On families of constrictions in model of overdamped Josephson junction and Painlev\'e 3 equation

The tunneling effect predicted by B.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 modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as a function of $(B,A)$ with fixed $\omega$. The phase-lock areas are the level sets $L_r:=\{\rho=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$ lie in its axis $\{ B=\omega r\}$ (confirming a conjecture of Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$ (confirming another conjecture). We first prove deformability of each constriction to another one, with arbitrarily small $\omega$, of the same $\rho$, $\ell:=\frac B\omega$ and type (positive or not), using equivalent description of model by linear systems of differential equations on $\bar{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlev\'e 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $\rho\neq\ell$, or of non-positive type) with a given $\ell$ for small $\omega$ is proved by slow-fast methods. In Section 6 we present applications of results and elaborated methods and open problems.

ω , or of non-positive type) with a given for small ω is proved by slow-fast methods. The Josephson effect is a tunnelling effect in superconductivity predicted theoretically by B.Josephson in 1962 [41] (Nobel Prize in physics, 1973) and confirmed experimentally by P.W.Anderson and J.M.Rowell in 1963 [1]. It concerns the so-called Josephson junction: a system of two superconductors separated by a very narrow dielectric fiber. The Josephson effect is the existence of a supercurrent crossing the junction (provided that the dielectric fiber is narrow enough), described by equations discovered by Josephson.
Here θ(τ ) is a general R-valued solution of the first equation in (1.4) whose parameter is the initial condition for τ = 0. Recall that the rotation number is independent on the choice of the initial condition, see [2, p.104]. The parameter B is called abscissa, and A is called the ordinate. Recall the following well-known definition. provided that it has a non-empty interior.
Remark 1.2 : phase-lock areas and Arnold tongues. H.Poincaré introduced the rotation number of a circle diffeomorphism. The rotation number of the flow of the field (1.4) on T 2 equals (modulo Z) the rotation number of the circle diffeomorphism given by its time 2π flow mapping restricted to the cross-section S 1 θ × {0}. In Arnold family of circle diffeomorphisms x → x + b + a sin x, x ∈ S 1 = R/2πZ the behavior of its phase-lock areas for small a demonstrates the tongues effect discovered by V.I. Arnold [2, p. 110]. That is why the phase-lock areas became "Arnold tongues", see [28, definition 1.1].
Recall that the rotation number has physics meaning of the mean voltage over a long time interval up to known constant factor.
Relation of phase-lock effect in model (1.1) to dynamical systems on torus was discovered in [15]. In physics papers earlier than [15] the phase-lock area effect dealt with convergence of differences φ(t + (n + 1)T ) − φ(t + nT ), where φ(t) is a solution of (1.1) and T = 2π ω . This effect was defined there as a phenomenon of convergence of the above differences to 2πk, k ∈ Z, on an open subset of the parameter space. It was observed in [15] that their convergence is equivalent to the statement that the rotation number of the corresponding dynamical system (1.4) is equal to k.
2) The boundary of the r-th phase-lock area consists of two analytic curves, which are the graphs of two functions B = G r,α (A), α = 0, π, (see [18]; this fact was later explained by A.V.Klimenko via symmetry, see [44]).
3) The latter functions have Bessel asymptotics , as s → ∞ (1.5) (observed and proved on physics level in [59], see also [46, p. 338], [6, section 11.1], [16]; proved mathematically in [44]). 4) Each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction. In this chain each two subsequent domains are separated by one point. This was proved in [44] using the above statement 3). Those separation points that lie on the horizontal Baxis, namely A = 0, were calculated explicitly, and we call them the growth points, see [18, corollary 3]. The other separation points, which lie outside the horizontal B-axis, are called the constrictions. 5) For every r ∈ Z the r-th phase-lock area is symmetric to the −r-th one with respect to the vertical A-axis.
6) Every phase-lock area is symmetric with respect to the horizontal B-axis. See Figures 1-3 below. and we will call it the axis of the phase-lock area L r .
The figures for the phase-lock areas obtained experimentally are given in the physics books on Josephson effect, see [47, p. 193, fig. 11.4], [49, p. 88, fig. 5.2], [46, p. 339, fig. 11.4] (which refers to physics paper [48]). They had shown that in each phase-lock area L r all the constrictions should lie on  The main results of the paper are the two following theorems. The first theorem confirms the above constriction alignment phenomena. The second one confirms another, positivity property of constrictions that can be also seen in the figures from physics books mentioned in the above paragraph. phase-lock area L r lie in its axis Λ r . Remark 1.5 It was proved in [28, theorem 1.2] that for every r ∈ Z the constrictions in L r have abscissas B = ω, ∈ Z, ≡ r(mod 2), ∈ [0, r]. For further results and discussion of the constriction alignment conjecture see [13, section 5] and [28,29,30]. Remark 1.8 It was shown in [29, theorem 1.8] that each constriction is either positive, or negative: there are no neutral constrictions. Positivity of constrictions was stated there as [29, conjecture 1.13]. It was also shown in [29] that Theorem 1.7 would imply Theorem 1.4. 1.2 Method of proof of Theorem 1.10. Plan of the paper We prove Theorem 1.10 in two steps given by the two following theorems. To state them, let us introduce the following notation. We set η := ω −1 . For every fixed ∈ Z we consider the set Theorem 1.12 For every ∈ Z the subset Constr ⊂ R 2 + is a regular one-dimensional analytic submanifold in R 2 + . The restriction of the coordinate η to each its connected component is unbounded from above (i.e., ω is unbounded from below). The rotation number and the type of constriction (positive or negative, see Remark 1.8) are constant on each component. Theorem 1.13 For every ∈ Z there are no ghost constrictions in the axis Λ := {B = ω } whenever ω > 0 is small enough (dependently on ). Theorem 1.13 will be proved in Section 5 by methods of the theory of slow-fast families of dynamical systems. Theorem 1.10 immediately follows from Theorems 1.12 and 1.13, see the Subsection 5.6.
The proof of Theorem 1.12 is sketched below. It is based on the following equivalent description of model of Josephson junction by a family of twodimensional linear systems of differential equations on the Riemann sphere, see [15,17,20,26,34,36], [12, subsection 3.2]. The variable change Recall that = B ω , µ = A 2ω , see (1.2). Equation (1.6) is the projectivization of the two-dimensional linear system in the following sense: a function Φ(z) is a solution of (1.6), if and only if Φ(z) = v u (z), where the vector function Y (z) = (u(z), v(z)) is a solution of system (1.7). For µ > 0 system (1.7) has two irregular nonresonant singular points at 0 and at ∞. Its monodromy operator acts on the space C 2 of germs of its solutions at a given point z 0 ∈ C * by analytic extension along a counterclockwise circuit around zero. Remark 1.14 The variable change E(z) := e µz v(z) transforms the family of systems (1.7) to the following family of special double confluent Heun equations, see [66], [19]- [23]: We will also deal with the so-called conjugate Heun equation obtained from (1.8) by change of sign at : Using this relation to well-known class of Heun equations a series of results on phase-lock area portrait of model of Josephson junction and related problems were obtained in [66], [19]- [23], [12,13]. See also a brief survey in the next subsection.
Recall that an isomonodromic family of linear systems is a family in which the collection of residue matrices of formal normal forms at singular points, Stokes matrices and transition matrices between canonical solution bases at different singular points remain constant (up to appropriate conjugacies).
We denote by Jos the three-dimensional family of systems (1.7), which will be referred to, as systems of Josephson type. For the proof of Theorem 1.12 we study their isomonodromic deformations in the four-dimensional space J N (R + ) of linear systems of the so-called normalized R + -Jimbo type here the matrix elements * may be arbitrary.
Step 1. We study the real one-dimensional analytic foliation of the space J N (R + ) by isomonodromic families of linear systems. These isomonodromic families are given by differential equation (3.17). They are obtained (by gauge transformations and rescaling of the variable z) from well-known Jimbo isomonodromic deformations [38], which are given by real solutions of Painlevé 3 equation (P3). Namely, the function w(τ ) = − R 12 (τ ) τ K 12 (τ ) = R 21 (τ ) should satisfy the P3 equation along the isomonodromic leaves. We show that the hypersurface Jos ⊂ J N (R + ) corresponds to poles of order 1 with residue 1 of solutions of (1.13). This implies that Jos is transversal to the isomonodromic foliation. This is the key lemma in the proof. The only role of the Painlevé 3 equation in the proof is the above transversality statement. Relation to Painlevé 3 equation led us to a series of new open problems presented in Section 6.
Step 2. We consider the subset Σ ⊂ J N (R + ) of systems (1.10) with trivial monodromy. We show that ∈ Z for these systems, and their germs at 0 and at ∞ are analytically equivalent to their diagonal formal normal forms. Using this fact, we show that Σ is a real two-dimensional analytic submanifold in J N (R + ) with the following properties: 2.1) Σ is a union of leaves of the isomonodromic foliation; 2.2) (the key theorem in the proof) there exists a submersive projection R : Σ → R x given by an analytic invariant R of linear systems, the so-called transition cross-ratio, that is constant along the leaves. Statement 2.1) follows from definition. Statement 2.2) is proved by showing that (x, τ ), x = R, form local analytic coordinates on Σ. Fix an ∈ Z, and let Σ ⊂ Σ denote the subset of systems with the given value of . For every pair (x 0 , τ 0 ) corresponding to a system from Σ realization of any pair (x, τ ) close to (x 0 , τ 0 ) by a system from Σ can be viewed as a solution of the Riemann-Hilbert type problem. It is proved via holomorphic vector bundle argument, as in famous works by A.A.Bolibruch on the Riemann-Hilbert Problem and related topics: see [9]- [11] and references therein. We glue a holomorphic vector bundle with connection on C realizing given (x, τ ) from the two trivial bundles: one over the disk D 2 ⊂ C, and the other one on the complement of the closed disk D 1 2 ⊂ C. The connections on the latter trivial bundles are given by the diagonal normal forms prescribed by and τ . The gluing matrix, which is holomorphic on the annulus D 2 \ D 1 2 , depends analytically on (x, τ ). The bundle thus obtained is trivial for (x 0 , τ 0 ) (by definition). It remains trivial for all (x, τ ) close enough to (x 0 , τ 0 ). This follows from the classical theorem stating that a holomorphic vector bundle close to a trivial one is also trivial [11, appendix 3, lemma 1, theorem 2], [56, theorem 2.3], [31]. The connection on the trivial bundle thus obtained is given by a meromorphic system with order two poles at 0 and at ∞ and the same normal forms. Its gauge equivalence to a normalized R + -Jimbo type system (1.10), (1.11) is proved by a symmetry argument.
The submanifold Jos is transversal to Σ, by the result of Step 1 and Statement 2.1). Therefore, the intersection Jos∩Σ is a real one-dimensional submanifold in J N (R + ). It is transversal to the isomonodromic foliation of Σ (Step 1), and hence, is locally diffeomorphically projected to an open subset in R by the mapping R. The above intersection is identified with Constr . This implies that Constr is a one-dimensional submanifold; each its connected component is analytically parametrized by an interval I = (a, b) of values of the parameter x = R and hence, is non-compact.
Step 3. We show that the coordinate η = ω −1 is unbounded from above on each component C in Constr . Assuming the contrary, i.e., that η is bounded on C, we have that for every c ∈ {a, b} at least one of the functions µ ±1 , η −1 (depending on the choice of c) should be unbounded, as x → c. Boundedness of µ is proved by using Klimenko-Romaskevich Bessel asymptotics of boundaries of the phase-lock areas [44]. For c = 0 we prove boundedness of the functions µ −1 , η −1 , as x → c, by studying accumulation points of the set Constr in the union of coordinate axes {η = 0} ∪ {µ = 0}.
Afterwards, to finish the proof of Theorem 1.12, it remains to show that the rotation number and type of constriction are constant on each connected component in Constr . We deduce constance of type from the fact that no constriction can be a limit of the so-called generalized simple intersections: those points of intersections Λ ∩ ∂L r , r ≡ (mod 2), that are not constrictions and do not lie in the abscissa axis. This, in its turn, is implied by the two following facts: -the generalized simple intersections correspond to Heun equations (1.9) having a polynomial solution [13, theorem 1.15]; this remains valid for their limits with A = 0; -no constriction can correspond to a Heun equation (1.9) with polynomial solution [12, theorems 3.3, 3.10].

