Pith. sign in

REVIEW 2 major objections 5 minor 44 references

Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that deforming the RSJ Josephson model to a general-Heun family preserves rotation number quantization but breaks every constriction, so phase-lock-area interiors become connected.

desk verdict New families connecting Heun equations to torus dynamics are solid and worth publishing, but the headline constriction-breaking result rests on an unproved extension of a prior lemma. read the letter →

arxiv 2507.07282 v3 pith:DZ6FLROR submitted 2025-07-09 math.DS math-phmath.MPnlin.AO

Artem Alexandrov , Alexey Glutsyuk This is my paper · ORCID
classification math.DSmath-phmath.MPnlin.AO MSC 34M3537E1037E45
keywords Heunequationphase-lockareasrotationnumberquantizationconstrictionsJosephsonjunctiontorusdynamicalsystemsmonodromyRiccati
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces the deformed RSJ family of torus dynamical systems and proves that two hallmarks of the original RSJ model separate: the rotation-number quantization effect survives, while the constrictions that pinch each phase-lock area into a garland disappear. Concretely, in the family $\frac{d\theta}{d\tau} = \frac{\cos\theta + B + A\sin\tau}{\omega(1-\delta\cos\tau)} + D$ with $\delta \in (0,1)$, phase-lock areas still exist only at integer rotation numbers, but for $A \neq 0$ the Poincar\'e first-return map is never the identity, so no constrictions can occur. The paper also builds a confluent-Heun companion family and identifies the precise curves on the $A=0$ axis where identity maps and growth points can appear. A sympathetic reader would care because this is the first qualitative separation between the RSJ model and a Heun-equation deformation of it, and it gives a monodromy-based mechanism for when phase-lock areas are connected versus chained.

What carries the argument

The engine of the argument is the class of torus dynamical type linear systems: Fuchsian (or confluent) systems on the Riemann sphere whose projectivized Riccati equation preserves the product of unit circles $S^1_\Phi \times S^1_z$. Restricting to those circles via $\Phi=e^{i\theta}$, $z=e^{i\tau}$ turns the Riccati equation into the torus ODE (2.1). Because the Poincar\'e map of such an ODE is the projectivized monodromy of the linear system, it is a M\"obius transformation, and an identity Poincar\'e map is equivalent to scalar monodromy. The proof of Theorem 24 compares residue eigenvalues at the singularities $0$ and $\alpha$, obtaining the relation $(\nu+n)^2=(\bar{\nu}+c)^2-4b^2$, which forces $\operatorname{Im}\nu=0$ and hence $A=0$; this is what rules out off-axis identity maps. The constriction-breaking conclusion then rides on the lemma that constrictions sit only at trivial Poincar\'e maps.

What would settle it

Compute the Poincaré map of (2.1) numerically for fixed $\delta=0.5$, $\omega=1$, $D=0$ with $A=0.5$ and scan $B$ over a fine grid; if any $B$ gives the identity map, Theorem 24 fails. Alternatively, for the same parameters plot the $r$-th phase-lock area and check whether its intersection with a horizontal line $A=0.5$ is disconnected by a point; a surviving constriction would disprove Corollary 25. A direct check of the $A=0$ prediction $B^2-1=\omega^2(1-\delta^2)(D-n)^2$ by integrating (5.14) also settles the identity-map locus.

Watch

Extended reading notes

Core claim

The central claim is that the deformed RSJ family (2.1) inherits the rotation number quantization of the RSJ model but loses its constrictions. Theorem 2 states that phase-lock areas exist only for integer rotation numbers, because the family is strictly increasing in $B$ and its Poincar\'e maps are M\"obius transformations. Theorem 24 states that for $\delta \in (0,1)$ and $A \neq 0$ the Poincar\'e map is never the identity, and that for $A=0$ identity maps occur exactly along $B^2-1=\omega^2(1-\delta^2)(D-n)^2$ with $n \in \mathbb{Z}$. Since a constriction, by the lemma quoted from [20], can occur only at a trivial Poincar\'e map, Corollary 25 concludes that in the dRSJ family all constrictions break and the intersections of phase-lock-area interiors with the upper and lower half-planes become connected. On the $A=D=0$ axis the paper also computes the remaining growth points and the rotation number $\rho = \frac{\sqrt{B^2-1}}{\omega\sqrt{1-\delta^2}}$.

