Pith. sign in

REVIEW 3 major objections 5 minor 49 references

Robust computation of higher-dimensional invariant tori from individual trajectories

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A single short trajectory can determine a higher-dimensional invariant torus without initial guesses or continuation.

desk verdict A practically valuable pipeline for computing higher-dimensional invariant tori from single short trajectories, with an honest but real theoretical gap in the frequency-convergence step. read the letter →

arxiv 2505.08715 v1 pith:TEJGKMQU submitted 2025-05-13 math.DS physics.comp-ph

Maximilian Ruth , Jackson Kulik , Joshua Burby This is my paper · ORCID
classification math.DSphysics.comp-ph MSC 37J4037M1065P10
keywords invarianttoriBirkhoffreducedrankextrapolationrotationvectorBayesianmaximumaposterioriestimationKorkine-Zolatarevlatticereductioncoupledstandardmaprestrictedthree-bodyproblemquasiperiodicmotion
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 presents a frequency-based method for computing invariant tori of dimension greater than one from a single short trajectory of a dynamical system. No continuation, no initial guess, and no preferred coordinate system are required, which is what makes the method practical when little is known about the system in advance. The pipeline first extracts the trajectory's dominant frequencies with Birkhoff reduced rank extrapolation, then infers a valid rotation vector by Bayesian maximum a posteriori estimation, then selects short, nearly orthogonal loops around the torus by Korkine-Zolatarev lattice reduction, and finally fits Fourier coefficients by least squares. On a coupled standard map the authors report successfully parameterizing 95 percent of the initialized tori, and on three-dimensional Earth-Moon restricted three-body tori they report KAM residuals near $10^{-5}$ to $10^{-6}$ from trajectories of a few thousand iterates. If these results hold, a short trajectory is enough to recover both the geometry and the rotation vector of a higher-dimensional invariant torus.

What carries the argument

The load-bearing object is the Birkhoff reduced rank extrapolation (Birkhoff RRE) filter: a palindromic set of coefficients $c_j$ that extrapolates the trajectory, whose residual $R_{\mathrm{RRE}}$ classifies the trajectory as integrable or chaotic, and whose filter roots $\psi_j$ produce high-accuracy frequency estimates $\Omega_j\approx\omega\cdot k_j$. Around this sit two further mechanisms: a Bayesian maximum a posteriori estimator that assigns wavenumbers to the measured frequencies and recovers a valid rotation vector, and a Korkine-Zolatarev lattice reduction on the averaged metric $G=\sum_k \|h_k\|^2 kk^\top$ that selects the homology generators giving the most compact Fourier representation. The final least-squares coefficient fit is what converts the trajectory and rotation vector into an explicit torus parameterization.

What would settle it

Run the frequency-extraction step on a trajectory of a known smooth two-dimensional torus with a prescribed rotation vector, increasing the filter size $J$, and check whether the computed frequencies approach the exact values $\omega\cdot k$; the central claim fails if the top frequencies plateau away from machine precision or never contain a pair of wavenumbers with determinant $\pm1$.

Watch

Extended reading notes

Core claim

The central claim is that the rotation vector and Fourier parameterization of a $d$-dimensional invariant torus can be recovered from a single finite trajectory, without continuation or initial guesses, by combining temporal frequency estimation with a Bayesian labeling of wavenumbers. Specifically, Birkhoff RRE returns high-precision frequencies $\Omega_j$ and magnitudes $H_j$; the MAP step identifies which lattice wavenumbers $k_j$ produced those frequencies and thereby finds a valid rotation vector $\omega$; the KZ reduction replaces $\omega$ by an equivalent rotation vector whose loops on the torus are short and nearly orthogonal; and a least-squares fit of the trajectory against modes $e^{2\pi i k\cdot\theta}$ yields the torus Fourier coefficients. The authors demonstrate the pipeline on random initial conditions of a weakly coupled standard map, on island chains, and on three-dimensional tori in the Earth-Moon restricted three-body problem, reporting KAM residuals as low as $3.63\times10^{-6}$ with trajectory lengths on the order of $10^3$--$10^4$.