Historical remarks
Model (1.1) of overdamped Josephson junction was studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi and other mathematicians and physicists, see [12]- [23], [45], [28]- [30], [34,36,43,44,65,66] and references therein. Hereby we present a brief survey of results that were not mentioned in the introduction. Recall that the rotation number quantization effect for a family of dynamical systems on T 2 containing (1.4) was discovered in [17]. I.A.Bizyaev, A.V.Borisov, and I.S.Mamaev noticed that a big family of dynamical systems on torus in which the rotation number quantization effect realizes was introduced by W.Hess (1890). It appears that in classi-cal mechanics such systems were studied in problems on rigid body movement with fixed point in works by W.Hess, P.A.Nekrassov, A.M.Lyapunov, B.K.Mlodzejewski, N.E.Zhukovsky and others. See [8,51,54,55,67] and references therein. P.A.Nekrasov observed in [55] that the above-mentioned big family of systems considered by Hess can be equivalently described by a Riccati equation (or by a linear second order differential equation).
Transversal regularity of the fibration by level sets ρ(B, A) = const / ∈ Z with fixed ω on the complement to the union of the phase-lock areas was proved in [13, proposition 5.3]. Conjectures on alignment and positivity of constrictions (now Theorems 1.4 and 1.7 respectively) were stated respectively in [28] and [29] and studied respectively in [28,29] and [29], where some partial results were obtained. Theorem 1.4 for ω ≥ 1 was proved in [28]. For further survey on these conjectures see [28,13,29] and references therein. A conjecture saying that the semiaxis Λ + := Λ ∩ {A > 0} intersects the corresponding phase-lock area L by a ray explicitly constructed in [29] was stated in [29, conjecture 1.14]. It was shown in [29, theorem 1.12] that the ray in question indeed lies in L . An equivalent description of model (1.1) in terms of a family of special double confluent Heun equations (1.8) was found by S.I.Tertychnyi in [66] and further studied in a series of joint papers by V.M.Buchstaber and S.I.Tertychnyi [19]- [23]. They have shown that the constrictions are exactly those parameter values (B, A; ω) for which the corresponding double confluent Heun equation (1.8) has an entire solution: holomorphic on C [20]. Using this observation they stated a conjecture describing ordinates of the constrictions lying in a given axis Λ as zeros of a known analytic function constructed via an infinite matrix product [20]. This conjecture was studied in [20,21] and reduced to the conjecture stating that if the Heun equation (1.8) has an entire solution, then the conjugate Heun equation (1.9) cannot have polynomial solution. Both conjectures were proved in [12]. New automorphisms of solution space of Heun equations (1.8) were discovered and studied in [22,23].
In [19] V.M.Buchstaber and S.I.Tertychynyi described those (B, A; ω), for which conjugate Heun equation (1.9) has a polynomial solution. Namely, for a given = B ω ∈ N their set is a remarkable algebraic curve, the so-called spectral curve (studied in [19,30]): zero locus of determinant of appropriate three-diagonal matrix with entries being linear non-homogeneous functions in the coefficients of equation (1.9). The fact that those points (B, A; ω) for which (1.9) has a polynomial solution are exactly the generalized simple intersections is a result of papers [19,13], stated and proved in [13].
There exists an antiquantization procedure that associates Painlevé equations to Heun equations; double confluent Heun equations correspond to Schurov made numerical experiences that have shown that as ω → 0, the "upper" part of the phase-lock area portrait converges to a kind of parquet in the renormalized coordinates ( , µ): the renormalized phase-lock areas tend to unions of pieces of parquet, and gaps between the phase-lock areas tend to zero. See Fig. 3 and the paper [43]. This is an open problem. In [43] V.A.Kleptsyn, O.L.Romaskevich, and I.V.Schurov proved some results on smallness of gaps and their rate of convergence to zero, as ω → 0, using methods of slow-fast systems.
A subfamily of family (1.3) of dynamical systems on 2-torus was studied by J.Guckenheimer and Yu.S.Ilyashenko in [35] from the slow-fast system point of view. They obtained results on its limit cycles, as ω → 0.
An analogue of the rotation number integer quantization effect in braid groups was discovered by A.V.Malyutin [52].
2 Preliminaries: irregular singularities, normal forms, Stokes matrices and monodromy-Stokes data of linear systems

Normal forms, canonical solutions and Stokes matrices
All the results presented in this subsection are particular cases of classical results contained in [3,4,5,37,42,60].
Recall that two germs of meromorphic linear systems of differential equations on a n-dimensional vector function Y = Y (z) at a singular point (pole), say, 0 are analytically equivalent, if there exists a holomorphic GL n (C)valued function H(z) on a neighborhood of 0 such that the Y -variable change Y = H(z) Y sends one system to the other one. Two systems are formally equivalent, if the above is true for a formal power series H(z) with matrix coefficients that has an invertible free term.
Consider a two-dimensional linear system on a neighborhood of 0; here the matrix K has distinct eigenvalues λ 1 = λ 2 , and O(1) is a holomorphic matrix-valued function on a neighborhood of 0. Then we say that the singular point 0 of system (2.1) is irregular nonresonant of Poincaré rank 1. Then K is conjugate to K = diag(λ 1 , λ 2 ), , and one can achieve that K = K by applying the constant linear change (gauge transformation) Y = H Y. System (2.1) is formally equivalent to a unique formal normal form 3) The matrix coefficient K in system (2.1) and the corresponding matrix K in (2.2) are called the main term matrices, and R, R the residue matrices. Generically, the normalizing series H(z) bringing (2.1) to (2.2) diverges. At the same time, there exists a covering of a punctured neighborhood of zero by two sectors S 0 and S 1 with vertex at 0 in which there exist holomorphic matrix functions H j (z), j = 0, 1, that are C ∞ smooth on S j ∩ D r for some r > 0, and such that the variable changes Y = H j (z) Y transform (2.1) to (2.2). This Sectorial Normalization Theorem holds for the so-called good sectors (or Stokes sectors.) Namely, consider the rays issued from 0 and forming the set {Re λ 1 −λ 2 z = 0}. They are called imaginary dividing rays (or Stokes rays). A sector S j is good, if it contains one imaginary dividing ray and its closure does not contain the other one.
Let W (z) = diag( Y 1 (z), Y 2 (z)) denote the canonical diagonal fundamental matrix solution of the formal normal form (2.2); here Y (z) are solutions of its one-dimensional equations. The matrices X j (z) := H j (z)W (z) are fundamental matrix solutions of the initial equation (2.1) defining solution bases in S j called the canonical sectorial solution bases. In their definition we choose the branches W (z) = W j (z) of the (a priori multivalued) matrix function W (z) in S j , j = 0, 1, so that W 1 (z) is obtained from W 0 (z) by counterclockwise analytic extension from S 0 to S 1 . And in the same way we define yet another branch W 2 (z) of W (z) in S 2 := S 0 that is obtained from W 1 (z) by counterclockwise analytic extension from S 1 to S 0 . This yields another canonical matrix solution X 2 := H 0 (z)W 2 (z) in S 0 , which is obtained from X 0 (z) by multiplication from the right by the monodromy matrix exp(2πi R) of the formal normal form (2.2). Let S j,j+1 denote the connected component of intersection S j+1 ∩S j , j = 0, 1, that is crossed when one moves from S j to S j+1 counterclockwise, see Fig. 5. The transition matrices C 0 , C 1 between thus defined canonical solution bases X j , are called the Stokes matrices.
Example 2.1 Let A = diag(λ 1 , λ 2 ), and let λ 2 − λ 1 ∈ R. Then the imaginary dividing rays are the positive and negative imaginary semiaxes. The good sectors S 0 and S 1 covering C * satisfy the following conditions: -the sector S 0 contains the positive imaginary semiaxis, and its closure does not contain the negative one; -the sector S 1 satisfies the opposite condition. See Example 2.2 Let us numerate the sectors S 0 , S 1 and the eigenvalues λ 1 , λ 2 so that This holds, e.g., in the conditions of the above example, if λ 2 − λ 1 > 0. The canonical solutions of the formal normal form (2.2) are given by the solutions 2). They are numerated by indices k = 1, 2 of the eigenvalues λ k of the main term matrix K. The corresponding solutions of the initial system (2.1) in S j , j = 0, 1, 2, i.e., the columns of the fundamental matrix X j (z), are also numerated by the same index k and will be denoted by f kj (z). The norm ||f 1j (z)|| is asymptotically dominated by ||f 2j (z)|| in S 0,1 , as z → 0, and the converse asymptotic domination statement holds on S 1,2 . This implies (and it is well-known) that f 10 ≡ f 11 on S 0,1 and f 22 ≡ f 21 on S 1,2 . The Stokes matrices C 0 and C 1 are unipotent: C 0 is upper-triangular and C 1 is lower-triangular. If the numeration of either eigenvalues, or sectors (but not both) is opposite, or if the singular point under question is ∞, not zero (see Remark 2.5 below), then the Stokes matrices are unipotent but of opposite triangular type.

Remark 2.3
The tautological projection C 2 \{0} → CP 1 = C sends canonical sectorial basic solutions f kj (z) of system (2.1) to canonical sectorial solutions q kj (z) of its projectivization: the corresponding Riccati equation. These are the unique C-valued holomorphic solutions of the Riccati equation in the sector S j that extend C ∞ -smoothly to S j ∩ D r for a sufficiently small r > 0. Their values at 0 are the projections of the eigenlines of the main term matrix K with eigenvalues λ k .
Theorem 2.4 [3,4,5,37,42,60] A germ of linear system at an irregular nonresonant singular point is analytically equivalent to its formal normal form, if and only if it has trivial Stokes matrices. Two germs of linear systems as above are analytically equivalent, if and only if their formal normal forms are the same and their Stokes matrix collections are equivalent in the following sense: they are simultaneously conjugated by one and the same diagonal matrix (independent on the choice of sector S j,j+1 ).
Recall that the monodromy operator of a germ of linear system at 0 acts on the space of germs of its solutions at a point z 0 = 0 sending a local solution to the result of its counterclockwise analytic extension along a circuit around the origin. Let the origin be an irregular nonresonant singular point of Poincaré rank 1, and let S 0 , S 1 be the corresponding good sectors. Let M be the monodromy matrix written in the canonical sectorial basis of solutions in S 0 . Let M norm denote the diagonal monodromy matrix of the formal normal form in the canonical solution basis with diagonal fundamental matrix. We will call M norm the formal monodromy. Recall that M norm = exp(2πi R). The matrix M is expressed in terms of the formal monodromy M norm and the Stokes matrices C 0 and C 1 via the following well-known formula [37, p.35]: Remark 2.5 We will also deal with the case, when the singular point under question is ∞, and the above statements hold in the local coordinate z := 1 z .
In the coordinate z the corresponding equation and formal normal form take the form The matrices K, K are called the main term matrices, and R, R the residue matrices of the corresponding systems at ∞. Let λ 1 , λ 2 be the eigenvalues of the matrix K. An imaginary dividing ray at infinity is a ray issued from 0 and lying in the set {Re(λ 1 − λ 2 )z = 0}. This yields the definition of good sectors "at infinity". The Sectorial Normalization and Analytic Classification Theorems and the definition of Stokes matrices at infinity are stated in the same way, as above; the sectors S 0 , S 0,1 , S 1 , S 1,2 at infinity are also numerated counterclockwise. Formula (2.6) also holds at ∞.

