Pith. sign in

REVIEW 3 major objections 8 minor 1 cited by

The topology of non-resonant stellarator divertors

T0 review · 3 major / 8 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Stellarator exhaust can be driven by unpaired X-points whose manifolds reach far beyond the confined plasma.

desk verdict Solid topological explanation for the NRD Hamiltonian's unpaired X-points, but the QUASR-based 'general feature' claim outruns the evidence and needs softening. read the letter →

arxiv 2501.18293 v1 pith:ONCLMSHC submitted 2025-01-30 physics.plasm-ph

Robert Davies , Christopher B. Smiet , Alkesh Punjabi , Allen Boozer , Sophia A. Henneberg This is my paper · ORCID
classification physics.plasm-ph
keywords stellaratordivertornon-resonantunpairedX-pointwindingnumbermagnetictopologyfield-linemapquasi-symmetricfixed-pointindex
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

This paper claims that exhaust in the non-resonant stellarator divertor and in several realistic stellarator fields is governed by X-points that have no compensating O-points. Because an unpaired X-point has topological index -1 and nothing inside the mapped region cancels it, its stable and unstable manifolds are not closed into a finite island-chain separatrix and instead extend far from the confined region, guiding field lines outward. The same winding-number method that reveals this in the NRD Hamiltonian is applied to a large database of optimised quasi-symmetric stellarators, where three configurations are shown to divert through the same mechanism. If the claim is right, it gives plasma physicists a simple topological design rule for stellarator exhaust and widens the range of divertor geometries available to future reactors.

What carries the argument

The central tool is the winding number of a closed contour under the field-line map: for a loop $\eta$, $W(\eta)$ equals the sum of the topological indices of the fixed points inside, with X-points contributing $-1$ and O-points $+1$. The paper defines the maximal contractible $n$-mapping set $M_n$, the largest simply connected region in which every point can be mapped forward for $n$ field periods, and computes the winding number on its boundary contour. A value below $+1$ over such a contour proves that negative-index fixed points outnumber positive-index ones, i.e. that unpaired X-points are present. The Jacobian trace classification (elliptic, parabolic, hyperbolic) then identifies each fixed point's nature, and tracing dense blobs of initial conditions reveals the stable and unstable manifolds that carry the diverted field lines.

What would settle it

For any claimed unpaired X-point, trace the field-line map on a contour enclosing both the X-point and the region where a paired O-point would have to lie; if the winding number is not $-1$ but rather $0$ or $+1$, the X-point is paired and the mechanism fails for that configuration. For the configuration whose winding number is reported as $W=-2$, repeating the calculation at finer grid spacing with the full four-period map would settle whether the missed fixed point is an X-point (then $W=-3$) or an O-point (then $W=-1$, falsifying the unpaired claim).

Watch

Extended reading notes

Core claim

The central discovery is that unpaired X-points — fixed points of the field-line map with negative topological index that have no positive-index O-point partner in the same mapping domain — produce the diverting structure of the NRD Hamiltonian and of several quasi-symmetric stellarator configurations. In the NRD Hamiltonian, four $\iota=0$ X-points sit just beyond the last good magnetic surface, and their manifolds coincide with the surfaces where the poloidal velocity vanishes, guiding trajectories away in two collimated bundles per X-point. A search of the optimised stellarator database found a configuration whose top and bottom X-points resemble a tokamak double-null divertor without any plasma current, a configuration with four unpaired $\iota=1$ X-points whose winding number is recorded as $W=-2$ because the contour resolution misses one fixed point, and a configuration that combines an unpaired $\iota=0$ X-point with a four-period island chain whose O-points are themselves hyperbolic rather than elliptic. The paper concludes that unpaired X-points are a general feature of realistic stellarator fields, not an artifact of the toy Hamiltonian.

Load-bearing premise

The claim that the apparent X-points are genuinely unpaired rests on the assumption that the winding number computed on the resolution-limited boundary of the maximal contractible $n$-mapping set captures every fixed point inside it, so that no compensating O-point is missed by the grid or lies outside the contour.