Load-bearing premise

The whole method rests on an unproved assumption: the frequencies that the extrapolation step returns really converge to the true frequencies of the torus for tori of dimension two or higher, and the strongest recovered frequencies contain enough independent directions to form a valid rotation vector.

Editorial extensions

If this is right

  • For the weakly coupled standard map with 1000 random initial conditions, 851 trajectories are classified as integrable by length 8001, and 95 percent of the initialized tori are successfully parameterized by the reported a posteriori measure.
  • The method computes three-dimensional tori of the Earth-Moon restricted three-body problem from trajectories of length 3335 to 8335, with KAM residuals of $3.63\times10^{-6}$, $5.70\times10^{-6}$, and $1.63\times10^{-5}$.
  • Birkhoff reduced rank extrapolation classifies the standard-map trajectories as integrable with roughly an order of magnitude shorter trajectories than weighted Birkhoff averaging requires.
  • The main failure modes are nearly resonant rotation vectors and strongly filamentary tori; the authors show these are detectable through the KAM residual and resonance-order diagnostics.
  • Island-chain trajectories are handled by treating the chain period as an added rational frequency, so a single pipeline covers tori, islands, and chaotic classification.

Reading between the lines

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

  • If a convergence proof for the frequency roots were supplied, the Bayesian uncertainty $\sigma_\omega$ could be set from theory, which would turn the pipeline into a method with end-to-end error estimates for the recovered torus.
  • Because only injectivity of the observable $h(S)$ is required, the same steps should work with delay embeddings or other observables, extending the method to systems whose natural state space is not Euclidean.
  • Since trajectory generation is often the dominant cost, the short-trajectory property suggests this approach could map out many tori in parameter scans where continuation-based methods struggle.
  • The KZ step is a lattice shortest-vector problem on the torus's average metric; the same reduction could be used to identify slow and fast directions in higher-dimensional systems for model reduction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a fully data-driven method for computing invariant tori of dimension d > 1 from a single, relatively short trajectory, without continuation or an initial guess. The pipeline has four stages: (i) Birkhoff reduced rank extrapolation (RRE) to extract many temporal frequencies and their amplitudes from the trajectory; (ii) a Bayesian maximum a posteriori (MAP) problem, Eq. (25), that selects a valid rotation vector from among the measured frequencies; (iii) a Korkine-Zolotarev lattice reduction that chooses a homology basis adapted to the shape of the torus; and (iv) an adaptive least-squares Fourier parameterization, Eq. (7) and Sec. 2.2. The method is tested on a coupled standard map (1000 random initial conditions, with 851 classified as integrable), on island chains, and on three three-dimensional tori in the cislunar elliptic restricted three-body problem. The authors report KAM residuals RKAM between 3.63e-6 and 1.63e-5 for the 3D examples and a 95% success rate for the standard-map ensemble, and they discuss two failure modes: nearly resonant rotation vectors and highly anisotropic (filamentary) tori.

Significance. If the central claims hold, this would be a practically valuable tool for astrodynamics and plasma physics, since it removes the need for continuation or a good initial guess and works from a single trajectory, including for tori of dimension three. The paper's strengths are its honest validation methodology: the KAM residual in Eq. (29) is evaluated on a uniform grid rather than on the fitting trajectory, and the validation error Rh in Eq. (9) uses held-out trajectory points. The authors also explicitly acknowledge the main theoretical gap, namely that convergence of Birkhoff RRE frequencies to the true Fourier frequencies omega·k is unproven for d > 1. The numerical experiments are extensive and the code is publicly available in SymplecticMapTools.jl. The main risk is that the entire method depends on that unproven convergence, and the reported residuals, while encouraging, are integrated checks that do not isolate whether the frequency estimates converge.

