On a model of Josephson effect, dynamical systems on two-torus and double confluent Heun equations V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi International Conference dedicated to G.M. Henkin, Quasilinear equations, inverse problems and their applications Moscow Institute of Physics and Technology Dolgoprudny, 12 - 15 Sept. 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 1 / 37
Authors V.M.Buchstaber – Steklov Mathematical Institute (Moscow), All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements (VNIIFTRI, Mendeleevo), Russia. Supported by part by RFBR grant 14-01-00506. A.A.Glutsyuk – CNRS, France (UMR 5669 (UMPA, ENS de Lyon) and UMI 2615 (Lab. J.-V.Poncelet)), National Research University Higher School of Economics (HSE, Moscow, Russia). Supported by part by RFBR grants 13-01-00969-a, 16-01-00748, 16-01-00766 and ANR grant ANR-13-JS01-0010. S.I.Tertychnyi – All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements (VNIIFTRI, Mendeleevo), Russia. Supported by part by RFBR grant 14-01-00506. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 2 / 37
Superconductivity Occurs in some metals at temperature T < T crit . The critical temperature T crit depends on the metal. Carried by coherent Cooper pairs of electrons. Josephson effect (B.Josephson, 1962) Let two superconductors S 1 , S 2 be separated by a very narrow dielectric, thickness ≤ 10 − 5 cm ( << distance in Cooper pair). There exists a supercurrent I S through the dielectric. S 1 I S S 2 Quantum mechanics. State of S j : wave function Ψ j = | Ψ j | e i χ j ; χ j is the phase , ϕ := χ 1 − χ 2 . . . . . . . . . . . . . . . . . . . . . Josephson relation: I S = I c sin ϕ , I c ≡ const . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 3 / 37
Josephson effect Let two superconductors S 1 , S 2 be separated by a very narrow dielectric, thickness ≤ 10 − 5 cm ( << distance in Cooper pair). There exists a supercurrent I S through the dielectric. S 1 I S S 2 Quantum mechanics. State of S j : wave function Ψ j = | Ψ j | e i χ j ; χ j is the phase , ϕ := χ 1 − χ 2 . Josephson relation I S = I c sin ϕ , I c ≡ const . RSJ model T < T crit , but T crit − T << 1. T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 4 / 37
Equivalent circuit of real Josephson junction See Barone, A. Paterno G. Physics and applications of the Josephson effect 1982, Figure 6.2. This scheme is described by the equation 2 e C d 2 φ � dt 2 + � 1 d φ dt + I c sin φ = I dc 2 e R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 5 / 37
Overdamped case This scheme is described by the equation 2 e C d 2 φ 1 d φ � dt 2 + � dt + I c sin φ = I dc 2 e R τ 1 = Ω t = 2 e Set � R I c t ) 2 ( 2 e ϵ = � C = 2 e � R I c � ( C R )( R I c ) 2 e I c ϵ d 2 φ + d φ + sin φ = I − 1 c I dc d τ 2 d τ 1 1 Overdamped case: | ε | << 1 . In the case, when I − 1 c I dc = B + A cos ωτ 1 , we obtain d ϕ = − sin ϕ + B + A cos ωτ 1 ( 1 ) d τ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 6 / 37
Equation (1) in other domains of mathematics In the case, when I − 1 c I dc = B + A cos ωτ 1 , we obtain d ϕ = − sin ϕ + B + A cos ωτ 1 . (1) d τ 1 Equation (1) occurs in other domains of mathematics. It occurs, e.g., in the investigation of some systems with non-holonomic connections by geometric methods. It describes a model of the so-called Prytz planimeter. Analogous equation describes the observed direction to a given point at infinity while moving along a geodesic in the hyperbolic plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 7 / 37
Reduction to a dynamical system on 2-torus Set τ = ωτ 1 , f ( τ ) = cos τ . { ˙ ϕ = − sin ϕ + B + Af ( τ ) ( ϕ, τ ) ∈ T 2 = R 2 / 2 π Z 2 . , (2) τ = ω ˙ System (2) also occurred in the work by Yu.S.Ilyashenko and J.Guckenheimer from the slow-fast system point of view. They have obtained results on its limit cycles, as ω → 0. Consider ϕ = ϕ ( τ ) . The rotation number of flow: ϕ ( 2 π n ) ρ ( B , A ; ω ) = lim , (3) n n → + ∞ Problem Describe the rotation number of flow ρ ( B , A ; ω ) as a function of the parameters ( B , A , ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 8 / 37
Rotation number of circle diffeomorphism V. I. Arnold introduced rotation number for circle diffeomorphisms g : S 1 → S 1 . Consider the universal covering p : R → S 1 = R / 2 π Z . Every circle diffeomorphism g : S 1 → S 1 lifts to a line diffeomorphism G : R → R such that g ◦ p = p ◦ G . G is uniquely defined up to translations by the group 2 π Z . The rotation number of the diffeomorphism g : G n ( x ) ρ := 1 lim (4) 2 π n n → + ∞ It is well-defined, independent on x , and ρ ∈ S 1 = R / Z . Example Let g ( x ) = x + 2 πθ . Then ρ ≡ θ ( mod Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 9 / 37
Arnold Tongues Properties in general case: ρ = 0 < == > g has at least one fixed point. ρ = p q < == > g has at least one q − periodic orbit ordered similarly to an orbit of the rotation x �→ x + 2 π p q . Arnold family of circle diffeomorphisms: g a ,ε ( x ) = x + 2 π a + ε sin x , 0 < ε < 1. V.I.Arnold had discovered Tongues Effect for given family g a ,ε : for small ε the level set { ρ = r } ⊂ R 2 a ,ε has non-empty interior, if and only if r ∈ Q . He called these level sets with non-empty interiors phase-lock areas. Later they have been named Arnold tongues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 10 / 37
Arnold family of circle diffeomorphisms: g a ,ε ( x ) = x + 2 π a + ε sin x , 0 < ε < 1. Arnold Tongues Effect for given family of diffeomorphisms g a ,ε : for small ε the level set { ρ = r } ⊂ R 2 a ,ε has non-empty interior, if and only if r ∈ Q . Arnold called these level sets with non-empty interiors phase-lock areas. Later they have been named Arnold tongues. The tongues are connected and start from ( p q , 0 ) . See V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften, Vol. 250, 1988, page 110, Fig. 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 11 / 37
Arnold family and dynamical system (2) { ˙ ϕ = − sin ϕ + B + Af ( τ ) ( ϕ, τ ) ∈ T 2 = R 2 / 2 π Z 2 . , ( 2 ) τ = ω ˙ Consider ϕ = ϕ ( τ ) . The rotation number of flow: ρ ( B , A ; ω ) = 1 ϕ ( 2 π n ) lim , 2 π n n → + ∞ It is equivalent ( mod 1 ) to the rotation number of the flow map for the period 2 π . Problem How the rotation number of flow depends on ( B , A ) with fixed ω ? The ε from Arnold diffeomorphisms family corresponds to the parameter A in (2). Arnold family is a family of diffeomorphisms arbitrarily close to rotations. The time 2 π flow diffeomorphisms of the system (2) for A = 0 are not rotations and even not simultaneously conjugated to rotations: for A = B = 0 we obtain ˙ ϕ = − sin ϕ : the flow map has attractive fixed point 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.M.Buchstaber, A.A.Glutsyuk, S.I.Tertychnyi Josephson effect 12 - 15 Sept. 2016 12 / 37
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