On the adjacency quantization in an equation modeling the Josephson effect

Abstract

In the present paper we investigate two-parametric family of nonautonomous ordinary differential equations on the two-torus that model the Josephson effect from superconductivity. We study its rotation number as a function of parameters and its Arnold tongues (also called phase locking domains): the level sets of the rotation number that have non-empty interior. The Arnold tongues of the equation under consideration have many non-typical properties: the phase locking happens only for integer values of the rotation number [5, 11]; the boundaries of the tongues are given by analytic curves [4, 7], the tongues have zero * Laboratoire J.-V.Poncelet (UMI 2615 du

On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction

“…The variable change t ↦ τ transforms (1.7) to a non-autonomous ordinary differential equation on the two-torus - [12], [19,22,23,26,27] and references therein. It is known that the phaselock areas exist only for integer values of the rotation number function [9], contrarily to the Arnold tongues picture [1, p.110].…”

“…One of the main open conjectures on the geometry of phase-lock area portrait is the following. Conjecture 1.13 (experimental fact, see [19]). In every phase-lock area L ρ all the constrictions lie in the line {B = ρω}.…”

“…It was shown in [19] that at all the constrictions in L ρ one has l = B ω ∈ Z, l ≡ ρ(mod 2) and l ∈ [0, ρ]. See a survey on Conjecture 1.…”

On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

“…Recall that the rotation number has physical meaning of the mean voltage over a long time interval. The phase-lock areas of the family (1.2) were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi et al, see [6]- [16], [26], [20] and references therein. It is known that the following statements hold:…”

“…Numerical experiences made by V.M.Buchstaber, S.I.Tertychnyi, V.A.Kleptsyn, D.A.Filimonov, I.V.Schurov led to the following conjecture, which was stated and partially investigated in [20], see also [7, section 5]. [7, conjecture 5.17]).…”