Load-bearing premise

The load-bearing premise is that a constriction of a phase-lock area can occur only where the Poincaré first-return map is the identity; the paper extends this lemma from the RSJ case to the deformed family by assertion rather than by a written proof.

Editorial extensions

If this is right

  • For every nonzero $A$, the Poincar\'e map of the deformed RSJ family is not the identity when $\delta \in (0,1)$, so the family has no constrictions.
  • The interiors of each phase-lock area in the upper and lower half-planes become connected in the dRSJ family, unlike the garland chains of the RSJ model.
  • Rotation number quantization survives, so phase-lock areas in the deformed family still exist only for integer rotation numbers.
  • On the axis $A=0$, identity Poincar\'e maps occur exactly along $B^2-1=\omega^2(1-\delta^2)(D-n)^2$, locating the growth points of the phase-lock areas in the $(B,D)$ plane.
  • For $A=D=0$ and $B>1$, the rotation number is explicitly $\rho = \frac{\sqrt{B^2-1}}{\omega\sqrt{1-\delta^2}}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same monodromy-and-constriction logic likely applies to the confluent companion family (2.7), so its phase-lock areas may also have connected interiors; the paper does not state this.
  • The explicit identity-condition curves give a testable numerical signature: for fixed $\omega$, $\delta$, and $D$, scanning $B$ at $A=0$ should show degenerate or growth points exactly at $B^2-1=\omega^2(1-\delta^2)(D-n)^2$, and deviation would indicate that the extended constriction lemma fails.
  • If a physical Josephson device realizes the dRSJ denominator as a time-modulated critical current, Shapiro steps should appear with connected step interiors rather than the familiar chained garlands, a distinction visible in current-voltage curves.
  • Because constrictions correspond to collisions of monodromy data in the RSJ case, their breaking suggests that the isomonodromic foliation of dRSJ parameters has a different topology, possibly traceable through Painlev\'e-type equations.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper constructs two families of torus dynamical systems associated, respectively, with general and confluent Heun equations, obtained by projectivizing linear systems of a special 'torus dynamical type.' The main object is the deformed RSJ family (2.1), dθ/dτ = (cosθ + B + A sinτ)/(ω(1 − δ cosτ)) + D. The authors prove that this family is equivalent to a family of general Heun equations (Theorem 17 and Proposition 22) and that a confluent analogue exists (Theorem 20 and Proposition 23). They also prove rotation-number quantization for (2.1) (Theorem 2) and a monodromy criterion for the identity Poincaré map (Theorem 24). Corollary 25 then asserts that, for δ ≠ 0, all constrictions break and the phase-lock areas become connected in the upper and lower half-planes. The paper also contains a computation of growth points (Theorem 26) and a rotation-number formula for the restricted family (Proposition 27).

Significance. If the constriction-breaking claim is fully established, the paper identifies a clean qualitative difference between the RSJ model and its general-Heun deformation, and it connects Heun-equation monodromy to phase-lock topology in a way that goes beyond the previously studied double-confluent case. The paper contains substantial direct algebraic work: the residue eigenvalue computation in Theorem 24, the reduction of linear systems to Heun equations, and the explicit rotation-number formula are concrete and mostly verifiable by direct calculation. The numerical figures provide a falsifiable prediction (absence of constrictions for δ ≠ 0). The main monodromy theorem is not circular and appears sound. However, the headline geometric conclusion is conditional on an unproved structural lemma, so the significance of the paper is not yet fully realized.