Editorial extensions

If this is right

  • In the NRD Hamiltonian, the outgoing turnstile bundles are located by the zeros of the poloidal and radial velocities, $d\theta/d\xi = d\psi_t/d\xi = 0$, so diversion can be predicted from derivatives of the Hamiltonian rather than from island-chain resonances.
  • Stellarator divertors can be built around $\iota=0$ unpaired X-points that need no plasma current, giving a tokamak-like double-null exhaust geometry in a stellarator.
  • Unpaired X-points are not tied to a rational surface, so their strike locations may stay resilient when the rotational transform profile changes.
  • Realistic quasi-symmetric configurations can contain exotic edge topologies, including island chains whose O-points are hyperbolic, which may offer new divertor geometries.
  • The automated winding-number scan over $M_n$, which takes about 100 seconds per configuration, can screen large stellarator databases for promising divertor topologies.

Reading between the lines

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

  • If unpaired $\iota=0$ X-points survive the addition of plasma current and equilibrium evolution, they could be deliberately engineered in reactor designs, with strike points placed on outboard plasma-facing components.
  • The winding-number condition $W(M_n) < +1$ could be turned into a topological objective for stellarator optimization, scanning the roughly 300,000-configuration database to map which coil shapes produce unpaired X-points.
  • The same contour-index method could be applied to island-divertor configurations to quantify when an island chain transitions into a non-resonant divertor as its O-points bifurcate into hyper-hyperbolic points.
  • A direct test of the 'arbitrarily far' claim would be to follow the manifolds of the innermost NRD X-point for many field periods; if they are eventually blocked by a cantorus or fold back, the practical exhaust picture would change.
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

3 major / 8 minor

Summary. The paper applies winding-number (topological index) computations to the Poincaré map of magnetic field lines to analyze stellarator divertor topology. It first studies the non-resonant divertor (NRD) Hamiltonian, finding four ι=0 X-points that are not paired with O-points, and argues that their manifolds guide field lines away from the confined region over arbitrarily large distances. It then presents an automated search of the QUASR database of quasi-symmetric stellarator vacuum fields, reporting three configurations whose edge fields appear to contain unpaired X-points, including novel examples with ι=1 X-points and hyper-hyperbolic island chains. The paper concludes that unpaired X-points are a general feature of realistic stellarators and may offer a new divertor concept.

Significance. If the claims hold, the paper provides a simple topological mechanism for non-resonant diversion: X-points with index −1 that lack compensating +1 O-points have manifolds that do not close into finite island separatrix loops, so field lines can be expelled over long distances. This is a conceptually useful explanation for the NRD Hamiltonian and could guide stellarator edge design. The QUASR analysis is an original application of index-theoretic methods to a large database, and the identification of ι=1 unpaired X-points and hyperbolic-only island chains extends the known topological repertoire of stellarator edges. The authors honestly discuss computational limitations (e.g., resolution-dependent winding numbers) and the small fraction of configurations analyzed.

