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A parameterization Newton scheme computes reducible quasiperiodic tori and continues them through saddle-node bifurcations by correcting parameters and an unfolding coordinate.

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2026-07-12 02:04 UTC pith:LTMEWUR3

load-bearing objection Solid continuous-time Newton–KAM for reducible quasiperiodic tori that actually crosses saddle-nodes via an explicit unfolding; real algorithmic contribution, cleanly scoped. the 2 major comments →

arxiv 2607.03498 v1 pith:LTMEWUR3 submitted 2026-07-03 math.DS nlin.CD

Numerical Computation of Quasiperiodic Reducible Saddle-Node Bifurcations: a Parameterization Method Approach

Jordi-Llu\'is Figueras , Joan Gimeno , Jeremy P. Parker This is my paper
classification math.DS nlin.CD MSC 37M2034C4534C2337C5565P30
keywords Invariant torusNormally hyperbolic invariant manifoldQuasiperiodic saddle-node bifurcationParameterization methodKAM theoryPseudo-arclength continuationReducible normal bundle
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper gives a practical Newton–KAM method that finds normally hyperbolic invariant tori whose internal motion is a prescribed quasiperiodic frequency, together with the system parameters that make those tori exist. It works by parameterizing both the torus and its normal directions, then solving the invariance and reducibility defects mode by mode in Fourier space so that small-divisor equations stay tractable. Near a saddle-node the usual continuation in a physical parameter becomes singular, so the authors introduce an artificial unfolding coordinate given by the average projection of the torus onto a distinguished normal fiber; Newton then stays regular while that coordinate is continued. The method recovers the local saddle-node normal form after the objects are computed, without requiring the original vector field to be put into normal form first. Two synthetic ODE models confirm quadratic residual decay and show that the distinguished normal rate changes sign exactly when the branch turns.

Core claim

Reducible normally hyperbolic quasiperiodic tori in autonomous ODEs, and the saddle-node bifurcations at which they appear or disappear, can be computed by a single parameterization Newton scheme that simultaneously corrects the torus embedding, a reducible normal frame, selected system parameters, and an unfolding scalar that regularizes the fold.

What carries the argument

The parameterization method of KAM theory, realized as projected cohomological equations on a moving frame, plus the unfolding condition ς = average of the torus embedding against the distinguished normal direction; together they convert the singular saddle-node problem into a regular continuation problem.

Load-bearing premise

The linearized normal dynamics on the torus must be reducible to a constant diagonal hyperbolic matrix with pairwise distinct real eigenvalues, and a sufficiently accurate initial guess for both the torus and its normal bundle must already be known.

What would settle it

On either model ODE, run the published algorithms from the given explicit initial data and check whether the torus and reducibility residuals fail to decay quadratically, or whether the distinguished normal rate fails to change sign while the continuation parameter traverses the fold predicted by the exact unperturbed solution.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops a parameterization-method Newton–KAM scheme for computing reducible normally hyperbolic quasiperiodic invariant tori in autonomous ODEs, simultaneously correcting the embedding, a reducible normal bundle, and system parameters so that a prescribed Diophantine frequency is realized. The main algorithmic contribution is an adapted correction (Algorithm 1) that introduces an unfolding/pseudo-arclength scalar ς = ⟨K, v_c⟩ associated with a distinguished near-neutral normal direction, allowing continuation through saddle-node bifurcations of tori without requiring a preliminary normal-form reduction. Explicit algorithms for the standard and saddle-node cases are given, and the methods are demonstrated on two synthetic models (a 5D toy system and a 3D saddle-node model), with multiprecision runs showing quadratic residual decay and successful fold crossing via a sign change of the distinguished normal rate λ_c.

