frequencies is proved by slow-fast methods.
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Dates and versions

ensl-03041406 , version 1 (04-12-2020)

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Alexey Glutsyuk, Yulia Bibilo. On families of constrictions in model of overdamped Josephson junction.. Russian Mathematical Surveys, 2020, 76 (2), pp.179-180. ⟨10.1070/RM9982⟩. ⟨ensl-03041406⟩
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