major comments (3)
  1. [Section 4 and 5] The conclusion that the QUASR configurations contain genuinely unpaired X-points is only established inside the resolution-limited maximal contractible n-mapping set M_n, not for the full magnetic field. In §4.2 the authors state that spatial resolution caused W=-2 instead of W=-3 for configuration 74609, and in §5 they list 'What can be said about fixed points outside of Mn?' as an open question. For configuration 104183, the measured W=-1 over M_n plus the two manually located X-points only proves a net deficit of one positive-index fixed point inside M_n; it does not rule out a compensating O-point outside M_n or missed by the 1 cm grid and the adaptive contour. If such an O-point exists, the X-points would belong to an island chain and the proposed 'unpaired' divertor mechanism would not be established for that configuration. Therefore the statement in §5 that 'unpaired X-points are a general feature of realistic stellarators' goes beyond what the evidence supports. Please either soften the claim to apply to fixed points within M_n, or provide additional evidence (e.g., a systematic search for O-points in a domain enclosing M_n, or a study of field-line behavior outside M_n).
  2. [Section 3] The proof that the four NRD Hamiltonian X-points are unpaired is not fully explicit. The winding numbers W=(+1,-2,-3,-3) on the ψt/ψ̄g = (1,10,100,1000) contours fix only the difference between the number of positive- and negative-index fixed points in each annular region. The manual search finds four X-points, each with W=-1 on a small enclosing loop, and the text argues that 'the sign of radial velocity dψt/dξ for the manifolds cannot vary when ψt is sufficiently large' (Section 3, paragraph after Figure 6), which would exclude the O-points required to close an island-chain separatrix. However, this statement is not demonstrated; an ι=0 elliptic fixed point is not obviously forbidden by the argument as written. Since the paper's first central claim is that NRD diversion is caused by unpaired X-points, the authors should provide a more rigorous argument or a numerical scan that rules out compensating O-points between the listed contours.
  3. [Section 4, automated method] The maximal contractible n-mapping set M_n is defined informally, and the automated contour finding on a 1 cm grid may not capture the true domain boundary, especially where the boundary lies close to X-point manifolds, as noted in §4.1. This makes the winding-number computation vulnerable to missing fixed points, a failure explicitly acknowledged in §4.2 for configuration 74609. A more robust definition of M_n or a convergence study (e.g., varying dR=dZ and the upsampling tolerance) would strengthen the reliability of the reported winding numbers and the conclusions drawn from them.
minor comments (8)
  1. [Section 2.2] The typesetting '1 1/2-dimensional Hamiltonian' appears malformed in the text; please correct it to '1 1/2-dimensional Hamiltonian'.
  2. [Appendix A] Configuration '74608' appears in the table and its surrounding text, while the main text refers to the same configuration as '74609'; please make the numbering consistent.
  3. [Reference [34]] Reference [34] is a bare URL for the pyoculus package; please provide a full bibliographic entry or a versioned citation.
  4. [Section 4.1] The phrase 'The contour stretches over the stable manifolds' is unclear; the intended meaning is likely that the contour straddles or crosses the stable manifolds.
  5. [Figure 4 caption] The caption says 'Maximum displacement of the forwards and backwards maps' but the figure appears to show the magnitude of the displacement; please adjust the wording.
  6. [Section 3] The sentence 'By following tracing the X-points' should be 'By tracing the X-points'.
  7. [Section 4] The phrase 'with using nmap = 4' should be 'using nmap = 4'.
  8. [Section 2.1] The stray footnote '∥ note: not sub groups' appears to be an editing artifact and should be removed or integrated into the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's calculations and database scans are self-contained, and the flagged limitations are correctness risks, not circular reductions.

full rationale

The paper's derivation chain is not circular in any load-bearing step. In the NRD Hamiltonian study, the winding numbers W=(+1, -2, -3, -3) are computed on fixed contours of a fixed Hamiltonian, and the X-points are located by direct numerical search and verified by local winding-number checks; these are computations, not parameters fitted to the conclusion. The statement that the X-points are 'unpaired' follows from the index balance within the computed contours and from the absence of compensating O-points in the searched region, which is a mathematical/topological conclusion rather than a definitional renaming. The choice to use the same Hamiltonian parameters as Punjabi and Boozer [8,9] is a parameter selection from prior work, and it does not encode or assume the paper's new claim that unpaired X-points cause the diversion. In the QUASR study, the winding number is computed from Biot-Savart fields of configurations taken from an external database; nothing is fitted to force W<1. The paper explicitly acknowledges the resolution dependence of the contour (Section 4.2) and lists as an open question 'what can be said about fixed points outside of M_n?' (Section 5), which are genuine limitations on the generality of the claim, but they do not make the argument circular. There is no self-citation chain carrying the central result, no imported uniqueness theorem, and no ansatz smuggled in via citation. Concerns about sample size, spatial resolution, or fixed points outside the maximal mapping set belong to correctness risk, not to circularity. The appropriate finding is no significant circularity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claims rest on hand-chosen parameters in the toy Hamiltonians and on numerical resolution choices in the QUASR scan. The topology theorems are standard; the main unverified input is that the finite mapping domain M_n and the grid resolution capture all relevant fixed points.