major comments (3)
  1. [Sec. 3.1 and Sec. 5] The load-bearing assumption of the method is that the Birkhoff RRE frequencies Omega_j converge to the true temporal frequencies omega·k_j for d > 1. The paper explicitly states in Sec. 3.1 that 'There is currently no theory for the convergence of the frequencies Omega_j to true values omega·k_j as J→∞' and repeats in the Conclusion that this is an open question. The MAP step in Sec. 3.2, Eq. (25), takes the measured Omega_j as input and restricts candidate rotation vectors to subsets of Omega, so if this convergence fails, the method collapses even on a perfectly smooth torus. The empirical evidence in Sec. 4 is integrated: the reported RKAM residuals are computed with the same estimated frequencies used to construct the parameterization, and a wrong but nearby frequency could still yield small residuals on a finite trajectory sampled on a 25×25 grid. I therefore ask for a targeted numerical test that isolates frequency convergence, for example comparing Birkhoff RRE frequencies against frequencies obtained from a high-accuracy parameterization-method torus for one or two d = 2 and d = 3 examples, with controlled J, T, and analytic or high-precision ground truth. Without such a test, the central methodological claim is not fully supported.
  2. [Sec. 3.2, Eq. (20) and Eq. (25)] The prior hyperparameters C and r in sigma(k) = C e^{-r||k||} are fit by least squares to the same trajectory's sorted Fourier magnitudes H_j, Eq. (20), and then used as the prior in the MAP objective (25). This is an empirical-Bayes procedure that uses the data twice: the same trajectory both determines the prior and is the evidence in the posterior. This double use can overstate posterior confidence and may bias the wavenumber assignment toward the fitted smoothness model. The paper does not report sensitivity of the final rotation vector or RKAM to changes in C and r, nor does it compare with a fixed, cross-validated choice. I request a sensitivity study that perturbs the fitted C and r (or holds out a fraction of the H_j when fitting them) and reports how often the rotation vector and the final RKAM change. The order-statistic model in Eq. (22) also assumes no measurement error in the H_j, which is an additional idealization that should be stated and tested.
  3. [Sec. 4.1, Fig. 7 caption and Introduction] The paper claims in the Introduction that 'we successfully parameterize 95% of the initialized tori for the weakly coupled standard map,' but the success criterion is not defined: no threshold on RKAM (or Rh) is given, and it is unclear whether the denominator is 1000 initialized trajectories, the 851 classified as integrable, or the 853 appearing in the Fig. 7 caption. The count inconsistency (851 in the text versus 853 in the caption) also needs correction. Since this 95% claim is a headline result, the definition of success and the exact counts must be made precise and reproducible.
minor comments (5)
  1. [Abstract and Sec. 3.3] The name 'Korkine-Zolatarev' is a misspelling; the standard spelling is Korkine-Zolotarev.
  2. [Sec. 2.1, Eq. (9)] The definition of R_h^2 writes R2_h = 1/R2_h0 sum ||h(F^t(x)) - hat h(theta + omega t)||^2, but the second factor R2_h0 is defined immediately after; please make the dependence on the theta initialization explicit, since the expression as written is ambiguous.
  3. [Sec. 3.4] In the sentence 'Typical values of these constants are pmax = 10 and epsilon_ada = 10^-8,' the symbol epsilon_ada appears to be a typo for epsilon_isl, which is the island tolerance defined earlier in the same paragraph.
  4. [Sec. 2.3] There is a duplicated word in 'a resolution of (K1, K2) = (24, 18) would be needed to capture all of the same Fourier modes for for the sheared torus.'
  5. [Sec. 4.1, Fig. 6] The statement that 'the weighted Birkhoff classification rate approximately matches the Birkhoff RRE classification rate in the right panel' is made without a quantitative comparison; reporting the actual classification counts for the weighted Birkhoff average at the same trajectory lengths would make the comparison in Fig. 6 more informative.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional circularity; one mild empirical-Bayes double-use in the MAP prior, while the rotation-vector search and KAM validation are independent.