Systems with two irregular singularities. Monodromy-Stokes data
Definition 2.6 By H 1 0,∞ we will denote the class of linear systems on the Riemann sphere having two singular points, at zero and at infinity, such that both of them are irregular nonresonant of Poincaré rank 1. Each system from the class H 1 0,∞ has the type where each one of the main term matrices K and N at zero and at ∞ has distinct eigenvalues.
Definition 2.7 Consider a linear system L ∈ H 1 0,∞ . Fix a point z 0 ∈ C * and two pairs of good sectors (S 0 0 , S 0 1 ), (S ∞ 0 , S ∞ 1 ) for the main term matrices at 0 and ∞ respectively, see Remark 2.5. Fix two paths α p in C * numerated by p = 0, ∞, going from the point z 0 to a point in S p 0 . Let f 1p , f 2p be a canonical sectorial solution basis for the system L at p in S p 0 . Consider the analytic extensions of the basic functions f kp to the point z 0 along paths α −1 p . Let π : Let M denote the monodromy operator of the system L acting on the local solution space at z 0 (identified with the space C 2 of initial conditions at z 0 ) by analytic extension along counterclockwise circuit around zero. The tuple taken up to the next equivalence is called the monodromy-Stokes data of the system L. Namely, two tuples (q, M ), (q , M ) ∈ (CP 1 ) 4 × GL 2 (C) are called equivalent 4 , if there exists a linear operator H ∈ GL 2 (C) whose projectivization sends q kp to q kp and such that H −1 • M • H = M . We will also deal with the transition matrix Q comparing the canonical bases at 0 and at ∞ at

Remark 2.8
The monodromy-Stokes data of a system L depends only on the homotopy class of the pair of paths (α 0 , α ∞ ) in the space of pairs of paths in C * with a common (variable) starting point z 0 and with endpoints lying in given sectors S 0 0 and S ∞ 0 respectively. Indeed, let a homotopy between two pairs of paths, (α 0 , α ∞ ) with base point z 0 and (α 0 , α ∞ ) with base point z 0 , move z 0 to z 0 along a path β in C * . Let X(z) be the germ of fundamental matrix of the system L at z 0 such that X(z 0 ) = Id. Let H = X(z 0 ) denote the value at z 0 of the analytic extension of the fundamental matrix function X(z) along the path β. Then H transforms the monodromy-Stokes data corresponding to z 0 and the path pair (α 0 , α ∞ ) to that corresponding to z 0 and the path pair (β −1 α 0 , β −1 α ∞ ), as at the end of the above definition. Proposition 2.9 One has q 1p = q 2p for every p = 0, ∞. The monodromy-Stokes data of a system L ∈ H 1 0,∞ determines the collection of formal monodromies M norm,p , the Stokes matrices C jp at p = 0, ∞, j = 0, 1, and the transition matrix Q uniquely up to the following equivalence. Two collections (M norm,p , C jp , Q) and (M norm,p , C jp , Q ) are equivalent, if M norm,p = M norm,p and there exists a pair of diagonal matrices D 0 , D ∞ such that The inequality q 1p = q 2p follows from linear independence of the basic functions f 1p , f 2p , which implies independence of their values at z 0 . A given pair of distinct points q 1p , q 2p ∈ CP 1 defines a basis (v 1p , v 2p ) in C 2 (whose vectors are projected to q jp ) uniquely up to multiplication of vectors by constants. Recall that in the basis (f 1p , f 2p ) of the local solution space at z 0 with f kp (z 0 ) = v kp the monodromy matrix is given by formula (2.6): Here the Stokes matrices C 0p , C 1p are unipotent of opposite triangular types (determined by the main term matrix of the system L at p). Let they be, say, upper and lower triangular respectively with the corresponding triangular elements c 0 and c 1 . Recall that the formal monodromy matrix M norm,p is diagonal, set M norm,p = diag(m 1p , m 2p ); m jp = 0. Then For simplicity everywhere below whenever we work with global systems on C we omit the word "analytically" ("constant"), and "gauge equivalence" means "constant gauge equivalence".
Theorem 2.11 Two systems L 1 , L 2 ∈ H 1 0,∞ are gauge equivalent, if and only if they have the same formal normal forms at each singular point and the same monodromy-Stokes data. In this case each linear automorphism of the fiber {z = z 0 } C 2 sending the monodromy-Stokes data of one system to that of the other system extends to a gauge equivalence of systems. Here both monodromy-Stokes data correspond to the same sectors and path collections.
Proof The statement of the theorem holds if one replaces the monodromy-Stokes data by collection of Stokes matrices and the transition matrix up to equivalence from the above proposition, see [39, proposition 2.5, p.319]. The collection of Stokes and transition matrices (taken up to the latter equivalence) is uniquely determined by the monodromy-Stokes data, by the same proposition. Conversely, the monodromy-Stokes data can be restored from the formal monodromy and Stokes and transition matrices. Namely, the monodromy matrix M in the basis (f 10 , f 20 ) is found from (2.10). Let us choose coordinates on C 2 in which f 10 (z 0 ) = (1, 0), f 20 (z 0 ) = (0, 1). Then one has q 10 = [1 : 0], q 20 = [0 : 1], and q 1∞ , q 2∞ are the projections of the columns of the transition matrix Q. Theorem 2.11 is proved. 2

Isomonodromic deformations and Painlevé equation
Here we introduce general Jimbo's isomonodromic deformations of linear systems in H 1 0,∞ , which form a one-dimensional holomorphic foliation of the space H 1 0,∞ (Subsection 3.2). Afterwards we study its restriction to the socalled Jimbo type systems, where isomonodromic deformations are described by solutions of Painlevé 3 equation (3.14) (Subsection 3.3). In Subsection 3.4 we consider the space of real Jimbo type systems (i.e., defined by real matrices) with R 21 > 0 > R 12 . We introduce the space J N (R + ) of their appropriate normalizations with R 21 = −R 12 > 0 by gauge transformations and variable rescalings: the so-called normalized R + -Jimbo type systems. Their space J N (R + ) contains the space Jos of systems (1.7) and is foliated by isomonodromic families obtained from Jimbo deformations by normalizations. We show that the family Jos is transversal to the isomonodromic foliation of J N (R + ), and it corresponds to poles of order 1 with residue 1 of solutions of Painlevé equations (3.14) (Subsection 3.5). A background material on isomonodromic deformations is recalled in Subsection 3.1.

Isomonodromic deformations: definition and Frobenius integrability sufficient condition
Let us give the following definition of isomonodromic family of linear systems in H 1 0,∞ , which is equivalent to the classical definition, by Proposition 2.9.
Definition 3.1 A family of systems in H 1 0,∞ is isomonodromic, if the residue matrices of formal normal forms at their singular points and the monodromy-Stokes data remain constant: independent on the parameter of the family. Remark 3.2 If a family of systems in question is continuously parametrized by a connected parameter space, then constance of the monodromy-Stokes data automatically implies constance of the residue matrices R p of the formal normal forms. Indeed, constance of formal monodromies M norm,p follows by Proposition 2.9. The formula M norm,p = exp(2πi R p ) implies that the residue matrices R p are uniquely determined by M norm,p up to addition of integer diagonal matrices. Hence, they are constant, by continuity and connectivity.
is isomonodromic if there is a rational in z (with possible poles only at z = 0, ∞) and analytic in t matrix differential 1-form Ω = Ω(z, t) on C × D such that 3) means that Ω is integrable in the Frobenius sense. See, e.g., [11, proof of theorem 13.2].
An isomonodromic deformation with a scalar parameter is often defined by a system of PDEs [25] ∂Y
The deformation space will be a simply connected domain D ⊂ C * containing R + . Though the deformation in [38] was written in a seemingly special case, it works in the following general case. We are looking for isomonodromic families of systems L(t) ∈ H 1 0,∞ given by system (3.4) of the following type: After the time variable change t = e s (which cancels "t" in the latter denominator), the integrability condition (3.5) takes the form of a system of autonomous polynomial ordinary differential equations on matrix coefficients in K, R, N (here and in what follows In the initial time variable t and the new matrix variable K := 1 t K system (3.7) takes the following simplified, though non-autonomous, form: is a polynomial vector field on the space H 1 0,∞ identified with a connected open dense subset in the space C 12 with coordinates being matrix coefficients. Its complex phase curves form a onedimensional holomorphic foliation of the space H 1 0,∞ by isomonodromic families. The corresponding system (3.8) considered as a non-autonomous differential equation in the linear-system-valued function L(t) ∈ H 1 0,∞ has the following first integrals: -the matrix N ; -the conjugacy class of the matrix K = 1 t K; -the residue matrices of the formal normal forms of L(t) at 0 and at ∞; -the conjugacy class of the monodromy operator of the system L(t). Invariance of residues and monodromy follows from Theorem 3.3. Invariance of residues can be also deduced directly from (3.8) and (2.3). The proposition follows immediately from expression (3.7).

Isomonodromic deformations of special Jimbo type systems
Definition 3.6 A special Jimbo type linear system is a system of type such that there exists a matrix G ∈ GL 2 (C) for which In (3.9) and (3.10) the symbol * stands for an arbitrary unknown matrix element. Here all the matrices are complex.
Remark 3.7 The formal normal form at ∞ of a system (3.9) is Condition (3.10) is equivalent to the statement saying that its formal normal form at 0 is Proposition 3.8 The space of Jimbo type systems (3.9), (3.10) corresponding to a given ∈ C is invariant under the flow of field (3.7) and hence, is a union of its phase curves. The number is a first integral.
The proposition follows from Remarks 3.4 and 3.7.
We study Jimbo's isomonodromic families of systems (3.9) given by (3.8), which take the following form: We denote the upper right entries of K(t) and R(t) by K 12 (t) and R 12 (t) respectively.

Isomonodromic families of normalized R + -Jimbo systems
Definition 3.11 An R + -Jimbo type system is a system (3.9) given by real matrices K, R satisfying (3.10) with R 12 < 0 < R 21 and t > 0. (The matrix G, whose inverse diagonalizes K, can be chosen real and unimodular.) The space of R + -Jimbo type systems will be denoted by J(R + ). A linear system is of normalized R + -Jimbo type, if it has the form (3.16) where K and R satisfy (3.10). The space of normalized R + -Jimbo type systems will be denoted by J N (R + ).