major comments (2)
  1. [§5.3, Corollary 25] The central claim that all constrictions break depends entirely on the assertion that [20, Proposition 2.2] 'remains valid in full generality together with its proof.' This is a load-bearing unproved premise. Theorem 24(1) excludes only identity Poincaré maps, whereas a constriction of a phase-lock area in a general strictly B-monotone family of Möbius circle maps could, in principle, occur at a nonidentity parabolic map with a single double fixed point. The RSJ proof of [20, Proposition 2.2] uses special structure (for example, constriction alignment B = rω), so the generalization is not self-evident. Please either supply a proof of the generalized constriction lemma or weaken Corollary 25 accordingly.
  2. [§5.3, Corollary 25] The connectedness conclusion of Corollary 25 also assumes, without proof, that the phase-lock areas of the deformed family (2.1) retain the RSJ 'garland' topology, in which constrictions are the only possible separators of components in the upper and lower half-planes. Even if one granted the absence of constrictions, the interiors of phase-lock areas could in principle remain disconnected for other reasons. A separate argument, at least for sufficiently small δ, is needed to justify the transition from 'no constrictions' to 'connected intersections with half-planes.'
minor comments (5)
  1. [§5.3, Theorem 24(1)] The wording 'Let δ ∈ [0, 1) and A ∈ R be both non-zero' is ambiguous and, if read literally as allowing δ = 0, contradicts the known existence of identity Poincaré maps at RSJ constrictions. The statement should say δ ∈ (0, 1) and A ≠ 0.
  2. [§5.3, Theorem 24(2)] Theorem 24(2) does not state a domain for δ, but the proof uses δ ∈ (0, 1). Please state the hypotheses explicitly, since the residue computation does not cover the δ = 0 case.
  3. [References] References [7] and [8] are identical; one of them should be removed or replaced by the intended distinct citation.
  4. [§5.3, Proposition 27] The proof of the rotation-number formula (5.24) is sketched via an argument modulo Z and sign followed by monotonicity; it would be helpful to spell out the continuity and monotonicity argument in a few more lines.
  5. [§2.1, Eq. (2.1)] The parameter δ is introduced as belonging to (0, 1) in (2.1), but later results and the RSJ limit use δ = 0; making the domain of δ consistent throughout would remove a source of confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 24's monodromy computation and Theorem 2's quantization argument are independent; the constriction-breaking conclusion of Corollary 25 leans on an unproved extension of the authors' own earlier [20, Prop. 2.2].

  1. self citation load bearing [Section 5.3, Corollary 25 (proof)]
    "Consider an arbitrary family of dynamical systems depending on parameters ( B, A) and depending strictly monotonously on B. Then each constriction of a phase-lock area (if any) always corresponds to a dynamical system with trivial Poincaré map, see [20, proposition 2.2]. Though this proposition was stated in a special case, it remains valid in full generality together with its proof."

    The corollary's conclusion that all constrictions break down is obtained by combining Theorem 24 (no identity Poincaré map) with the claim that in every strictly B-monotone family every constriction corresponds to a trivial Poincaré map. That claim is not proved in this paper; it is imported from [20], whose author list includes the present co-author A. Glutsyuk. The text explicitly admits that [20, Prop. 2.2] was 'stated in a special case' and asserts that it 'remains valid in full generality together with its proof' without supplying that proof. Thus the central qualitative result depends on an unproved generalization of the authors' own earlier proposition rather than on a derivation contained in this paper.

full rationale

The paper's computational core is self-contained and not circular. Theorem 24 is a direct monodromy computation: assuming an identity Poincaré map forces Im ν = 0 and hence A = 0, contradicting A ≠ 0; Theorem 26 and Proposition 27 are explicit computations in the A = D = 0 slice. Theorem 2 cites the external quantization result [11] for strictly monotone families of Möbius circle diffeomorphisms; that citation is not self-referential and is independent support for the quantization statement. The genuine circularity concern is isolated to Corollary 25: the headline claim that constrictions break in dRSJ needs the lemma that constrictions coincide with trivial Poincaré maps in the whole dRSJ family, and the proof delegates this to [20, Prop. 2.2] with the remark that it 'remains valid in full generality together with its proof' without giving that proof. Because [20] is the authors' own earlier special-case result, the inference is load-bearing on a self-citation whose generalization is asserted rather than demonstrated. The corollary also assumes without proof that dRSJ phase-lock areas retain the RSJ garland topology in which constrictions are the only separators of components in the upper and lower half-planes; this is a separate missing-support issue rather than a circularity. Overall, the central computation is independent, but the concluding constriction-breaking claim is partially circular because it rests on an unproved extension of the authors' prior work.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper fits no constants to data: the new families are defined by free parameters (ω, δ, B, A, D), and the Heun equation coefficients are expressed through the linear system parameters without any optimization. The central claims rest on standard monodromy and Moebius theory and on the cited quantization theorem [11], plus one unproved extension of the authors' prior constriction lemma [20, Prop 2.2], which is the main additional assumption.