  1. other [Section 3.2, Eqs. (20)-(25)]
    "To find these parameters, consider the idealized case that H consist of a sorted list of the exact expected norms H2 k = E[∥hk∥2] = Dσ(k)2. ... we heuristically fit a line to the log-norm of the coefficients H ... (20) ... αj(Hj,π(j)) = P(H2 π(j)=H2 j)/P(H2 π(j)≤H2 J0). (25)"

    The same sorted magnitudes H_j are used twice: first to calibrate the smoothness-prior hyperparameters C and r in Eq. (20), and then as the 'data' in the likelihood ratio α_j in Eq. (25). Thus the Bayesian prior is not independent of the observations it scores; it is an empirical-Bayes fit to the very H_j used in the MAP objective. This is a mild in-sample calibration rather than a definitional equivalence: the frequency likelihood P(Ω_j|ω,π(j)) with very small σω dominates the label assignment, and the det±1 and KZ checks enforce that the selected pair is a genuine integer homology change. The final KAM residual is also an independent uniform-grid check, so this double-use does not force the central claim.

full rationale

The central derivation is self-contained. Birkhoff RRE solves the constrained least-squares problem (17)-(18) for a filter, and the frequencies Ω_j are roots of the associated linear difference equation, not a fitted parameter renamed as output. The MAP step (25) combines these measured frequencies with a smoothness prior; the only in-sample element is Eq. (20), where C and r are fit to the same H_j that Eq. (25) later scores via α_j. This is an empirical-Bayes double-use, but it does not determine the rotation vector by itself. The frequency likelihood with σω of order 10^-10 dominates the label assignment, and the det±1 condition plus KZ reduction ensure that the selected pair is a genuine integer change of homology. The final parameterization is validated with the held-out trajectory error Rh (9) and the uniform-grid KAM residual RKAM (29), both independent of the least-squares fit. The paper's own admission that convergence of the Birkhoff RRE roots to ω·k for d>1 is open (Sec. 3.1 and Sec. 5) is a rigor gap, not circularity, because the method does not assume that theorem as an input. Accordingly, the only mild circularity signal is the fitted smoothness prior, worth a 2 rather than a higher score.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The method rests on standard ergodic theory and Diophantine assumptions, plus problem-specific modeling choices: Bayesian independence, uniform priors, order statistics, and a smoothness prior whose hyperparameters are fit to the same data. No new physical entities are introduced. The main unproven premise is the convergence of Birkhoff RRE frequencies for d > 1, stated as an open question in Section 3.1 and the Conclusion.