Remark 3.12
The space J(R + ) of R + -Jimbo type systems is a union of real isomonodromic families L(t), real phase curves of vector field (3.7). (Proposition 3.5). Each R + -Jimbo type system can be transformed to a normalized one by composition of a unique diagonal gauge transformation Example 3.13 The space Jos of linear systems of Josephson type, i.e., family (1.7), is contained in J N (R + ). Its natural inclusion to J N (R + ) transforms a system (1.7) with parameters (µ, , ω) to a system (3.16) with K = diag( 1 2 , 0) and the parameters τ = 2µ, , R 21 = 1 2ω .
Proposition 3.14 (Rigidity). No two distinct systems in J N (R + ) are gauge equivalent.
Proof A gauge equivalence must be diagonal: it should keep the main term matrix at ∞ diagonal. It should also preserve the equality R 12 = −R 21 and the inequality R 21 > 0. Therefore, it is a constant multiple of identity, and leaves the system in question invariant. This proves the proposition. 2 It carries a real analytic foliation by isomonodromic families (which will be referred to, as normalized real isomonodromic families) given by the differential equation Along its solutions the function w(τ ) = − R 12 (τ ) τ K 12 (τ ) = R 21 (τ ) τ K 12 (τ ) satisfies Painlevé 3 equation (3.14). The above foliation will be denoted by F.
Proof Let us show that the closed subset J N (R + ) ⊂ H 1,0 0,∞ is a 4-dimensional submanifold. For every matrix K ∈ Mat 2 (C) with distinct eigenvalues and any their fixed order (λ 1 , λ 2 ), the matrix G such that G −1 KG = diag(λ 1 , λ 2 ) is uniquely defined up to multiplication from the right by a non-degenerate diagonal matrix. We will cover H 1,0 0,∞ by two open subsets W 1 , W 2 ⊂ H 1,0 0,∞ : Let us show that the intersection J N (R + ) ∩ W 1 is a 4-dimensional submanifold in W 1 . Then we prove the similar statement for the intersection J N (R + ) ∩ W 2 . For every system in J N (R + ) ∩ W 1 the corresponding matrix G can be normalized as above in a unique way so that Hence, its matrix K is defined by two parameters G 12 and G 21 , and the correspondence (G 12 , G 21 ) → K is bijective. Let us write the second equation in (3.10) for a normalized system with R 12 = −R 21 . It says that the matrix has zero right-lower element. This is the equation which is equivalent to the equation saying that G 12 is a known rational function of three independent variables G 21 , R 21 , . The latter equivalence holds outside the exceptional set where the numerator and the denominator in (3.20) vanish simultaneously. Vanishing of the numerator is equivalent to vanishing of G 21 (since R 21 = 0, by assumption), and in this case the denominator equals R 21 = 0. Thus, the exceptional set is empty. This implies that Let us now prove the above statement for J N (R + ) ∩ W 2 . If G 12 = 0, then we can normalize the matrix G in a unique way so that Then the second equation in (3.10), which says that the matrix which is equivalent to the equation Now it suffices to show that the above numerator and denominator cannot vanish simultaneously, as in the previous discussion. Indeed, their vanishing means that The first statement of the lemma is proved. The space J(R + ) of R + -Jimbo type systems is a manifold projected to the space J N (R + ) via the diagonal gauge normalizations from Remark 3.12. The projection is an analytic bundle with fiber R + , by the same remark and since J N (R + ) is a submanifold. It sends isomonodromic families in J(R + ) given by (3.11) to normalized isomonodromic families in J N (R + ). Let us find the differential equation describing them. Fix a τ 0 > 0 and matrices K 0 , R 0 defining a system in J N (R + ). Set t 0 = τ 2 0 . Consider the R + -Jimbo type system (3.9) defined by the same matrices. Let L(t) be its isomonodromic deformation given by equation (3.11), and let K(t), R(t) denote the corresponding matrices. Let L(t) denote its projection to J N (R + ), which is given by a gauge transformation family (Y 1 , Y 2 ) → (Y 1 , λ(t)Y 2 ) and z-variable rescalings z = τ ζ: the matrices defining the systems L(t) are In the second equation in (3.22) R 21 = −R 12 , since R 12 ≡ −R 21 . This yields Substituting the above formula for ν to (3.22), replacing the matrix diag(0, 1) in the commutators by diag(−1, 0) = diag(0, 1) − Id, and changing the variable t to τ = √ t yields (3.17). The function w(τ ) defined in (3.12) for the family L(t) coincides with the analogous function defined for the family L(t), since R 12 . It satisfies equation (3.14), by Theorem 3.9. Lemma 3.15 is proved. Proof It is well-known that non-zero singular points of solutions of equation (3.14) are poles of order 1 with residues ±1 [33, p.158]. Let us check that systems in Jos correspond to poles with residue 1. Consider a system in J N (R + ) with τ = τ 0 and K 12 = 0 (e.g., a system lying in Jos) and its isomonodromic deformation given by (3.17). The upper triangular term in the second matrix equation in (3.17) has the form If the initial system corresponding to τ = τ 0 lies in Jos, then (1)), and w has simple pole with residue 1 at τ 0 .
Conversely, let w(τ ) have a simple pole with residue 1 at τ 0 . Then (3.24). Note that the trace of the matrix K is constant and equal to 1 2 . Hence, K 22 (τ 0 ) = 0, K 11 (τ 0 ) = 1 2 . Now to show that the system in question lies in Jos, it suffices to prove that K 21 (τ 0 ) = 0. Suppose the contrary: K 21 (τ 0 ) = 0. Then the matrix G, whose inverse conjugates K(τ 0 ) to diag( 1 2 , 0), is lower triangular with G 21 = 0. We normalize it by constant factor to have G 11 = 1. Equation The contradiction thus obtained proves that K(τ 0 ) = diag( 1 2 , 0) and the system in question lies in Jos. Lemma 3.16 is proved. Proof Way 1 of proof. In an isomonodromic family given by (3.17) the derivative K 12 is non-zero at τ corresponding to a system lying in Jos, see (3.23). Hence, this family is transversal to the hypersurface Jos.  Proof Proposition 3.2 from [28] states that a point is a constriction, if and only if (1.7) has projectively trivial monodromy: the monodromy matrix is a scalar multiple of identity. Another criterion given by [28, lemma 3.3] states that a point is a constriction, if and only if ∈ Z and the germ of linear system (1.7) at the origin is analytically equivalent to its formal normal form. In this case system (1.7) and its formal normal form have the same monodromy matrices in appropriate bases. The monodromy of the normal form is given by the diagonal matrix diag(e −2πi , 1), which is identity if ∈ Z. Proposition 4.1 is proved. For every system L ∈ J N (R + ) let us choose good sectors S 0 and S 1 that contain the upper (respectively, lower) half-plane punctured at 0, see Fig.  5. Consider its monodromy-Stokes data (q 10 , q 20 , q 1∞ , q 2∞ ; M ) defined by the base point z 0 = 1 ∈ S 1,2 ⊂ S 0 ∩ S 1 and trivial paths α 0 , α ∞ ≡ 1. Set We will call R(L) the transition cross-ratio of the system L. It depends only on the monodromy-Stokes data and not on choice of its representative. For the proof of Theorem 1.12 we first prove the following theorem and lemma in Subsections 4.1 and 4.2 respectively. Theorem 4.3 (Key Theorem). The subset Σ ⊂ J N (R + ) is a two-dimensional analytic submanifold, a union of leaves of the real isomonodromic foliation F from Lemma 3.15. One has ∈ Z for every system in Σ. The function R is constant on leaves of F in J N (R + ). The restriction R| Σ is real-valued; it is an analytic submersion For every ∈ Z by Σ ⊂ Σ we denote the subset of systems with given .

Lemma 4.4
For every ∈ Z the subset Constr ⊂ (R 2 + ) (µ,η) , η = ω −1 , is a real-analytic one-dimensional submanifold identified with the intersection Jos ∩ Σ . The restriction of the function R to the latter intersection yields a mapping Constr → R \ {0, 1} that is a local analytic diffeomorphism.
Afterwards in Subsection 4.3 we prove the following more precise version of the first two statements of Theorem 1.12. 2) Let C := R −1 : I → C denote the inverse function. For every c ∈ {a, b} \ {0} there exists a sequence x n ∈ I, x n → c, as n → ∞, such that η n = η(C(x n )) → ∞, i.e., ω(C(x n )) → 0.
In Subsection 4.4 we prove constance of the rotation number and type of constriction on each connected component in Constr and finish the proof of Theorem 1.12.

Systems with trivial monodromy. Proof of Theorem 4.3
In the proof of Theorem 4.3 we use a series of propositions. Proposition 4.6 Every system L ∈ H 1 0,∞ with trivial monodromy (e.g., every system in Σ) has trivial Stokes matrices and trivial formal monodromies at both singular points 0, ∞. In particular, the residue matrices of its formal normal forms have integer elements. If L ∈ Σ, then one has ∈ Z.
Proof The proof repeats arguments from [28, proof of lemma 3.3]. Triviality of the Stokes matrices follows from formulas (2.11) and (2.12). Then M = M norm = Id, by (2.6). Hence, ∈ Z, if L ∈ Σ. 2 Proposition 4.7 Let in a system L ∈ H 1 0,∞ , see (2.7), the matrices K, R, N be real, and let each one of the matrices K, N have distinct real eigenvalues. Let the Stokes matrices of the system L at 0 and at ∞ be trivial. Then the transition cross-ratio R(L) is either real, or infinite.
Proof One has q kp = g k0,p (1) = π • f k0,p (1), where f 10,p , f 20,p form the canonical basis of solutions of the system in S 0 . Their linear independence implies linear independence of their values at z = 1, and hence, the inequality q 1p = q 2p . Let us now prove that among the points q kp there are at least three distinct ones. To do this, we use the fact that g k,p (z) := g k0,p (z) = g k1,p (z) are two meromorphic functions on C * ∪ {p}, p = 0, ∞. Meromorphicity on C * follows from Proposition 4.6 and the proof of Proposition 4.7. Meromorphicity at p follows from Remark 2.3. Suppose the contrary: there are only two distinct points among q kp . Then g 1,0 ≡ g k 1 ,∞ , g 2,0 ≡ g k 2 ,∞ , where (k 1 , k 2 ) is some permutation of (1, 2). Therefore, g 1,p and g 2,p are meromorphic on C, by the above discussion.
Their graphs are disjoint, since so are graphs of their restrictions to C * (being phase curves of the Riccati foliation on Proof Fix a neighborhood V such that 3 distinct points in q 0 remain distinct in each collection from V. Let us normalize them by the PSL 2 (C) action in such a way that these points be 0, 1, ∞: such normalization is unique. Then the fourth point is uniquely determined by the cross-ratio. 2 Proof of Theorem 4.3. A system L ∈ J N (R + ) is uniquely defined by the formal invariants , τ and the monodromy-Stokes data (Theorem 2.11 and Proposition 3.14). Let now M (L) = Id. Then the latter data are reduced to the PSL 2 (C)-orbit of the collection (q 10 , q 20 , q 1∞ , q 2∞ ). The latter collection consists of at least three distinct points (Proposition 4.8). Therefore, each system L ∈ Σ has a neighborhood W = W(L) ⊂ Σ such that two systems in W have the same monodromy-Stokes data, if and only if the corresponding cross-ratios R are equal. This follows from Proposition 4.9 and the above discussion. One has (R, τ )(L) ∈ RP 1 × R, by Propositions 4.6 and 4.7. This together with the above statement on unique local determination by R imply that the mapping Π : (R, τ ) : Σ → RP 1 × R is locally injective.
Proof We have to realize each T = (R, τ ) close to T 0 by a linear system from Σ. To this end, we first realize T by an abstract two-dimensional holomorphic vector bundle over C with connection. Namely, we take two linear systems defined by the given formal normal forms at 0 and ∞ respectively: We consider the following trivial bundles with connections over discs covering C: the bundle ) equipped with the system H ∞ . The bundle realizing T is obtained by the following gluing F 0 and F ∞ over the annulus (1, 0), v 2 = (0, 1) denote the standard basis in C 2 . For every R close enough to R 0 fix a linear isomorphism L 1 : C 2 → C 2 such that the tautological projection to CP 1 = C of the collection of vectors L 1 v 1 , L 1 v 2 , v 1 , v 2 has the given cross-ratio R and L 1 depends analytically on R. Let W 0 (z) = diag(e (4.4) Let E = E(R, τ ) denote the disjoint union F 0 F ∞ pasted by the following identification: for every z ∈ A the point (Y 0 , z) ∈ F 0 is equivalent to The space E inherits a structure of holomorphic vector bundle over C with a well-defined meromorphic connection induced by the formal normal forms H 0 , H ∞ in the charts F 0 and F ∞ (which paste together by L z to the same connection over A). This connection has two Poincaré rank 1 irregular nonresonant singular points at 0 and ∞ where it is analytically equivalent to H 0 and H ∞ . Note that the monodromy-Stokes data and the transition cross-ratio are well-defined for bundles with connections as well, provided that the singularities at 0 and at ∞ are irregular nonresonant of Poincaré rank 1. The transition cross-ratio of the bundle E(R, τ ) coincides with R, by construction.
Let now V 2 be a small ball centered at T 0 = (R 0 , τ 0 ) in the complex product C R × C τ (in its local chart centered at T 0 ). Set E := (R,τ )∈ V 2 E(R, τ ). This is a holomorphic vector bundle over the product C × V 2 .
Claim 1. The bundle E is trivial, if the ball V 2 is small enough. Proof The bundle E(T 0 ) is trivial, since it has the same monodromy-Stokes data and formal normal forms, as the system L 0 (which is a connection on trivial bundle), and by Theorem 2.11 (which remains valid for bundles with connections). It is glued from two trivial bundles over the domains D 2 and C \ D 1 2 by the transition matrix function L z,T 0 . Triviality implies that there exist (and unique) GL 2 (C)-valued matrix functions U 0 (z) and U ∞ (z) holomorphic on D 2 and C \ D 1 2 respectively such that U 0 (z) = U ∞ (z)L z,T 0 on A and U ∞ (∞) = Id. They are holomorphic on bigger domains D 3 D 2 , C\D 1 3 C\D 1 2 , by the above statement applied to the latter bigger domains and holomorphicity of the transition matrix function L z,T 0 on C * . Consider the following new trivializations of the trivial bundles C 2 In the new coordinates Y 0 and Y ∞ the fiber identifications gluing E of the above trivial bundles over points (z, T ) ∈ A × V 2 become the following: a point ( Therefore, E can be viewed as the bundle glued from two trivial bundles on D 2 × V 2 and (C \ D 1 2 ) × V 2 by the transition matrix function M (z, T ) holomorphic on A × V 2 . One has M (z, T 0 ) = Id, by construction. Choosing V 2 small enough, one can make M (z, T ) continuous on A × V 2 and make the C 0 -norm ||M (z, T ) − Id|| on A × V 2 arbitrarily small. Therefore, the bundle E glued by M (z, T ) is "close to trivial", and hence, is trivial, whenever V 2 is small enough, by [11, appendix 3, lemma 1]. (Formally speaking, this lemma should be applied after rescaling the coordinates in the chart containing V 2 in the parameter space to make V 2 the unit ball.) The claim is proved. 2 Let V 2 ⊂ V 2 be the subset of real points of the complex ball V 2 , which is a real planar disk. The claim implies that the family E(T )| T ∈V 2 yields a family of connections on the trivial bundle C 2 × C depending analytically on the parameter T ∈ V 2 . They should be linear systems in H 1 0,∞ , since the singularities at 0 and ∞ are irregular non-resonant of Poincaré rank 1. This yields an analytic map g : 0,∞ such that for every (R, τ ) ∈ V 2 the system g(R, τ ) has trivial monodromy, transition cross-ratio equal to R, and is analytically equivalent to formal normal forms (4.2), (4.3) near 0 and ∞ respectively. Without loss of generality we consider that g(T 0 ) = L 0 , applying a gauge transformation independent on (R, τ ). For every system in g(V 2 ) the corresponding points q kp ∈ C Φ from the monodromy-Stokes data given by the base point z 0 = 1 and trivial paths α 0 ≡ α ∞ ≡ 1 lie on the same circle, since their cross-ratio R lies in R ∪ {∞}. The latter circle is unique, since there are at least three distinct points q kp : this is true for T = T 0 (Proposition 4.8) and remains valid for all T ∈ V 2 , provided that V 2 is chosen small enough . We normalize the systems in g(V 2 ) so that the latter circle is the real line, applying an analytic family of gauge transformations depending on (R, τ ).
Claim 2. The systems in g(V 2 ) are defined by real matrices. Proof The transformation σ : (Y 1 , Y 2 ; z) → (Y 1 , Y 2 ;z) applied to systems in g(V 2 ) preserves formal normal forms and monodromy-Stokes data, by construction and the above normalization. Therefore, it sends each system in g(V 2 ) to a system gauge equivalent to it, and the collections of points q kp in the fiber C × {1} are the same for both systems. Their gauge equivalence restricted to the fiber C 2 × {1} should fix the lines corresponding to q kp . Hence, it is identity up to scalar factor, since the number of distinct points q kp is at least three. Therefore, the systems in question coincide. Thus, σ fixes each system in g(V 2 ), which means that its matrices are real. 2 The main term matrix N at ∞ of each system in g(V 2 ) is real, and its eigenvalues are − τ 2 , 0. It is close to diag(− τ 2 , 0), if V 2 is small enough. Therefore, it is conjugated to the diagonal matrix diag(− τ 2 , 0) by a real matrix H close to the identity. The matrix H is unique up to left multiplication by a real diagonal matrix. It can be chosen in a unique way so that the gauge transformation Y = H −1 Y makes R 21 = −R 12 > 0. This yields a family of gauge transformations sending systems in g(V 2 ) to systems lying in J N (R + ), and hence, in Σ (triviality of monodromy). From now on, the mapping V 2 → J N (R + ) thus constructed will be denoted by g. By construction, its image lies in Σ , and for every (R , τ ) ∈ V 2 the transition cross-ratio R and the formal invariant τ of the system g(R , τ ) are respectively R and τ . Conversely, every system L ∈ Σ close enough to L 0 has invariants (R, τ ) lying in V 2 , and hence L = g(R, τ ), by construction, Theorem 2.11 and Proposition 3.14. This proves Proposition 4.10. 2 The mapping g is an immersion, since the projection L → (R, τ )(L) is real-analytic and (R, τ )•g = Id. This together with Proposition 4.10 implies that Σ is a 2-dimensional submanifold, and (R, τ ) : Σ → RP 1 × R is a local diffeomorphism. Hence, the projection R : Σ → RP 1 (which is constant along isomonodromic leaves) is a submersion. Theorem 4.3 is proved. The space of systems (1.7) with given is identified with (R + ) 2 µ,η , η = ω −1 . They are represented as systems in Jos ⊂ J N (R + ) with parameters τ = 2µ, , R 21 = η 2 . The constriction subset Constr ⊂ (R + ) 2 µ,η is thus identified with the intersection Jos ∩ Σ , by Proposition 4.1. The latter intersection is transversal, since Σ is a union of leaves of the isomonodromic foliation F and Jos is transversal to F (Lemma 3.17). Therefore, Constr is a onedimensional submanifold transversal to the isomonodromic foliation on Σ . Hence, R : Constr → RP 1 is a local analytic diffeomorphism (submersivity of the projection R : Σ → RP 1 , see Theorem 4.3). It remains to show that R = 0, 1, ∞ on Constr . Proof One has q 1p = q 2p . Hence, the only a priori possible coincidences are the following. Case 1): q k0 = q k∞ for some k. Then the same equality holds for the other k, by symmetry (Φ, z) → (Φ −1 , z −1 ) of the corresponding Riccati equation (1.6). Thus, the collection of points q kp consists of two distinct points. This contradicts to Proposition 4.8.
Case 2): q k0 = q (3−k)∞ for some k. This means that the transition matrix between the canonical solution base of system (1.7) at 0 and the canonical base at ∞ taken in inverse order is a triangular matrix. But this contradicts to [29, theorem 2.10, statement (2. 19)].

