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

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 the level sets *L**r *:= {ρ = *r*} with non-empty interiors; they exist for *r *∈ Z (Buchstaber, Karpov, Tertychnyi). Each *L**r *is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which *A is non-zero* are called *con**strictions*. We show that: (1) *all the constrictions in L**r **lie on the axis *{*B *= ω*r*}; (2) *each constriction *is *positive*: this means that some its punctured neighbourhood on the axis {*B *= ω*r*} lies in Int(*L**r*). 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 l:= *B/ω*, using equivalent description of model by linear systems of differential equations on C (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 l = *B/ω *or of non-positive type) with a given l for small ω is proved by slow-fast methods.