free parameters (6)
  • Prior decay rate r and scale C in sigma(k) = C e^{-r||k||} = Fit by linear regression to the trajectory's sorted Fourier magnitudes (Eq. 20)
    These hyperparameters define the smoothness prior used in the MAP rotation-vector estimation; they are fit to the same data from which the torus is inferred, so they are in-sample fitted values.
  • MAP frequency uncertainty sigma_omega = Chosen as 1e-10 in Sec. 4.1; range 1e-7 to 1e-11 suggested in Sec. 3.2
    The circular-normal likelihood width determines which wavenumbers can explain each measured frequency; the claimed rotation-vector validity depends on this hand-set number.
  • Number of frequencies J0 used in MAP = 30 for standard map (Sec. 4.1); 20, 18, 8 for ER3BP examples
    The algorithm assumes a valid rotation vector appears among the top J0 frequencies; for anisotropic tori this assumption fails, as acknowledged for the failure in Fig. 8(b).
  • Wavenumber search bound P = Scales as gamma (J0)^(1/d) with gamma about 10
    The MAP search is restricted to an infinity-norm grid; if the true wavenumbers lie outside this grid, the rotation vector can be misidentified.
  • Birkhoff RRE filter length J and trajectory length T = Chosen per example; default (J, T) = (2000, 4000) for standard map; (2500, 3334) for L4 ER3BP
    The method's accuracy and integrability classification depend on these; the paper adjusts them to reach an RRE residual below 5e-14.
  • Maximum Fourier modes Kmax = 2000 (Sec. 4.1)
    Cap on the adaptive parameterization; part of the algorithm's settings.
assumptions (7)
  • domain assumption Diophantine condition on the rotation vector omega
    Section 2 assumes |omega dot k|_T >= c / ||k||^nu, which guarantees ergodicity and frequency separation; near-resonant tori are a known failure mode.
  • domain assumption Smoothness, analytic or C^M, of the torus and observable with rapidly decaying Fourier coefficients
    Used to justify weighted Birkhoff convergence rates and the Gaussian/exponential prior on Fourier coefficients in Eq. (20).
  • domain assumption Injectivity of the observable embedding h(S)
    Stated in Section 2: 'As long as h(S) is injective, the map h can be pulled back' to recover the torus on the state space.
  • ad hoc to paper Uniform prior on the rotation vector and independence assumptions in the Bayesian model
    Section 3.2 sets P(omega) = 1 and assumes the frequencies Omega_j are independent of the coefficients H and of each other; these are modeling choices, not derived facts.
  • standard math Birkhoff ergodic theorem and convergence of weighted Birkhoff averages
    Used in Eqs. (4) and (5) to justify the ergodic projection formula for Fourier coefficients; standard result.
  • ad hoc to paper Existence of a valid rotation vector among the observed frequency set Omega
    The MAP optimization restricts omega to entries of Omega (Section 3.2); if no such pair or triple exists, the method fails, as seen for highly anisotropic tori.
  • ad hoc to paper Order-statistic model with no measurement error on the coefficient magnitudes
    Section 3.2 states 'we assume that extrapolation step returns an order statistic pi on the magnitudes ... with no measurement error', used to derive Eq. (22).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Robust computation of higher-dimensional invariant tori from individual trajectories." pith.science (2026). https://pith.science/paper/TEJGKMQU

@misc{pith2026250508715,
  author       = {Pith},
  title        = {Pith review of: Robust computation of higher-dimensional invariant tori from individual trajectories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TEJGKMQU}},
  note         = {Machine review of arXiv:2505.08715}
}
read the original abstract

We present a method for computing invariant tori of dimension greater than one. The method uses a single short trajectory of a dynamical system without any continuation or initial guesses. No preferred coordinate system is required, meaning the method is practical for physical systems where the user does not have much \textit{a priori} knowledge. Three main tools are used to obtain the rotation vector of the invariant torus: the reduced rank extrapolation method, Bayesian maximum a posteriori estimation, and a Korkine-Zolatarev lattice basis reduction. The parameterization of the torus is found via a least-squares approach. The robustness of the algorithm is demonstrated by accurately computing many two-dimensional invariant tori of a standard map example. Examples of islands and three-dimensional invariant tori are shown as well.

Figures

Figures reproduced from arXiv: 2505.08715 by the authors.

Figure 1
Figure 1. A trojan invariant torus for the ER3BP plotted in the ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The validation error (9) for the (left) projection and (right) least-squares methods, for both the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A schematic of how the Fourier coefficients for the ER3BP example shear under the transformation [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Four projections of a trajectory of the coupled standard map (11). [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: A C∞-windowed discrete Fourier transform of a length 10000 trajectory of the 2D standard map (11) shown in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: (Left) The Birkhoff RRE residual (17) as a function of trajectory length [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Distribution of the output quantities (a) [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: (a-b) Two examples of poorly approximated invariant tori projected onto the ( [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Projections of two successfully computed island trajectories into the ( [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: Visualizations of three d = 3 invariant tori of the earth-moon ER3BP. In each column, the three￾dimensional torus is “sliced” in each of the three angular coordinates. For instance, in the first column, the tori slices (h ◦ S) (1,j) (θ2, θ3) = (h ◦ S)(πj/4, θ2, θ3) fo…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 47 canonical work pages

  1. [1]

    Murray and Stanley F

    Carl D. Murray and Stanley F. Dermott. Solar System Dynamics . Cambridge University Press, Cam- bridge, 2000

    work page 2000

  2. [2]