free parameters (6)
  • NRD Hamiltonian rotational transform iota_0 = 0.15
    Chosen from Punjabi and Boozer [8,9]; the fixed-point structure of the NRD toy model depends on this value.
  • NRD Hamiltonian shaping amplitudes epsilon_0, epsilon_t = 0.5
    Same prior-work values; these control the theta-dependent terms that create the poloidal velocity reversal responsible for unpaired X-points.
  • NRD Hamiltonian epsilon_x = -0.31
    Same prior-work values; sign and magnitude set the location of the X-points.
  • Axisymmetric Hamiltonian amplitude A = 0.05
    Hand-chosen in Section 3 to mimic the NRD velocity reversal; the existence of the two X-points depends on this choice.
  • QUASR field grid resolution dR=dZ = 1 cm
    Chosen in Section 4; the paper states W can be -2 instead of -3 because resolution misses a fixed point.
  • Contour upsampling tolerance = 2*pi/100
    Chosen in Section 4 as the maximum allowed change in displacement direction between successive contour points; controls whether winding numbers are correct.
assumptions (4)
  • domain assumption Divergence-free magnetic field implies the Jacobian of the Poincare map has unit determinant, so fixed points are classified by the trace of the SL(2,R) matrix.
    Used in Section 2.1 to connect Greene's residue and topological index to magnetic fixed points; standard plasma physics.
  • standard math Winding number over a closed curve equals the sum of topological indices of fixed points inside the curve.
    Used throughout Sections 3 and 4 as the main counting tool.
  • domain assumption In the relevant edge region the magnetic field is equivalent to a 1.5-dimensional Hamiltonian system, so field line dynamics is area-preserving and Poincare sections are well defined.
    Invoked in Section 2.2 with limitations acknowledged; the domain of the Poincare map in real stellarators is finite and not explicitly characterized.
  • domain assumption The vacuum magnetic fields in QUASR configurations represent buildable stellarator edge fields.
    Assumed in Section 4 when transferring the Hamiltonian picture to optimized configurations; no plasma response is included.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The topology of non-resonant stellarator divertors." pith.science (2026). https://pith.science/paper/ONCLMSHC

@misc{pith2026250118293,
  author       = {Pith},
  title        = {Pith review of: The topology of non-resonant stellarator divertors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ONCLMSHC}},
  note         = {Machine review of arXiv:2501.18293}
}
read the original abstract

We apply topological methods to better understand how the magnetic field in the stellarator edge can be diverted away from the confined region. Our primary method is calculating the winding numbers of closed contours, which gives information on the number and nature of fixed points within a bounded region. We first apply this to the non-resonant divertor (NRD) Hamiltonian system, and present a simple explanation for the system's diversion: trajectories are guided away from the confined region by X-points which are "unpaired" i.e. do not have corresponding O-points and therefore do not resemble an island chain. We show how similar phenomena can occur in a similar, axisymmetric Hamiltonian system. Secondly, we find examples of neoclassically optimised stellarators in the QUASR database which divert the magnetic field via unpaired X-points. We present and discuss three examples, each containing novel phenomena which might be desirable for stellarator divertors. These findings broaden the horizons of how magnetic fields can be diverted in realistic stellarators, and may be attractive for future experiments and stellarator reactor design.

Figures

Figures reproduced from arXiv: 2501.18293 by the authors.