assumptions (2)
  • standard math Rotation number quantization holds in any family of Moebius circle diffeomorphisms that is strictly monotone in a parameter (cited to [11]).
    Used in Theorem 2 to establish quantization in family (2.1).
  • ad hoc to paper At a constriction of a phase-lock area in a family strictly increasing in B, the Poincare map is trivial (identity); asserted to extend [20, Proposition 2.2] from the RSJ special case to full generality, without proof.
    Load-bearing for Corollary 25: constriction breaking is deduced from Theorem 24 via this lemma. The extension beyond the published special case is unproved in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking." pith.science (2026). https://pith.science/paper/DZ6FLROR

@misc{pith2026250707282,
  author       = {Pith},
  title        = {Pith review of: Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZ6FLROR}},
  note         = {Machine review of arXiv:2507.07282}
}
read the original abstract

The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on torus that can be equivalently described by a family of general Heun equations (GHE), with four singular points, and confluent Heun equations, with three singular points. The first family, related to GHE, is a deformation of the RSJ model, which will be denoted by dRSJ. The phase-lock areas of a family of dynamical systems on the torus are those level subsets of the rotation number function that have nonempty interiors. It is known that for the RSJ model, the rotation number quantization effect occurs: phase-lock areas exist only for integer rotation number values. Moreover, each phase-lock area is a chain of domains separated by points. Those separation points that do not lie on the abscissa axis are called constrictions. In the present paper, we study phase-lock areas in the new family dRSJ. The quantization effect remains valid in this family. On the other hand, we show that in the new family dRSJ the constrictions break down.

Figures

Figures reproduced from arXiv: 2507.07282 by the authors.

Figure 1
Figure 1. Phase-lock areas of the RSJ model (1.1) for different ω. White crosses correspond to constriction points (only three constrictions are shown), white upper triangles correspond to growth points. Color bar represents the values of the largest Lyapunov exponent. Here φ is the phase difference between two superconductors with insulator link between them, B and A are dimensionless amplitudes of DC and AC currents applied… view at source ↗
Figure 2
Figure 2. Phase-lock areas of system (2.1) in (B, A)-plane without constrictions. Here ω = 1, D = 0. Color bar represents the values of the largest Lyapunov exponent. −4 −2 0 2 4 B −4 −2 0 2 4 D δ = 0.5 0 1 2 3 4 5 6 7 −4 −2 0 2 4 B −4 −2 0 2 4 δ = 0.8 0 2 4 6 8 10 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Phase-lock areas of system (2.1) with A = 0, ω = 1 in the (B, D)-plane. Color bar represents the values of the largest Lyapunov exponent. Corollary 25. Let D = 0, ω ∈ R \ {0} be fixed. As we perturb system (5.13) to sys￾tem (2.1)ω,δ,D with small fixed δ ̸= 0 and arbitrary small D ∈ R, then all the constrictions break down. Thus, the intersections of the interiors of the phase-lock areas with upper and lower half-pla… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 34 canonical work pages

  1. [1]

    Phase-locking in dynamical systems and quantum mechanics

    A. Alexandrov, A. Glutsyuk, and A. Gorsky. Phase-locking in dynamical systems and quantum mechanics, 2025. URL https://arxiv.org/abs/2504.20181

    work page Pith review arXiv 2025

  2. [2]

    Aminov, A

    G. Aminov, A. Grassi, and Y. Hatsuda. Black hole quasinormal modes and Seiberg- Witten theory. Annales Henri Poincare , 23(6):1951–1977, 2022. doi: 10.1007/ s00023-021-01137-x. URL https://doi.org/10.1007/s00023-021-01137-x

    doi:10.1007/s00023-021-01137-x 1951

  3. [3]

    V. Arnold. Geometrical methods in the theory of ordinary differential equations , volume

    work page

  4. [4]