    R. D. Hazeltine and J. D. Meiss. Plasma confinement. Dover Publications, Mineola, N.Y, 2003

    work page 2003

  3. [3]

    An Introduction to Stellarators: From Magnetic Fields to Symmetries and Optimization

    Lise-Marie Imbert-Gerard, Elizabeth Paul, and Adelle Wright. An Introduction to Stellarators: From Magnetic Fields to Symmetries and Optimization . Society for Industrial and Applied Mathematics, Philadelphia, PA, 2024

    work page 2024

  4. [4]

    The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations , volume 195 of Applied Mathematical Sciences

    Alex Haro, Marta Canadell, Jordi-Lluis Figueras, Alejandro Luque, and Josep Maria Mondelo. The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations , volume 195 of Applied Mathematical Sciences. Springer International Publishing, Cham, 2016

    work page 2016

  5. [5]

    Haro and R

    A. Haro and R. de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results. Journal of Differential Equations , 228(2):530–579, September 2006. 31

    work page 2006

  6. [6]

    A parameterization method for the computation of invariant tori andtheir whiskers in quasi-periodic maps: Numerical algorithms

    Alex Haro and Rafael de la Llave. A parameterization method for the computation of invariant tori andtheir whiskers in quasi-periodic maps: Numerical algorithms. Discrete and Continuous Dynamical Systems - B , 6(6):1261–1300, August 2006

    work page 2006

  7. [7]

    Haro and R

    A. Haro and R. de la Llave. A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity. SIAM Journal on Applied Dynamical Systems , 6(1):142–207, January 2007

    work page 2007

  8. [8]

    A tutorial on KAM theory

    Rafael de la LLave. A tutorial on KAM theory . American Mathematical Society; Oxford University Press, Providence, R.I., Oxford, 2004

    work page 2004

Show all 49 references
  1. [9]

    On conservation of conditionally periodic motions for a small change in Hamilton’s function

    Andrey Nikolaevich Kolmogorov. On conservation of conditionally periodic motions for a small change in Hamilton’s function. In Dokl. akad. nauk Sssr , volume 98, pages 527–530, 1954

    1954

  2. [10]

    Proof of a theorem of AN Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian

    Vladimir I Arnold. Proof of a theorem of AN Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965 , pages 267–294, 2009

    1957

  3. [11]

    On invariant curves of area-preserving mappings of an annulus

    Jurgen M¨ oser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. G¨ ottingen, II, pages 1–20, 1962

    1962

  4. [12]

    Figueras, A

    J.-Ll. Figueras, A. Haro, and A. Luque. Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach. Foundations of Computational Mathematics , 17(5):1123–1193, October 2017

    2017

  5. [13]

    Olikara, and Daniel J

    Nicola Baresi, Zubin P. Olikara, and Daniel J. Scheeres. Fully Numerical Methods for Continuing Families of Quasi-Periodic Invariant Tori in Astrodynamics. The Journal of the Astronautical Sciences , 65(2):157–182, June 2018

    2018

  6. [14]

    Jeremy Kasdin, and Pini Gurfil

    Egemen Kolemen, N. Jeremy Kasdin, and Pini Gurfil. Multiple Poincar´ e sections method for finding the quasiperiodic orbits of the restricted three body problem. Celestial Mechanics and Dynamical Astronomy, 112(1):47–74, January 2012

    2012

  7. [15]

    On the computation of reducible invariant tori on a parallel computer

    Angel Jorba and Estrella Olmedo. On the computation of reducible invariant tori on a parallel computer. SIAM Journal on Applied Dynamical Systems , 8(4):1382–1404, 2009

    2009

  8. [16]

    Leveraging quasi-periodic orbits for trajectory design in cislunar space

    Brian P McCarthy and Kathleen C Howell. Leveraging quasi-periodic orbits for trajectory design in cislunar space. Astrodynamics, 5(2):139–165, 2021

    2021

  9. [17]

    R. L. Dewar and J. D. Meiss. Flux-minimizing curves for reversible area-preserving maps. Physica D: Nonlinear Phenomena, 57(3):476–506, August 1992