Asymptotics and unboundedness. Proof of Theorem 4.5
The subset Constr ⊂ (R 2 + ) µ,η is a submanifold that admits a locally diffeomorphic projection R to R \ {0, 1} (Lemma 4.4). This implies that it has no compact components, since no compact component can admit a locally diffeomorphic mapping to R. Therefore, each its component C is diffeomorphic to some interval I = (a, b) with coordinate x := R. This implies the first statement of Theorem 4.5. To prove its second statement, the existence of a sequence x n → c with η(C(x n )) → ∞ for c ∈ {a, b} \ {0}, C = R −1 , we will -use the following Klimenko-Romaskevich Bessel asymptotic result [44] to show that boundedness of η implies boundedness of µ; -prove that (µ, η)(x n ) cannot converge to (0, 0), by using solution of variational equation to (1.3) and studying local parametrization of the analytic subset in R 2 containing Constr ; -show that if η(x n ) → 0, then c = lim x n = 0. Let us recall that the boundary of the phase-lock area L r consists of two curves ∂L r,0 , ∂L r,π , corresponding to those parameter values, for which the Poincaré map of the corresponding dynamical system (1.4) acting on the circle {τ = 0} has fixed points 0 and π respectively. These are graphs ∂L r,α = {B = G r,α (A)}, G r,α are analytic functions on R; α = 0, π.
Theorem 4.12 [44, theorem 2]. There exist positive constants ξ 1 , ξ 2 , K 1 , K 2 , K 3 such that the following statement holds. Let r ∈ Z, A, ω > 0 be such that |rω| Let J r denote the r-th Bessel function. Then Proposition 4.13 Fix an ∈ Z. For every η 0 > 0 the intersection is a one-dimensional analytic submanifold with infinitely many connected components, and each component is bounded.
Proof Let u 1 < u 2 < . . . denote the sequence of points of local maxima of the modulus |J (−u)|, which tends to plus infinity.
Claim. Fix an η 0 > 0 and an ∈ Z. For every k ∈ N large enough (dependently on η 0 and ) the interval I k := {µ = u k 2 } × (0, η 0 ) does not intersect the constriction set Constr . Proof In the coordinates (µ, η) inequalities (4.5), (4.6) and (4.7) can be rewritten for r = respectively as For every k large enough the value µ = u k 2 satisfies inequality (4.8) for all η ∈ (0, η 0 ). Substituting µ = u k 2 to the right-hand side in (4.9) transforms it to a sequence of functions of η ∈ (0, η 0 ) with uniform asymptotics The values |J (−u k )| are known to behave asymptotically as 1 √ u k (up to a known constant factor). Therefore, they dominate the right-hand sides in (4.9) and (4.10). This together with (4.9), (4.10) implies that for every k large enough, set A k = u k η , as k → ∞, uniformly in ω > ω 0 = η −1 0 . This implies that ω lies between G ,0 (A k ) and G ,π (A k ) for large k. Therefore, for every k large enough and every ω > ω 0 the point ( ω, A k ; ω) lies in the interior of the phase-lock area L , and hence, is not a constriction. This proves the claim. 2 For every point q ∈ Constr and every k large enough dependently on q the connected component of the point q in Constr is separated from infinity by the segment I k from the above claim. This proves boundedness of connected components. Infiniteness of number of connected components follows from their boundedness and the fact that for every given ∈ Z and ω > 0 the vertical line Λ = {B = ω } contains an infinite sequence of constrictions with A-ordinates converging to +∞; the latter fact follows from  Lemma 4.14 For every ∈ Z the subset Constr ⊂ R 2 + does not accumulate to zero. That is, there exists no sequence of constrictions (B k , A k ; ω k ) with B k = ω k where ω k → +∞ and µ k : For the proof of Lemma 4.14 (given below) let us recall that the first equation in system (1.4) describing model of Josephson junction takes the following form in the new parameters µ and η: The constrictions correspond to those values of (µ, η) ∈ R 2 + for which the time 2π flow map h = h 2π = h µ,η of equation (4.11) acting on the θ-circle {τ = 0} is identity (Proposition 1.11). For the proof of the lemma it suffices to show that (0, 0) is an isolated point in the analytic subset {h µ,η = Id} ⊂ R 2 µ,η . This is done by using the following formulas for a solution θ(τ ) of (4.11) and its derivatives in parameters for η = 0.
where g(θ 0 ) is a non-constant function of θ 0 that is equal to either sin θ 0 , or cos θ 0 up to non-zero constant factor.
Claim. The above (k, 1)-th derivative is 2π-periodic in τ , if 1 ≤ k ≤ − 1. If k = , it is equal to g(θ 0 )τ plus a 2π-periodic function; here g(θ 0 ) has the same type, as in Proposition 4.16. Proof The (k, 1)-th derivative equals the primitive of the right-hand side in (4.17). The latter right-hand side is a linear combination of values of sin (or cos) of θ 0 + rτ , r ∈ Z, − k ≤ r ≤ + k. Moreover, the coefficient at the "lower term", the sin (cos) of θ 0 + ( − k)τ , is non-zero, by elementary trigonometry. Therefore, the primitive of the latter right-hand side in (4.17) is a linear combination of cos (sin) of the above arguments, except for a possible term with r = 0, which is τ cos θ 0 (τ sin θ 0 ) up to constant factor. For k ≤ − 1 the latter term does not arise. For k = it arises with a non-zero constant factor, by the above discussion. The claim is proved. 2 One has ∂ k+m θ ∂µ k ∂η m (0, θ 0 ; 0, 0) = 0, by definition. This together with the above claim and discussion implies the statements of Proposition 4.16. 2 Proof of Lemma 4.14. Suppose the contrary: the set Constr accumulates to zero. Recall that it lies in the ambient analytic subset in R × R defined by the equation h µ,η = Id. (The Poincaré map h µ,η is Möbius, being the restriction to S 1 = {|Φ| = 1} of the monodromy map of Riccati equation (1.6); the equation h µ,η = Id is written in the Lie group Aut(D 1 ) PSL 2 (R).) Therefore, the latter analytic subset contains an irreducible germ of analytic curve Γ at 0 with η| Γ , µ| Γ ≡ 0, since η, µ = 0 on Constr . Hence, Γ can be considered as a graph of (may be singular) analytic function µ = cη α (1 + o(1)), α > 0, c = 0. Substituting the latter expression for µ to the Taylor formula (4.19) yields The right-hand side in (4.20) should be identically equal to θ 0 , since h µ,η = Id for (µ, η) ∈ Γ. This together with (4.20) implies that its second and third terms should cancel out: 1 + α = 2 and g(θ 0 ) ≡ c − π . But we know that g(θ 0 ) ≡ const. The contradiction thus obtained proves Lemma 4.14. 2 Proof of the second statement of Theorem 4.5. Suppose the contrary: as x ∈ I tends to a non-zero endpoint c ∈ {a, b} of the interval I, the function η = η(C(x)) is bounded from above. But then µ(C(x)) is also bounded from above, by Proposition 4.13. The component C being a non-compact submanifold in R 2 + , it should go to "infinity" (to the boundary), as x → c. Therefore, there exists a sequence x k → c such that C(x k ) → C * ∈ {µη = 0 | µ, η ≥ 0} (boundedness of µ and η). One has C * = (0, 0), by Lemma 4.14. Let show that two other possible cases treated below are impossible. Case 1): C * = (0, η), η > 0. Then the equations (4.11) corresponding to C(x k ) = (µ k , η k ) have identity Poincaré map and limit to the equation which should also have identity Poincaré map. In the case, when = 0, this is obviously impossible, since the dynamical system on T 2 given by (4.21) is hyperbolic with an attracting periodic orbit θ ≡ π 2 . In the case, when ∈ N, the rotation number of the above system is an integer non-negative number ρ < . This follows from the fact that the -th phase-lock area L intersects the B-axis {A = 0} = {µ = 0} at the so-called growth point with known abscissa B( , ω) = √ 2 ω 2 + 1, ω = η −1 , see [18, corollary 3], while C(x k ) correspond to constrictions with the abscissas ω k < B( , ω k ). Therefore, the points C(x k ) ∈ Constr also correspond to the same rotation number ρ < , whenever k is large enough (continuity of the rotation number function and its integer-valuedness on the points (B k , A k ; ω k ) corresponding to C(x k )). Thus, the points C(x k ) correspond to constrictions lying on the axis Λ = {B = ω} with non-negative rotation number ρ < . But all the constrictions lying in Λ should correspond to rotation numbers no less than , by [28, theorem 1.2]. The contradiction thus obtained shows that the case under consideration is impossible. Case 2): C * = (µ, 0), µ > 0. Then the linear system (1.7) corresponding to C * is diagonal, and hence, has zero cross-ratio R = x. Hence, the crossratios x k corresponding to C(x k ) tend to zero. But their limit c is non-zero, by assumption. The contradiction thus obtained shows that Case 2) is also impossible and finishes the proof of Theorem 4.5.