    Bibilo and A

    Y. Bibilo and A. Glutsyuk. On families of constrictions in model of overdamped Josephson junction and Painlev´ e 3 equation. Nonlinearity, 35(10):5427, 2022. doi: 10.1088/1361-6544/ac8aee. URL https://doi.org/10.1088/1361-6544/ac8aee

    work page doi:10.1088/1361-6544/ac8aee 2022

  5. [5]

    Buchstaber and A

    V. Buchstaber and A. Glutsyuk. On determinants of modified Bessel functions and entire solutions of double confluent Heun equations. Nonlinearity, 29(12):3857, 2016. doi: 10.1088/0951-7715/29/12/3857. URL https://doi.org/10.1088/0951-7715/29/ 12/3857. 25

    work page doi:10.1088/0951-7715/29/12/3857 2016

  6. [6]

    Buchstaber and A

    V. Buchstaber and A. Glutsyuk. On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect.Proceedings of the Steklov Institute of Mathematics , 297:50–89, 2017. doi: 10.1134/S0081543817040046. URL https://doi.org/10.1134/S0081543817040046

    work page doi:10.1134/s0081543817040046 2017

  7. [8]

    Buchstaber and S

    V. Buchstaber and S. Tertychniy. Explicit solution family for the equation of the re- sistively shunted Josephson junction model. Theoretical and Mathematical Physics, 176 (2):965–986, 2013. doi: 10.1007/s11232-013-0085-2. URL https://doi.org/10.1007/ s11232-013-0085-2

    work page doi:10.1007/s11232-013-0085-2 2013

  8. [9]

    Buchstaber and S

    V. Buchstaber and S. Tertychnyi. Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction. Theoretical and Mathematical Physics, 182:329–355, 2015. doi: 10.1007/s11232-015-0267-1. URL https: //doi.org/10.1007/s11232-015-0267-1

    work page doi:10.1007/s11232-015-0267-1 2015

Show all 44 references
  1. [10]

    Buchstaber, O

    V. Buchstaber, O. Karpov, and S. Tertychnyi. On properties of the differential equation describing the dynamics of an overdamped Josephson junction. Russian Mathematical Surveys, 59(2):377, 2004. doi: 10.1070/RM2004v059n02ABEH000725. URL https: //doi.org/10.1070/RM2004v059n02A...

    2004 doi

  2. [11]

    Buchstaber, O

    V. Buchstaber, O. Karpov, and S. Tertychniy. Rotation number quantization effect.The- oretical and Mathematical Physics, 162:211–221, 2010. doi: 10.1007/s11232-010-0016-4. URL https://doi.org/10.1007/s11232-010-0016-4

    2010 doi

  3. [12]

    R. C. Dinsmore, M. Bae, and A. Bezryadin. Fractional order Shapiro steps in supercon- ducting nanowires. Applied physics letters , 93(19), 2008. doi: 10.1063/1.3012360. URL https://doi.org/10.1063/1.3012360

    2008 doi

  4. [13]

    Figueiredo

    B. Figueiredo. Schr¨ odinger equation as a confluent Heun equation. Physica Scripta, 99 (5):055211, 2024. doi: 10.1088/1402-4896/ad3510. URL https://doi.org/10.1088/ 1402-4896/ad3510

    2024 doi

  5. [14]

    Fiziev and D

    P. Fiziev and D. Staicova. Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes. Physical Review D , 84(12):127502,

  6. [15]

    R. Foote. Geometry of the Prytz planimeter. Reports on mathematical physics , 42(1- 2):249–271, 1998. doi: 10.1016/S0034-4877(98)80013-X. URL https://doi.org/10. 1016/S0034-4877(98)80013-X. 26

    1998 doi

  7. [16]

    Foote, M

    R. Foote, M. Levi, and S. Tabachnikov. Tractrices, bicycle tire tracks, hatchet planime- ters, and a 100-year-old conjecture. The American Mathematical Monthly , 120(3):199– 216, 2013. doi: 10.4169/amer.math.monthly.120.03.199. URL https://doi.org/10. 4169/amer.math.monthly.120.03.199

    2013 doi

  8. [17]

    Glutsyuk

    A. Glutsyuk. On constrictions of phase-lock areas in model of overdamped Josephson effect and transition matrix of the double-confluent Heun equation. Journal of Dynam- ical and Control Systems , 25(3):323–349, 2019. doi: 10.1007/s10883-018-9411-1. URL https://doi.org/10.1007/...