    1992

  10. [18]

    R. L. Dewar, S. R. Hudson, and P. F. Price. Almost invariant manifolds for divergence-free fields. Physics Letters A , 194(1):49–56, October 1994

    1994

  11. [19]

    S. R. Hudson and R. L. Dewar. Almost-invariant surfaces for magnetic field-line flows. Journal of Plasma Physics, 56(2):361–382, October 1996

    1996

  12. [20]

    S. R. Hudson, D. A. Monticello, and A. H. Reiman. Reduction of islands in full-pressure stellarator equilibria. Physics of Plasmas , 8(7):3377–3381, July 2001

    2001

  13. [21]

    Direct com- putation of magnetic surfaces in Boozer coordinates and coil optimization for quasisymmetry

    Andrew Giuliani, Florian Wechsung, Georg Stadler, Antoine Cerfon, and Matt Landreman. Direct com- putation of magnetic surfaces in Boozer coordinates and coil optimization for quasisymmetry. Journal of Plasma Physics , 88(4):905880401, August 2022

    2022

  14. [22]

    Introduction to Frequency Map Analysis

    Jacques Laskar. Introduction to Frequency Map Analysis. In Carles Sim´ o, editor,Hamiltonian Systems with Three or More Degrees of Freedom, pages 134–150. Springer Netherlands, Dordrecht, 1999

    1999

  15. [23]

    J. Laskar. The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones. Icarus, 88(2):266–291, December 1990. 32

    1990

  16. [24]

    Quantitative quasiperiodicity

    Suddhasattwa Das, Yoshitaka Saiki, Evelyn Sander, and James A Yorke. Quantitative quasiperiodicity. Nonlinearity, 30(11):4111–4140, November 2017

    2017

  17. [25]

    Sander and J.D

    E. Sander and J.D. Meiss. Birkhoff averages and rotational invariant circles for area-preserving maps. Physica D: Nonlinear Phenomena , 411:132569, October 2020

    2020

  18. [26]

    J. D. Meiss and E. Sander. Resonance and weak chaos in quasiperiodically-forced circle maps. Com- munications in Nonlinear Science and Numerical Simulation , 142:108562, March 2025

    2025

  19. [27]

    Exponential convergence of the weighted Birkhoff average

    Zhicheng Tong and Yong Li. Exponential convergence of the weighted Birkhoff average. Journal de Math´ ematiques Pures et Appliqu´ ees, 188:470–492, August 2024

    2024

  20. [28]

    Quantitative uniform exponential acceleration of averages along decaying waves, September 2024

    Zhicheng Tong and Yong Li. Quantitative uniform exponential acceleration of averages along decaying waves, September 2024. arXiv:2408.09398 [math]

    2024

  21. [29]

    David Blessing and J. D. Mireles James. Weighted Birkhoff Averages and the Parameterization Method. SIAM Journal on Applied Dynamical Systems , 23(3):1766–1804, September 2024

    2024

  22. [30]

    Ruth and D

    M. Ruth and D. Bindel. Finding Birkhoff averages via adaptive filtering. Chaos: An Interdisciplinary Journal of Nonlinear Science , 34(12):123109, December 2024

    2024

  23. [31]

    The High-Order Magnetic Near-Axis Expansion: Ill-Posedness and Regularization, March 2025

    Maximilian Ruth, Rogerio Jorge, and David Bindel. The High-Order Magnetic Near-Axis Expansion: Ill-Posedness and Regularization, March 2025. arXiv:2411.04352 [physics]

    2025 arXiv

  24. [32]

    J. W. Burby, I. A. Maldonado, M. Ruth, D. A. Messenger, and L. Carbajal. Nonperturbative Guiding Center Model for Magnetized Plasmas. Physical Review Letters, 134(17):175101, April 2025

    2025

  25. [33]