Constance of rotation number and type. Proof of Theorem 1.12
Without loss of generality we can and will restrict ourselves to the case, when ∈ Z ≥0 , due to symmetry. All the statements of Theorem 1.12 except for the last one follow immediately from Theorem 4.5. Let us prove its last statement: constance of rotation number and type. Fix a connected component C of the manifold Constr . Constance of the rotation number function on C follows from its continuity and integer-valuedness. Constance of the constriction type is obvious for = 0: the A-axis lies in L 0 , hence, all its constrictions are positive. Thus, everywhere below we consider that ∈ N (symmetry). To prove constance of type, we use the following proposition. To state it, let us recall that for every ω > 0 a generalized simple intersection is a point (B, A; ω) with = B ω ∈ Z, A = 0 and ρ = ρ(B, A; ω) ≡ (mod 2Z) that lies in the boundary of the phase-lock area L ρ = L ρ (ω) and that is not a constriction [30, definition 1.16]; they exist only for = 0. Proof One has = B ω ∈ Z. Without loss of generality we can and will consider that ≥ 1 (symmetry). Generalized simple intersections correspond to special double confluent Heun equations (1.9) having polynomial solution [13, theorem 1.15]. If, to the contrary, the constriction C were a limit of generalized simple intersections, then it would also corresponds to equation Let a constriction C(x 0 ) ∈ Constr be negative. Let us show that for every x close to x 0 the constriction C(x) = (B(x), A(x); ω(x)) is also negative: the case of positive constriction is treated analogously. (Note that each constriction is either positive, or negative, by [29, theorem 1.8].) Let ρ ∈ Z denote the rotation number of the constriction C(x 0 ). Set ω 0 := ω(x 0 ), Λ (ω) := {B = ω} ⊂ R 2 B,A . For every r > 0 let U r ⊂ R 2 denote the disk of radius r centered at (B(x 0 ), A(x 0 )). Fix an r > 0 such that (B(x 0 ), A(x 0 )) is the only point of intersection ∂L ρ (ω 0 ) ∩ Λ (ω 0 ) lying in U 2r . Such an r exists, since the latter intersection is discrete, by analyticity of the graphs ∂L ρ,0 , ∂L ρ,π forming ∂L ρ , and since none of these graphs is a vertical line.
Case 1). Let for every x close enough to x 0 the point (B(x), A(x)) be the only point of intersection ∂L ρ (ω(x)) ∩ Λ (ω(x)) lying in U r . Then all the above constrictions C(x) have the same, negative type, by definition.
Case 2). Let now the unique point C(x 0 ) of intersection ∂L ρ (ω(x 0 )) ∩ Λ (ω(x 0 )) split into several intersection points, as we perturb x = x 0 slightly. Then all these points are constrictions, by Proposition 4.17 and since ρ ≡ (mod 2), see [28, theorem 3.17]. Their number is finite, and they split the intersection Λ (ω(x))∩U r into a finite number of intervals. Any two adjacent division intervals either both lie outside the phase-lock area L ρ (ω(x)), or both lie inside L ρ (ω(x)), since the constriction separating them is either negative, or positive (see [29, theorem 1.8] and Remark 1.8). The division intervals adjacent to ∂U r should lie outside, since this is true for x = x 0 and by continuity. Therefore, all the above intervals lie outside. Hence, all the constrictions bounding them are negative. Theorem 1.12 is proved. Their absence follows from results of [29,30], which imply that the whole ray { ω}×[A 1 2 , +∞) lies in the phase-lock area L for small ω. In Subsection 5.5 we show that there are no constrictions ( ω, A) with A ∈ (1− ω, A 1 2 (ω)), whenever ω is small enough. This is done by studying family of systems (1.4) modeling Josephson junction as a slow-fast family of dynamical systems, with small ω and A = A α (ω) = 1 + ( − α)ω + o(ω). The corresponding background material on slow-fast systems is given in Subsection 5.2. The key lemma used in the proof of absence of the above-mentioned constrictions is the Monotonicity Lemma stated and proved in Subsection 5.4. It concerns a pair of slow-fast families (1.4) corresponding to two families of ordinates A α 1 and A α 2 as above with 0 < α 1 < α 2 . It deals with their Poincaré maps of the cross-section {τ = 0} lifted to the universal cover as maps of the line {τ = 0} to {τ = 2π}. The Monotonicity Lemma states that the Poincaré map of the system (1.4) corresponding to A α 2 is less than the analogous Poincaré map for A α 1 , whenever ω is small enough. Its proof is based on the Comparison Lemma on arrangement and disjointness of slow flowboxes of the systems in question (stated and proved in Subsection 5.3). To do this, let us recall the definition of the point P . Set

Absence of ghost constrictions with big ordinates
Consider the corresponding Heun equation (1.9). Fix an ω > 0. The value µ(P ) = A(P ) 2ω is the maximal number µ > 0 for which equation (1.9) has a polynomial solution, see [29, definition 1.9], [13, theorem 1.15]. It was shown in [19] that existence of polynomial solution is equivalent to the condition that the point (λ, µ) lies in a remarkable algebraic curve Γ ⊂ R 2 , the socalled spectral curve. Thus, for every ω > 0 the point (λ(P ), µ(P )) lies in Γ , and it is the point in Γ with the biggest coordinate µ. As ω → 0, one has 1 ω = 4(λ + µ 2 ) → ∞, thus, (λ, µ) → ∞. It is known that the complexified curve Γ intersects the complex infinity line in CP 2 at distinct regular real points. Their asymptotic directions correspond to the ratios λ µ equal to − 1, − 3, . . . , −( − 1), and the corresponding local branches are real. This was proved by I.V.Netay [30, proposition 1.10]. Therefore, as a point of the curve Γ tends to its infinite point, one has µ → ∞, ω . The latter ratio should tend to a number k as above. Therefore, as a point in Γ tends to infinity, one of the following asymptotics takes place: The asymptotics corresponding to points with the maximal possible A is given by m = − 1. This proves (5. Proposition 5.2 (see [43, proposition 2]). For every (A, B) ∈ R 2 + with |1 − B| < A < 1 + B the curve γ is a regular strictly convex contractible curve lying in the interior of the fundamental square [0, 2π] 2 of the torus T 2 . See Fig. 6a).

Remark 5.3
The curve γ is always symmetric with respect to the horizontal and vertical lines through the center of the latter square.
For completeness of presentation we give the proof of Proposition 5.2. Proof of Proposition 5.2. Let 1 − B < A < 1 + B. Let us now show that the curve γ does not intersect the boundary of the above fundamental square. Indeed, on the boundary either cos θ = 1, or cos τ = 1. If cos θ = 1, then f (θ, τ ) = cos θ + B + A cos τ ≥ 1 + B − A > 0. If cos τ = 1, then f (θ, τ ) ≥ −1 + B + A > 0. Therefore, f (θ, τ ) = 0 on the boundary of the fundamental square, and γ lies in its interior. For the proof of strict convexity it suffices to show that the value 6 of the Hessian form of the function f on its skew gradient tangent to its level curves is positive on γ. That is, One has P (v) > 0 for every v ∈ R, since its discriminant is negative, i.e., −2AB < A 2 + B 2 − 1 < 2AB. Indeed, the latter inequality can be rewritten as |A − B| < 1 < A + B, which is equivalent to the system of inequalities of the proposition for positive A and B. The proposition is proved. 2 Proposition 5.4 In the case, when A, B > 0 and A = 1 − B, the curve γ is regular, except for one singular point q = (π, 0) of type "transversal double self-intersection". Its intersection with the interior of the fundamental square [0, 2π] 2 is a convex curve. In the case, when B > 0 and 0 < A < 1−B, the curve γ is regular and consists of two non-contractible closed connected components of homological type (0, 1) in the standard basis in H 1 (T 2 θ,τ ). See Fig. 6b),c).
Proof Consider the first the case: A = 1−B. Convexity is preserved under passing to limit, as A > 1 − B tends to 1 − B. The singular point statement and uniqueness of singular point follow by straightforward calculation, the Implicit Function Theorem and Morse Lemma. In more detail, γ being a level curve of an analytic function f (θ, τ ), its singular points (if any) are the critical points of the function f contained in γ. The critical points are those with cos θ, cos τ = ±1. The only critical point in γ is the one with cos θ = −1, cos τ = 1, i.e., q = (π, 0). This is a Morse critical point with index −1, i.e., the Hessian form of the function f at q has eigenvalues of opposite signs: Hence, it is a transversal self-intersection singular point of the curve γ (Morse Lemma). See Fig. 6b).
As A and B vary, the topological type of the curve γ may change only near those parameter values, for which γ is a critical level curve of the function f (θ, τ ). It follows from the above critical point description that γ is a critical level curve, if and only if ±1 + B ± A = 0 for some of the four possible sign choices. Therefore, the topological type is constant in the domain {B > 0, 0 < A < 1 − B} in the parameter space. To find this topological type, fix a point (B 0 , A 0 ) ∈ R 2 + with A 0 = 1 − B 0 . We show that as A > A 0 decreases and crosses the value A 0 , a connected contractible curve γ B 0 ,A given by Proposition 5.2 is transformed to two disjoint curves isotopic to the τ -circle. For A close to A 0 the complement of each γ B 0 ,A to a small disk U centered at the singular point q is a regular curve depending analytically on the parameter A. It consists of two connected components γ B 0 ,A;± (U ) disjoint from the circle {θ = π} and projected diffeomorphically to an interval (ε, 2π − ε) of the τ -circle; here ε = ε(U ) is small. (The projection interval is the same for both components, since the symmetries θ → −θ, τ → −τ preserve each curve γ B,A .) The curves γ B 0 ,A ∩ U form a singular foliation in U by level curves of the function g(θ, τ ) := − 1 cos τ (cos θ+ B 0 ) with critical value A 0 corresponding to a Morse critical point q of index 1. The union of local branches of the singular curve γ B 0 ,A 0 at q is invariant under the above symmetries, and the local branches intersect transversally. Therefore, they are transversal to the circles {θ = π}, {τ = 0}. For A > A 0 close to A 0 the curve γ B 0 ,A is strictly convex, and its intersection with U is a union of two connected components separated by the circle {τ = 0}, by Proposition 5.2. This implies that for A < A 0 close to A 0 the local level curve γ B 0 ,A ∩ U consists of two components intersecting the circle {τ = 0}, diffeomorphically projected to an interval in the τ -circle and disjoint from the circle {θ = π}. Adding the latter components to γ B 0 ,A;± (U ) results in two closed curves in T 2 disjoint from the circle {θ = π} and projected diffeomorphically onto the τ -circle. See Fig. 6c). Thus, they are isotopic to the τ -circle. This proves the last statement of Proposition 5.4.
on T 2 with ω → 0. According to the commonly used terminology in the theory of slow-fast systems, see e.g. [35], we will call the curve the slow curve of family (5.7). Propositions 5.2 and 5.4 imply the following Corollary 5.5 For every fixed , α ∈ R + with α = 2 , for every ω small enough dependently on and α (i) if 0 < α < 2 , then the slow curve of system (5.7) is convex, regular, contractible and lies in the interior of the fundamental square [0, 2π] 2 ; (ii) if α > 2 , then the slow curve is regular and consists of two noncontractible closed connected components of homological type (0, 1).
We deal with the liftings to the universal cover R 2 over T 2 of vector fields (5.7) and their phase portraits. The lifted fields will be denoted by the same symbol (5.7). The slow curve γ α = γ α (ω) ⊂ T 2 will be identified with its lifting γ 0 α to the square [0, 2π] 2 ⊂ R 2 . Its other lifting, obtained from the latter one by translation by the vector (2π, 0) will be denoted by γ 1 α .