    2019 doi

  9. [18]

    Glutsyuk

    A. Glutsyuk. On germs of constriction curves in model of overdamped josephson junc- tion, dynamical isomonodromic foliation and painlev´ e 3 equation.Moscow Mathematical Journal, 23(4):479–513, 2023

    2023

  10. [19]

    Glutsyuk and L

    A. Glutsyuk and L. Rybnikov. On families of differential equations on two-torus with all phase-lock areas. Nonlinearity, 30(1):61, 2016. doi: 10.1088/0951-7715/30/1/61. URL https://doi.org/10.1088/0951-7715/30/1/61

    2016 doi

  11. [20]

    Glutsyuk, V

    A. Glutsyuk, V. Kleptsyn, D. Filimonov, and I. Schurov. On the adjacency quan- tization in an equation modeling the Josephson effect. Functional Analysis and Its Applications, 48(4):272–285, 2014. doi: 10.1007/s10688-014-0070-z. URL https: //doi.org/10.1007/s10688-014-0070-z

    2014 doi

  12. [21]

    Gr¨ uner and A

    G. Gr¨ uner and A. Zettl. Charge density wave conduction: A novel collective trans- port phenomenon in solids. Physics Reports , 119(3):117–232, 1985. doi: 10.1016/ 0370-1573(85)90073-0. URL https://doi.org/10.1016/0370-1573(85)90073-0

    1985 doi

  13. [22]

    Y. Hatsuda. Quasinormal modes of Kerr-de Sitter black holes via the Heun function. Classical and Quantum Gravity , 38(2):025015, 2020. doi: 10.1088/1361-6382/abc82e. URL https://doi.org/10.1088/1361-6382/abc82e

    2020 doi

  14. [23]

    Horta¸ csu

    M. Horta¸ csu. Heun functions and some of their applications in physics. Advances in High Energy Physics , 2018(1):8621573, 2018. doi: 10.1155/2018/8621573. URL https: //doi.org/10.1155/2018/8621573

    2018 doi

  15. [24]

    Ilyashenko, D

    Y. Ilyashenko, D. Ryzhov, and D. Filimonov. Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations. Functional Analysis and Its Applications, 45(3):192–203, 2011. doi: 10.1007/s10688-011-0023-8. URL https: //doi.org/10.1007...

    2011 doi

  16. [25]

    Ishkhanyan

    A. Ishkhanyan. Schr¨ odinger potentials solvable in terms of the confluent Heun func- tions. Theoretical and Mathematical Physics , 188(1):980–993, 2016. doi: 10.1134/ S0040577916070023. URL https://doi.org/10.1134/S0040577916070023

    2016 doi

  17. [26]

    Klimenko and O

    A. Klimenko and O. Romaskevich. Asymptotic properties of Arnold tongues and Joseph- son effect. Moscow Mathematical Journal , 14(2):367–384, 2014. 27

    2014

  18. [27]

    Likharev, K.K.; Ulrikh

    B. Likharev, K.K.; Ulrikh. Systems with Josephson junctions: Basic Theory. [In Rus- sian]. Izdat. MGU, Moscow, 1978

    1978

  19. [28]

    Likharev

    K. Likharev. Introduction to the dynamics of Josephson junctions. [In Russian]. Moscow, Nauka, 1985

    1985

  20. [29]

    Likharev

    K. Likharev. Dynamics of Josephson junctions and circuits . Routledge, 1986. doi: 10.1201/9781315141572. URL https://doi.org/10.1201/9781315141572

    1986 doi

  21. [30]

    Likharev, K.K.; Semenov

    V. Likharev, K.K.; Semenov. Electrodynamical properties of superconducting point contacts. [in russian]. Radiotechnika i Electronika, 16(11):2167, 1971

    1971

  22. [31]