    Numerical computation of rotation numbers of quasi-periodic planar curves

    Alejandro Luque and Jordi Villanueva. Numerical computation of rotation numbers of quasi-periodic planar curves. Physica D: Nonlinear Phenomena , 238(20):2025–2044, October 2009

    2025

  26. [34]

    SymplecticMapTools.jl

    Maximilian Ruth. SymplecticMapTools.jl. https://github.com/maxeruth/SymplecticMapTools.jl, 2025

    2025

  27. [35]

    Victor Szebehely and Giorgio E. O. Giacaglia. On the elliptic restricted problem of three bodies. The Astronomical Journal, 69:230, April 1964

    1964

  28. [36]

    Shan Zhao and G. W. Wei. A unified discontinuous Galerkin framework for time integration. Mathe- matical Methods in the Applied Sciences , 37(7):1042–1071, 2014

    2014

  29. [37]

    Geometricintegrators.jl: Geometric numerical integration in julia

    Michael Kraus. Geometricintegrators.jl: Geometric numerical integration in julia. https://github. com/JuliaGNI/GeometricIntegrators.jl, 2020

    2020

  30. [38]

    Trefethen and J

    Lloyd N. Trefethen and J. A. C. Weideman. The Exponentially Convergent Trapezoidal Rule. SIAM Review, 56(3):385–458, January 2014

    2014

  31. [39]

    H. A. David and H. N. Nagaraja. Order statistics. John Wiley, Hoboken, N.J, 3rd ed edition, 2003

    2003

  32. [40]

    Complexity of Lattice Problems

    Daniele Micciancio and Shafi Goldwasser. Complexity of Lattice Problems . Springer US, Boston, MA, 2002

    2002

  33. [41]

    M. Ajtai. Generating hard instances of lattice problems (extended abstract). In Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing , STOC ’96, pages 99–108, New York, NY, USA, July 1996. Association for Computing Machinery

    1996

  34. [42]

    A. K. Lenstra, H. W. Lenstra, and L. Lov´ asz. Factoring polynomials with rational coefficients. Mathe- matische Annalen, 261(4):515–534, December 1982

    1982

  35. [43]

    HKZ and Minkowski Reduction Algorithms for Lattice- Reduction-Aided MIMO Detection

    Wen Zhang, Sanzheng Qiao, and Yimin Wei. HKZ and Minkowski Reduction Algorithms for Lattice- Reduction-Aided MIMO Detection. IEEE Transactions on Signal Processing, 60(11):5963–5976, Novem- ber 2012. 33

    2012

  36. [44]

    J. D. Meiss and E. Sander. Birkhoff averages and the breakdown of invariant tori in volume-preserving maps. Physica D: Nonlinear Phenomena , 428:133048, December 2021

    2021

  37. [45]

    Dynamical systems, the three-body problem and space mission design

    Wang Sang Koon, Martin W Lo, Jerrold E Marsden, and Shane D Ross. Dynamical systems, the three-body problem and space mission design. In Equadiff 99: (In 2 Volumes) , pages 1167–1181. World Scientific, 2000

    2000

  38. [46]

    Numerical exploration of the restricted problem, v.Astronomy and Astrophysics, 1:223– 238, 1969

    Michel H´ enon. Numerical exploration of the restricted problem, v.Astronomy and Astrophysics, 1:223– 238, 1969

    1969

  39. [47]

    Exploration of distant retrograde orbits around moon

    Xu Ming and Xu Shijie. Exploration of distant retrograde orbits around moon. Acta Astronautica, 65(5-6):853–860, 2009

    2009

  40. [48]

    https://ssd.jpl.nasa.gov/tools/periodic_orbits.html

    Three-body periodic orbits. https://ssd.jpl.nasa.gov/tools/periodic_orbits.html. Jet Propul- sion Laboratory Solar System Dynamics

  41. [49]

    Golub and Charles F

    Gene H. Golub and Charles F. Van Loan. Matrix computations. Johns Hopkins studies in the mathe- matical sciences. The Johns Hopkins University Press, Baltimore, fourth edition, 2013. 34

    2013

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.