Definition 5.7
The interior component of the complement T 2 \ γ α is its connected component containing the point (π, π). Its liftings to the squares [0, 2π] 2 and [2π, 4π] × [0, 2π] will be called the interior components of the complements of the latter squares to the curves γ 0 α and γ 1 α respectively.

Remark 5.10
The phase-portrait of vector field (5.7) is symmetric with respect to the points C 0 := (π, π), C 1 := (2π, π), C 2 := (3π, π); the symmetry changes the sign (i.e., orientation) of the field. Let I − denote the horizontal segment symmetric to I + with respect to the point C 1 , see Fig.  7. The above construction applied to the inverse vector field, the segment I − and the heights h − j := 2π − h j yields the slow flowbox F α,− symmetric to F α,+ with respect to the point C 1 .
It suffices to prove the statement of the lemma for the flowboxes F α j ,+ , by symmetry (Remark 5.10). Here and below we use the next proposition.
At each point in the ω 8 -neighborhood of the flowbox F α 1 ,+ the angles between the vectors of the fields are greater than σ := arctan(2 + b) − arctan 2, b = cos h 0 2 (α 2 − α 1 ). The image of the flowbox F α 1 ,+ under the unit time flow map of the field (5.7) 2 is disjoint from F α 1 ,+ , and its intersection with the strip Π lies on the left from F α 1 ,+ .
Proof The vectors of the fields (5.7) 1 and (5.7) 2 have the same τ -component equal to ω. The difference of their θ-components is f α 1 (θ, τ ; ω)−f α 2 (θ, τ ; ω) = (α 2 − α 1 )ω(1 + o(1)) cos τ > 0 on W , whenever ω is small enough, since cos τ > 0 on W . Therefore, the vectors of the field (5.7) 2 are directed to the left from the vectors of the field (5.7) 1 on W , that is, the orientation statement of the proposition holds. For every ω small enough one has (5.8) by the above asymptotics, and also ω 4 Indeed, the flow lines of the field (5.7) 1 in F α 1 ,+ C 1 -converge to the line τ = θ − π (Propositions 5.8 and 5.9), hence f α 1 (θ, τ ; ω) ω on F α 1 ,+ . This together with (5.6), (5.7) and the obvious inequality | cos x| = | sin x| ≤ 1 implies (5.9). The angle lower bound statement of Proposition 5.12 follows from (5.8) and (5.9). Its last statement on image of the flowbox F α 1 ,+ under the unit time flow map of the field (5.7) 2 follows from the above angle bound and the fact that the vectors of the field (5.7) 2 have length no less than ω, while the width of the flowbox F α 1 ,+ is exponentially small (the last statement of Proposition 5.9). Proposition 5.12 is proved. Consider the analogous flowbox F α 2 ,+ and point q 2 := (χ 2 , h 1 ) for the field (5.7) 2 . One has χ 2 < χ 1 , by the claim. First suppose that q 2 / ∈ F α 1 ,+ . Then the lower base of the flowbox F α 2 ,+ is disjoint from the flowbox F α 1 ,+ and lies on its left. This together with the above claim implies that the flowboxes are disjoint. In the case, when q 2 ∈ F α 1 ,+ , the image q 2 of the point q 2 under the time 1 flow map of the field (5.7) 2 would lie strictly to the left from the flowbox F α 1 ,+ , by Proposition 5.12. Therefore, the positive orbit of the point q 2 also lies on its left, by the claim. Note that whenever ω is small enough. Therefore, the above positive orbit intersects the strip Π = {h 1 ≤ τ ≤ h 2 } by an arc of curve going from its lower base to its upper base and lying on the left from the flowbox F α 1 ,+ . The latter curve bounds F α 2 ,+ from the right, by construction. Hence, F α 2 ,+ is disjoint from F α 1 ,+ and lies on its left. The Comparison Lemma is proved. 2
We study their Poincaré maps P τ (5.10) Lemma 5.13 is proved below. In its proof we use the following proposition.  Fig. 7. Proof It suffices to prove the first statement of the proposition, due to symmetry (Remark 5.10). The segment I + defining the flowbox F α j ,+ is horizontal and is obtained from the segment [C 1 , C 2 ] by vertical shift up. The shift length is fixed and equal to h 0 − π > 0. Let J l and J r denote respectively the segment connecting C 1 (C 2 ) to the left (respectively, right) endpoint of the segment I + . One has f α j > 0 on J l and f α j < 0 on J r , which follows from Remark 5.6 and Proposition 5.8. Thus, on the segment J l (J r ) the vectors of the field (5.7) j are directed to the right (respectively, left). This implies that the time h 0 −π ω flow map of the field sends the segment [C 1 , C 2 ] strictly inside the segment I + . This together with the definition of the flowbox F α j ,ω implies the first statement of the proposition. 2 Proof of the Monotonicity Lemma. Set h + j := h j . One has Whenever ω is small enough, one has P is less than that of the latter, since cos τ > 0 on this set. This implies the inequalities of the claim. 2 Taking into account Claim 1 and (5.11), for the proof of the Monotonicity Lemma it suffices to prove the above inequality for the middle Poincaré maps in (5.11) for all ω small enough: P 2 (θ) < P 1 (θ) for every θ ∈ R. (5.12) Consider the horizontal lines L ± := {τ = h ± 1 }, which are the cross-sections for the Poincaré maps in question. We identify each their point with its θcoordinate. For every j = 1, 2 let b j± denote the point of intersection of the line L ± with the orbit of vector field (5.7) j through the point C 1 = (2π, π). Let a j± denote the analogous intersection points with the orbit through the point C 0 = (π, π). See Fig. 8.
Claim 2. One has ). The latter arcs are ωclose, which follows from the Implicit Function Theorem for the equations defining the curves γ 0 α j . This together with the above discussion proves O(ω)-closeness of the four points a j− and b j− , j = 1, 2. This proves the first part of inequality (5.13). The proof of its second part is analogous.
by (5.13). The latter segment-image in the right-hand side is disjoint from the former one and lies on the left from it, by (5.13). This proves inequality (5.12) on the segment [b 2− , a 1− + 2π].

Absence of constrictions with small ordinates
Here we prove the following theorem and then Theorem 1.13. Proof of Theorem 5.15. It suffices to prove the statement of the theorem for arbitrarily small β, e.g., β < 1 2 . Set B = ω. For every α > 0 set A α = A α (ω) = 1 + ( − α)ω. The family of systems (5.7) defined by this ordinate family A α will be denoted by (5.7) α .
Suppose the contrary: there exists a sequence ω k → 0 such that there exists a sequence of constrictions (B k , A α k ) with Passing to a subsequence, without loss of generality we can and will consider that α k converge to some α * ≥ β. Thus, the sequence of dynamical systems corresponding to the above (B k , A α k ) can be embedded into a continuous family of systems (5.7) with α replaced by α * . The latter new family of systems (5.7) will be denoted by (5.7) α * .
Fix an arbitrary α ∈ (0, β). Let P and P * denote respectively the Poincaré maps P 0,2π of the line {τ = 0} to the line {τ = 2π} defined by vector fields (5.7) α and (5.7) α * . For every ω small enough the point ( ω, 1 + ( − α)ω) lies in the phase-lock area L , by Lemma 5.1 and since α < β < 1 2 . Therefore, the corresponding system (5.7) α has a periodic orbit with rotation number . This means that there exists a point a in the θ-axis with P (a) = a + 2π . On the other hand, by the Monotonicity Lemma and since α * ≥ β > α. Therefore, the rotation number of system (5.7) α * is no greater than and a cannot be its periodic point with rotation number at least for small ω. In particular, the latter statements holds for the systems corresponding to the above constrictions (B k , A α k ) ∈ Λ . On the other hand, the dynamical system (1.4) corresponding to a constriction lying in Λ should have rotation number at least and all its orbits should be periodic with rotation number at least , see [28,  The Connectivity Conjecture, see [29, conjecture 1.14], states that the intersection L + ∩ Λ coincides with the ray S , and thus, is connected. Theorem 1.7 implies the following corollary Corollary 6.1 Let, to the contrary to the above conjecture, the intersection LΛ := L + ∩ Λ ∩ {0 < A < A(P )} be non-empty. Then its lowest point (i.e., its point with minimal ordinate A) is a generalized simple intersection.
Proof The lowest point P ∈ LΛ is well-defined, has positive ordinate and lies in ∂L , since the growth point in L , i.e., its intersection point with the abscissa axis, has abscissa √ 2 ω 2 + 1 > ω. Hence, it is either a constriction, or a generalized simple intersection, by definition. If P were a constriction, it would be negative, since its lower adjacent interval Λ ∩ {0 < A < A(P )} lies outside the phase-lock area L . But there are no negative constrictions, by Theorem 1.7. Therefore, P is a generalized simple intersection. 2 Remark 6.2 It is known that the generalized simple intersections ( ω, A) correspond to the parameters (λ, µ), µ = A 2ω , λ = 1 4ω 2 − µ 2 , of those special double confluent Heun equations (1.9) that have polynomial solutions. The set of the latter parameters (λ, µ) is a remarkable algebraic curve: the socalled spectral curve Γ ⊂ R 2 (λ,µ) introduced in [19] and studied in [19,30]. It is the zero locus of the polynomial from [19, formula (21)], which is the determinant of a three-diagonal matrix formed by diagonal terms of type λ + const and linear functions in µ at off-diagonal places. See also [30, formula (1.4)]. (The complexification of the spectral curve is known to be irreducible, see [30, theorem 1.3].) For every given ω > 0 the curve Γ contains at most points (λ, µ) corresponding to the given ω with µ > 0; the point with the biggest µ corresponds to the higher generalized simple intersection P . This follows from Bézout Theorem and the fact that the spectral curve Γ is the zero locus of a polynomial of degree in (λ, µ 2 ), see [19, p. 937]. Corollary 6.1 and the above remark reduce the Connectivity Conjecture to the following equivalent, algebro-geometric conjecture. Conjecture 6.3 For every ω > 0 the above real spectral curve Γ contains a unique point (λ, µ) with λ = 1 4ω 2 − µ 2 (up to change of sign at µ) for which the corresponding rotation number ρ = ρ( ω, 2µω) equals . (The point (B, A) = ( ω, 2µω) coincides with P , see the above remark.) Theorem 6.4 For every ∈ Z =0 and every positive ω < 1 | | the Connectivity Conjecture holds.
Indeed, the abscissa √ r 2 ω 2 + 1 of the growth point of the phase-lock area L r is greater than ω < 1. On the other hand, each boundary curve of the area L r contains constrictions, which lie in the axis Λ r , and hence, on the left from the axis Λ . Hence, each boundary curve intersects Λ + in at least one point (this statement is given by [30, theorem 1.18] for all ω small enough). It cannot be a common intersection point for both boundary curves, i.e., it cannot be a constriction, since r = ρ < and by Theorem 1.4. Therefore, the intersection ∂L r ∩ Λ + contains at least two distinct points. Analogously, ∂L 0 intersects Λ + in at least one point, since the point (1, 0) ∈ ∂L 0 lies on the right from the point ( ω, 0) ∈ Λ . If 0 ≤ r < and r ≡ (mod 2), then each point of intersection ∂L r ∩ Λ + is a generalized simple intersection. Taking these intersections for all latter r yields − 1 distinct generalized simple intersections lying in Λ + . But the total number of generalized simple intersections in Λ + is no greater than , see the above remark. Therefore, at most one of them may correspond to the rotation number , and hence, is reduced to the known generalized simple intersection P with ρ = . In particular, there are no generalized simple intersections in Λ ∩ L with 0 < A < A(P ). This together with Corollary 6.1 implies that L + ∩ Λ = S and proves the Connectivity Conjecture for 0 < ω < 1 | | . . Their experiences have shown that the interiors of the phase-lock areas tend to open subsets (the so-called limit rescaled phase-lock areas) whose connected components form a partition of the plane. In some planar region, the latter partition looks like a chess table turned by π 4 . It would be interesting to prove this mathematically and to find the boundaries of the limit phase-lock areas.
Some results on smallness of gaps between rescaled phase-lock areas for small ω were obtained in [43].
To our opinion, methods elaborated in [43] and in the present paper could be applied to study Problem 6.5.