    Lisovyy and A

    O. Lisovyy and A. Naidiuk. Perturbative connection formulas for Heun equations. Jour- nal of Physics A: Mathematical and Theoretical , 55(43):434005, 2022. doi: 10.1088/ 1751-8121/ac9ba7. URL https://doi.org/10.1088/1751-8121/ac9ba7

    2022 doi

  23. [32]

    Litvinov, S

    A. Litvinov, S. Lukyanov, N. Nekrasov, and A. Zamolodchikov. Classical conformal blocks and Painlev´ e VI. Journal of High Energy Physics , 2014(7):1–20, 2014. doi: 10.1007/JHEP07(2014)144. URL https://doi.org/10.1007/JHEP07(2014)144

    2014 doi

  24. [33]

    D. E. McCumber. Effect of AC impedance on DC voltage-current characteristics of superconductor weak-link junctions. Journal of Applied Physics , 39(7):3113–3118, 1968. doi: 10.1063/1.1656743. URL https://doi.org/10.1063/1.1656743

    1968 doi

  25. [34]

    Mishra, A

    S. Mishra, A. Ryabov, and P. Maass. Phase locking and fractional Shapiro steps in collec- tive dynamics of microparticles. Physical Review Letters, 134(10):107102, 2025. doi: 10. 1103/PhysRevLett.134.107102. URL https://doi.org/10.1103/PhysRevLett.134. 107102

    2025 doi

  26. [35]

    Panghotra, B

    R. Panghotra, B. Raes, C. C. de Souza Silva, I. Cools, W. Keijers, J. E. Scheerder, V. V. Moshchalkov, and J. Van de Vondel. Giant fractional Shapiro steps in anisotropic Josephson junction arrays. Communications Physics , 3(1):53, 2020. doi: 10.1038/ s42005-020-0315-5. URL ht...

    2020 doi

  27. [36]

    Pikovsky, M

    A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization: A Universal Concept in Nonlinear Sciences . Cambridge University Press, 2001. ISBN 9780521533522. doi: 10.1017/CBO9780511755743

    2001 doi

  28. [37]

    Reichhardt and J

    C. Reichhardt and J. Olson. Shapiro steps for skyrmion motion on a washboard potential with longitudinal and transverse ac drives. Physical Review B, 92(22):224432, 2015. doi: 10.1103/PhysRevB.92.22443. URL https://doi.org/10.1103/PhysRevB.92.22443

    2015 doi

  29. [38]

    M. J. Renne and D. Polder. Some analytical results for the resistively shunted Joseph- son junction. Revue de physique appliqu´ ee, 9(1):25–28, 1974. doi: 10.1051/rphysap: 019740090102500. URL https://doi.org/10.1051/rphysap:019740090102500. 28

    1974 doi

  30. [40]

    S. Shapiro. Josephson currents in superconducting tunneling: The effect of mi- crowaves and other observations. Physical Review Letters , 11(2):80, 1963. doi: 10.1103/PhysRevLett.11.80. URL https://doi.org/10.1103/PhysRevLett.11.80

    1963 doi

  31. [41]

    Slavyanov and W

    S. Slavyanov and W. Lay. Special Functions: A Unified Theory Based on Singularities . Oxford University Press, 2000. doi: 10.1093/oso/9780198505730.001.0001. URL https: //doi.org/10.1093/oso/9780198505730.001.0001

    2000

  32. [42]

    Tertychniy

    S. Tertychniy. The modelling of a Josephson junction and Heun polynomials, 2006. URL https://arxiv.org/abs/math-ph/0601064

    2006 arXiv

  33. [43]

    J. R. Waldram and P. H. Wu. An alternative analysis of the nonlinear equations of the current-driven Josephson junction. Journal of Low Temperature Physics , 47:363–374,

  34. [250]

    Springer Science & Business Media, 2012

    2012

  35. [1982]

    URL https://doi.org/10.1007/BF00683738

    doi: 10.1007/BF00683738. URL https://doi.org/10.1007/BF00683738. 29

    doi

  36. [2011]

    URL https://doi.org/10.1103/PhysRevD

    doi: 10.1103/PhysRevD.84.127502. URL https://doi.org/10.1103/PhysRevD. 84.127502

    doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.