Figure 1
Figure 1. Schematic illustration of the most popular methods of diversion in magnetically confined fusion plasmas. In each case, nested flux surfaces of the confined plasma are shown in red and solid components are shaded grey, with the plasma￾facing components (PFCs) outlined in dashed black. Upper left: A tokamak double null configuration. External coils (shown as orange squares) generate X-points (cyan crosses). This forms… view at source ↗
Figure 2
Figure 2. Illustration of the definition of the topological index of a singular fixed point in two dimensions. The topmost row corresponds to an X-point, the middle row to an O-point and the bottommost row to a hyper-hyperbolic fixed point. The left subplots show the fixed point x0 (blue dot), surrounded by a closed curve parametrized by η, that is traversed in clockwise direction (from blue to green to red). The displacement… view at source ↗
Figure 3
Figure 3. Poincar´e section for the non-resonant divertor Hamiltonian and displacement vector magnitude for the forwards map, |df | at toroidal location ξ = 0. Left: a zoom-in of the confined region. Right: arrows indicate direction of df at locations on closed contours (black lines), over which the winding number is calculated [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Maximum displacement of the forwards and backwards maps and closed contours over which winding numbers are calculated for the non-resonant divertor Hamiltonian. The left plot shows all four X-points of the system. The right plot shows a zoom-in of the innermost X-point…
Figure 5
Figure 5. Figure 5: Manifolds of the four X-points of the NRD Hamiltonian system shown in grey, blue, orange, green and purple. Left: manifolds of the innermost X-point when tracing forwards in toroidal position ξ (blue) and backwards in ξ (orange). These are found by tracing a block of 3…
Figure 6
Figure 6. Figure 6: Derivative information for the NRD Hamiltonian. Upper row: poloidal velocity dθ dξ at ξ = 0 (left) and radial velocity dψt dξ at ξ = 0 (right), also showing arrows indicating the direction of poloidal and radial travel as one advances in ξ, and manifolds of the innermo…
Figure 7
Figure 7. Figure 7: Properties of the axisymmetric Hamiltonian (8) system with similar properties to the NRD Hamiltonian system. Poloidal velocity dθ dξ is shown in filled colours and the contours over which the winding number is calculated are shown by blue, orange and green lines. A Poi…
Figure 8
Figure 8. Figure 8: X-points and manifolds for (quasi-axisymmetric) QUASR configuration 104183 at three toroidal locations (ϕ = 0, π/2, π). Poincar´e section is shown in black and the coil locations shown as colored squares (each square represents a point on the coil which is within 2.5 ◦…
Figure 9
Figure 9. Figure 9: Poincare (black points) and contour of the maximal mapping set (red line) for (quasi-axisymmetric) QUASR configuration 74609 at three toroidal locations (ϕ = 0, π/6, π/4). This configuration shows four unpaired X-points with rotational transform ι = 1, displayed as gre…
Figure 10
Figure 10. Figure 10: Poincare plot (black) and fixed point information for the (quasi￾helically symmetric) QUASR configuration 1258083 at four toroidal locations. This configuration contains a 4/4 island chain consisting of regular X-points (green crosses) and positive-index hyperbolic po…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Surfaces with Klein bottle topology occur in fusion reactor fields

    physics.plasm-ph 2025-06 conditional novelty 6.0 of 10

    Magnetic surfaces in fusion devices are not always tori: a period-doubling bifurcation at half-integer rotational transform creates separatrix surfaces with the topology of an immersed (lemniscate) Klein bottle, demon...