The dynamical isomonodromic foliation
Let us consider family (1.4) modeling overdamped Josephson junction as a three-dimensional family, with variable frequency ω. Its three-dimensional phase-lock areas in R 3 B,A,ω are defined in the same way, as in Definition 1.1. Each three-dimensional phase-lock area is fibered by two-dimensional phase-lock areas in R 2 B,A corresponding to different fixed values of ω. Linear systems (1.7) corresponding to (1.4) form a transversal hypersurface to the isomonodromic foliation of the 4-dimensional manifold J N (R + ) (Lemma 3.17). It appears that there is another four-dimensional manifold with the latter property that has the following advantage: it consists of linear systems on C coming from a family of dynamical systems on 2-torus. Namely, consider the following four-dimensional family of dynamical systems on T 2 containing (1.4): dθ dτ = ν + a cos θ + s cos τ + ψ cos(θ − τ ); ν, a, ψ ∈ R, s > 0, (a, ψ) = (0, 0).
(6.1) The variable changes Φ = e iθ , z = e iτ transform (6.1) to the Riccati equation A function Φ(z) is a solution of the latter Riccati equation, if and only if Φ(z) = Y 2 (z) Y 1 (z) , where Y = (Y 1 , Y 2 )(z) is a solution of the linear system The residue matrix of the formal normal forms of system (6.2) at 0 and at ∞ is the same and equal to diag(− , 0), Theorem 6.6 The four-dimensional family of linear systems (6.2) is analytically foliated by one-dimensional isomonodromic families defined by the following non-autonomous system of differential equations: 2s a s = −2sχ + a s ( + 2χa) s = 0 . (6.4) For every ∈ R the function w(s) := a(s) 2sχ(s) = a(s) ψ(s) (6.5) satisfies Painlevé 3 equation (3.14) along solutions of (6.4).
Proof The composition of variable rescalings z = s −1 ζ and gauge transformations sends family (6.2) to the following family of linear systems: Let J denote the space of special Jimbo type systems, see (3.9), (3.10), with real matrices. Systems (6.7) lie in J , since the formal normal forms of a system (6.2) at 0, ∞ have common residue matrix diag(− , 0). Every system L 0 of type (6.7) with χ = 0 has a neighborhood W = W (L) ⊂ J where family (6.7) forms a hypersurface X = {K 22 = 2K 2 12 } ∩ W so that each system L ∈ W can be projected to a system L * ∈ X by a diagonal gauge transformation (Y 1 , Y 2 ) → (Y 1 , λY 2 ), λ = λ(L). Projecting to X a Jimbo isomonodromic family given by (3.11) yields an isomonodromic family of systems (6.7). The differential equation satisfies by the projected isomonodromic families is found analogously to the proof of equation (3.17). To do this, fix a t 0 > 0 and matrices K(t 0 ), R(t 0 ) as in (6.7). Let K(t), R(t) be solutions of (3.11) with initial conditions K(t 0 ) and R(t 0 ) at t 0 , and let (Y 1 , Y 2 ) → (Y 1 , λ(t)Y 2 ) be the family of the above normalizing gauge transformations: the matrices K(t) = diag(1, λ(t)) K(t) diag(1, λ −1 (t)), R(t) = diag(1, λ(t)) R(t) diag(1, λ −1 (t)) are the same, as in (6.7), that is K 22 (t) = 2K 2 12 (t); λ(t 0 ) = 1. Set ξ = λ (t 0 ). The equation on the matrix function K(t) given by (3.11) yields K 12 (t 0 ) = −ξK 12 (t 0 ) + 1 t 0 [R(t 0 ), K(t 0 )] 12 , K 22 (t 0 ) = 1 t 0 [R(t 0 ), K(t 0 )] 22 , (6.8) Substituting K 22 = 2χ 2 , K 12 = χ to the second equation in (6.8) and changing the time parameter t to s = √ t yields formula for the derivative χ t = (K 12 ) t and the first equation in (6.4). Substituting thus found derivative K 12 to the first formula in (6.8) yields a linear equation on ξ, whose solution is ξ = − +2χa 2t 0 . The differential equation on the matrix R(t) in (3.11) yields the differential equation on R 12 analogous to the first equation in (6.8), which also includes the above already found value ξ. Substituting R 12 = − a 2 there yields the second equation in (6.4). Painlevé 3 equation (3.14) on w(s) along isomonodromic families thus constructed follows from Theorem 3.9, since diagonal gauge transformations do not change the ratio R 12 K 12 . Equation (3.14) can be also deduced directly from (6.4). 2 The foliation from Theorem 6.6 given by (6.4) induces a one-dimensional foliation in the 4-dimensional space of dynamical systems (6.1) given by the following non-autonomous system of equations obtained from (6.4) by change of the variable χ to ψ = 2sχ: ψ s = a + (1 − ) ψ s − aψ 2 s 2 a s = −ψ + a s + ψa 2 s 2 . (6.9) The latter foliation of family (6.1) given by (6.9) will be denoted by G and called the dynamical isomonodromic foliation. Proof The projectivized monodromy of linear system (6.2) is the complexification of the Poincaré map of the corresponding dynamical system (6.1). Therefore, constance of its conjugacy class along leaves implies constance of conjugacy class of the Poincaré map and hence, of the flow and of its rotation number; = const, by Theorem 6.6. Family of systems (1.4) coincides with the hypersurface {ψ = 0} ∩ {a > 0} in the parameter space. It is transversal to the vector field (6.9), since ψ = a > 0 at all its points. The characterization of systems (1.4) in terms of poles follows from construction and Lemma 3.16 and, on the other hand, immediately from (6.9): if ψ(s 0 ) = 0, then ψ(s) a(s 0 )(s − s 0 ), w(s) = a(s) ψ(s) 1 s−s 0 , as s → s 0 , and vice versa. 2 Recall that the growth point of a two-dimensional phase-lock area L r has abscissa sign(r) √ r 2 ω 2 + 1. For a given r ∈ Z =0 the latter growth points form a curve bijectively parametrized by ω > 0 in the three-dimensional parameter space, which will be called the r-th growth curve. We already know that the family of constrictions in R 2 B,A × (R + ) ω is a one-dimensional submanifold, by Theorem 1.12. Thus, it is a disjoint union of connected curves, which will be called the constriction curves.
In what follows the three-dimensional phase-lock area in R 2 B,A × (R + ) ω with a rotation number r ∈ Z will be denoted by L r . Conjecture 6.8 Each constriction curve is bijectively projected onto (R + ) ω . Each three-dimensional phase-lock area of family (1.4) is a countable garland of domains, where any two adjacent domains are separated either by the corresponding growth curve, or by a constriction curve. Remark 6.9 Conjecture 6.8 does not follow from known results on twodimensional phase-lock area. A priori, a constriction curve may be not bijectively projected to the ω-axis, and the projection may have some critical value ω 0 that is a local maximum (minimum). In this case the corresponding two-dimensional phase-lock area with ω less (greater) than ω 0 has two constrictions that collide for ω = ω 0 and disappear, when ω crosses the critical value ω 0 . Numerical experiences made by S.I.Tertychnyi, D.A.Filimonov, V.A.Kleptsyn, I.V.Schurov show that such a scenario does not arise. This can be viewed as a numerical confirmation of Conjecture 6.8. Remark 6.10 In each two-dimensional phase-lock area L r the constrictions lying in the half-plane {A > 0} are ordered by natural numbers k corresponding to their heights: the lowest constriction is ordered by 1, the second one by two, etc. If Conjecture 6.8 is true, then along each constriction curve C lying in the upper quarter-space {A > 0} the above height number is constant. In this case each constriction curve C = C ,k ⊂ {A > 0} is numerated by two integer numbers = ρ = B ω , k := the above height number.
Studying of the following two problems, which are of independent interest, would have important applications to Conjecture 6.8 and related problems.
Problem 6.11 Study the Poincaré map of the dynamical isomonodromic foliation G, see (6.9), acting on the transversal hypersurface given by family of systems (1.4). The Poincaré map sends the intersection of its definition domain with each three-dimensional phase-lock area in family (1.4) to the same phase-lock area, by constance of the rotation number along leaves. Study the action of the Poincaré map of the foliation G given by (6.9) on the three-dimensional phase-lock area portrait of family (1.4).
Problem 6.12 Is it true that the above Poincaré map is well-defined on each constriction curve C ,k and sends its diffeomorphically onto C ,k+1 ?
If Conjecture 6.8 is true, then for every ∈ Z and k ∈ N there is a unique connected component O ,k of the interior of the three-dimensional phase-lock area L that is adjacent to the constriction curves C ,k and C ,k+1 . Remark 6.14 The Poincaré map of the foliation G (where it is defined) can be viewed as the suspension over the map sending a given simple pole s 0 > 0 with residue 1 of solution w(s) of Painlevé 3 equation (3.14) to its next pole s 1 > s 0 of the same type (if any). Many solutions of (3.14) have an infinite lattice of simple poles with residue 1 converging to +∞. Our Painlevé 3 equations (3.14) admit a one-dimensional family of Bessel type solutions, see [24], whose poles are zeros of solutions of Bessel equation and are known to form an infinite lattice. Victor Novokshenov's recent numerical experience has shown that their small deformations also have an infinite lattice of poles. Few solutions, e.g., the tronquée solutions [50], are bounded on some semi-interval [C, +∞), and hence, do not have poles there.
Problem 6.15 Describe those parameter values of family (1.4) for which the corresponding solution w(s) of (3.14) is tronquée. Is it true that this holds for some special points of boundaries of the three-dimensional phaselock areas?Does this hold for the higher generalized simple intersections P discussed in Subsection 6.1? Problem 6.16 Study geometry of phase-lock areas 7 in four-dimensional family (6.1) of dynamical systems on T 2 . Study special points of boundaries of the phase-lock areas: analogues of growth points, constrictions and generalized simple intersections.
Let Σ denote the subfamily in (6.1) consisting of dynamical systems with trivial Poincaré map. The value = ν − ψa s corresponding to a system in Σ should be integer, as in Proposition 4.6, and its rotation number ρ is also integer. For every , ρ ∈ Z let Σ ,ρ ⊂ Σ denote the subset consisting of systems with given and ρ. Those systems (1.4) with given that correspond to constrictions are contained in Σ , , by Theorem 1.4. To our opinion, a progress in studying the above problems would have applications to problems on geometry of phase-lock areas, for example, to problems discussed in the previous subsection.