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Energy Exhaust from Tokamaks: Problems and Solutions

    Soboleva T. Energy Exhaust from Tokamaks: Problems and Solutions. Astrophysics and space science. 1997;256:247-62

    work page 1997

  2. [2]

    A tutorial on some basic aspects of divertor physics

    Stangeby P. A tutorial on some basic aspects of divertor physics. Plasma Physics and Controlled Fusion. 2000;42(12B):B271

    work page 2000

  3. [3]

    Stellarator design

    Boozer AH. Stellarator design. Journal of Plasma Physics. 2015;81(6):515810606

    work page 2015

  4. [4]

    Physical aspects and design of the Wendelstein 7-X divertor

    Renner H, Sharma D, Kisslinger J, Boscary J, Grote H, Schneider R. Physical aspects and design of the Wendelstein 7-X divertor. Fusion science and technology. 2004;46(2):318-26

    work page 2004

  5. [5]

    Helical divertor in the large helical device

    Ohyabu N, Noda N, Ji H, Akao H, Akaishi K, Ono T, et al. Helical divertor in the large helical device. Journal of nuclear materials. 1992;196:276-80

    work page 1992

  6. [6]

    Overview and future plan of helical divertor study in the Large Helical Device

    Masuzaki S, Morisaki T, Shoji M, Kubota Y, Watanabe T, Kobayashi M, et al. Overview and future plan of helical divertor study in the Large Helical Device. Fusion science and technology. 2006;50(3):361-71

    work page 2006

  7. [7]

    Simulation of stellarator divertors

    Boozer AH, Punjabi A. Simulation of stellarator divertors. Physics of Plasmas. 2018;25(9)

    work page 2018

  8. [8]

    Simulation of non-resonant stellarator divertor

    Punjabi A, Boozer AH. Simulation of non-resonant stellarator divertor. Physics of Plasmas. 2020;27(1)

    work page 2020

Show all 34 references
  1. [9]

    Magnetic turnstiles in nonresonant stellarator divertor

    Punjabi A, Boozer AH. Magnetic turnstiles in nonresonant stellarator divertor. Physics of Plasmas. 2022;29(1)

    2022

  2. [10]

    Magnetic field properties in non-axisymmetric divertors

    Boozer AH. Magnetic field properties in non-axisymmetric divertors. Physics of Plasmas. 2023;30(11)

    2023

  3. [11]

    HSX as an example of a resilient non-resonant divertor

    Bader A, Boozer A, Hegna C, Lazerson S, Schmitt J. HSX as an example of a resilient non-resonant divertor. Physics of Plasmas. 2017;24(3)

    2017

  4. [12]

    Resilient Stellarator Divertor Characteristics in the Helically Symmetric eXperiment

    Garcia K, Bader A, Boeyaert D, Boozer A, Frerichs H, Gerard M, et al. Resilient Stellarator Divertor Characteristics in the Helically Symmetric eXperiment. arXiv preprint arXiv:241110611. 2024

    2024

  5. [13]

    Exploration of non-resonant divertor features on the Compact Toroidal Hybrid

    Garcia K, Bader A, Frerichs H, Hartwell G, Schmitt J, Allen N, et al. Exploration of non-resonant divertor features on the Compact Toroidal Hybrid. Nuclear Fusion. 2023;63(12):126043

    2023

  6. [14]

    Studies of Non-Resonant Divertor Strike Line Resiliency in the Compact Toroidal Hybrid

    Allen N, Garcia K, Bader A, Schmitt J, Maurer D, Ennis D, et al. Studies of Non-Resonant Divertor Strike Line Resiliency in the Compact Toroidal Hybrid. Bulletin of the American Physical Society. 2024

    2024

  7. [15]

    Magnetic field line diversion in Helias stellarator configurations: perspectives for divertor operation

    Strumberger E. Magnetic field line diversion in Helias stellarator configurations: perspectives for divertor operation. Nuclear fusion. 1992;32(5):737

    1992

  8. [16]

    Topology of divertor field line mapping in HELIAS configurations

    Strumberger E. Topology of divertor field line mapping in HELIAS configurations. Contributions to Plasma Physics. 1992;32(3-4):212-8

    1992

  9. [17]

    Development of divertor concept for optimized stellarators

    N¨ uhrenberg J, Strumberger E. Development of divertor concept for optimized stellarators. Contributions to Plasma Physics. 1992;32(3-4):204-11

    1992

  10. [18]

    Flexible Stellarator Physics Facility

    Parra F, Baek SG, Churchill M, Demers D, Dudson B, Ferraro N, et al. Flexible Stellarator Physics Facility. arXiv preprint arXiv:240704039. 2024

    2024

  11. [19]

    The Compelling The topology of non-resonant stellarator divertors 23 Need for a Mid-Scale Stellarator Facility

    Pablant NA, Anderson D, Anderson J, Anderson S, Andruczyk D, Bader A, et al. The Compelling The topology of non-resonant stellarator divertors 23 Need for a Mid-Scale Stellarator Facility. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); 2024

    2024

  12. [20]

    Divertor design plans for Eos stellarator

    Tang L, Kumar S, Swanson C, Dudt D, Kruger T, Martin M, et al. Divertor design plans for Eos stellarator. Bulletin of the American Physical Society. 2024

    2024

  13. [21]

    Œuvres de Henri Poincare Tome 1

    Poincar´ e H. Œuvres de Henri Poincare Tome 1. Paris: Gauthier-Villars; 1928

    1928

  14. [22]

    Two-Dimensional Measure-Preserving Mappings

    Greene JM. Two-Dimensional Measure-Preserving Mappings. Journal of Mathematical Physics. 1968;9(5):760-8

    1968

  15. [23]

    Ordinary differential equations

    Arnold VI. Ordinary differential equations. Springer Science & Business Media; 1992

    1992

  16. [24]

    Plasma confinement in closed magnetic systems

    Solov’ev L, Shafranov V. Plasma confinement in closed magnetic systems. In: Reviews of Plasma Physics: Volume 5. Springer; 1970. p. 1-247

    1970

  17. [25]

    Bifurcations of the magnetic axis and the alternating-hyperbolic sawtooth

    Smiet C, Kramer G, Hudson S. Bifurcations of the magnetic axis and the alternating-hyperbolic sawtooth. Nuclear Fusion. 2020;60(8):084005

    2020

  18. [26]

    Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

    Wei W, Liang Y. Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion. Plasma Science and Technology. 2023;25(9):095105

    2023

  19. [27]

    Quantifying chaos with Lyapunov exponents

    Wolf A, et al. Quantifying chaos with Lyapunov exponents. Chaos. 1986;16:285-317

    1986

  20. [28]

    The influence of errors on plasma-confining magnetic fields

    Kerst D. The influence of errors on plasma-confining magnetic fields. Journal of Nuclear Energy Part C, Plasma Physics, Accelerators, Thermonuclear Research. 1962;4(4):253

    1962

  21. [29]

    Global realisation of magnetic fields as 1 2 D Hamiltonian systems

    Duignan N, Perrella D, Pfefferl´ e D. Global realisation of magnetic fields as 1 2 D Hamiltonian systems. arXiv preprint arXiv:240705692. 2024

    2024

  22. [30]

    Symplectic maps, variational principles, and transport

    Meiss J. Symplectic maps, variational principles, and transport. Reviews of Modern Physics. 1992;64(3):795

    1992

  23. [31]

    Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices

    Giuliani A. Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices. Journal of Plasma Physics. 2024;90(3):905900303

    2024

  24. [32]

    A comprehensive exploration of quasisymmetric stellarators and their coil sets

    Giuliani A, Rodr ´ ıguez E, Spivak M. A comprehensive exploration of quasisymmetric stellarators and their coil sets. arXiv preprint arXiv:240904826. 2024

    2024

  25. [33]

    Theory of plasma confinement in non-axisymmetric magnetic fields

    Helander P. Theory of plasma confinement in non-axisymmetric magnetic fields. Reports on Progress in Physics. 2014;77(8):087001

    2014

  26. [34]

    different

    Https://github.com/zhisong/pyoculus. Appendix A. Greene’s Residue and Lyapunov exponents of fixed points We use the pyoculus package to find the Jacobian M of each fixed point of the QUASR examples shown in section 4, and from this the Greene’s Residue and the Lyapunov exponen...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.