Significance. If the algorithms perform as claimed under the stated reducibility hypotheses, the work supplies a practical continuous-time tool for computing and continuing quasiperiodic NHIM tori through saddle-node bifurcations in dissipative autonomous systems, complementing existing map-based and Poincaré-section approaches and the discrete-time saddle-node study of Vitolo–Broer–Simó. Strengths include fully explicit algorithms, analytic derivative callbacks, multiprecision residual monitoring with clear quadratic Newton decay (Fig. 2), and successful traversal of the fold on both models. The scope is honestly limited to reducible tori with pairwise-distinct real normal rates and good initial guesses; a-posteriori validation is deferred to a companion paper. Within that scope the contribution is solid and useful for computational dynamical systems.

major comments (2)
  1. The numerical evidence is confined to two synthetic models that are explicit perturbations of closed-form unperturbed tori and bundles (§4.1–4.2, (39)–(43)). While residual decay and λ_c sign change are convincing under A1–A4, the paper does not show performance on a non-synthetic system or against an existing torus-continuation method (e.g. map-based or Poincaré-section schemes cited in the introduction). A short comparison or a third, less constructed example would substantially strengthen the claim of practical applicability.
  2. A-posteriori existence and quantitative validation of the computed objects are deferred entirely to the companion paper [Fig+] (Introduction and §2.2). The present manuscript therefore rests on residual smallness and multiprecision Newton behavior alone. For a numerical-methods journal this is acceptable if clearly flagged, but the abstract and conclusions should state more explicitly that existence is not proved here and that the algorithms are validated only by residual monitoring under the reducibility assumptions A1–A4.
minor comments (5)
  1. Assumption A4 (real Λ_N and real N) and the pairwise-distinctness condition A3 are essential for the real Fourier solves in Algorithms 1–3; a brief remark on how complex conjugate pairs would be handled (or a pointer to [BGJ]) would help readers who encounter non-real spectra.
  2. Figure 2 reports raw CPU times without parallelization for the 3D model; Table 2 (Appendix) shows modest OpenMP speed-ups limited by non-parallel FFT. A short note on the current computational bottleneck and expected scaling with mesh size would improve reproducibility.
  3. Notation for the frame and defects is dense (P_0, η_L/η_N, E_tor, E_red, etc.). A small summary table of symbols at the start of §2 would reduce the cognitive load when reading the correction equations.
  4. The companion paper is cited as “[Fig+] work in progress.” If a preprint or arXiv identifier becomes available before final acceptance, it should be updated; otherwise the dependence should be stated more carefully in the abstract.
  5. Typographical inconsistencies appear in a few places (e.g. “tacklereducible”, “ph ere 0 ∈ R^d”, mixed “˜ω” vs “ω”). A careful copy-edit pass is needed.

Circularity Check

0 steps flagged

No significant circularity: constructive Newton–KAM algorithms whose residuals are measured against the invariance/reducibility equations themselves; unfolding scalar is a free continuation coordinate.

full rationale

The paper derives explicit Fourier-mode Newton corrections (Algorithms 1–3) for the torus embedding K, system parameters, distinguished normal direction vc, reduced normal bundle W and rates Λ from the first-order expansion of the invariance equation Lω[K]+F(K;μ,ϑ)=Etor and the reducibility equation for the frame. The unfolding condition ς=⟨K,vc⟩ is introduced as a free pseudo-arclength coordinate that regularizes the fold; it is not fitted to data and does not force any residual to vanish by construction. Convergence is demonstrated by direct residual decay (quadratic in multiprecision) and by the sign change of the distinguished rate λc on two synthetic models that start from known exact solutions. Self-citations (parameterization literature, companion a-posteriori paper [Fig+], TorKam library) supply tools or deferred validation; none is used as a uniqueness premise that closes the derivation. The method is therefore self-contained against its own defining equations under the stated reducibility hypotheses A1–A4; no prediction reduces to an input by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The method rests on standard KAM/parameterization hypotheses (Diophantine frequencies, normal hyperbolicity, reducibility to constant coefficients) plus the usual smoothness needed for second-order Taylor expansions. No free parameters are fitted to external data; numerical tolerances and mesh sizes are ordinary discretization choices. The unfolding scalar ς is a standard pseudo-arclength device, not a new physical entity.

free parameters (3)
  • Fourier mesh size / truncation
    Chosen by the user (e.g. 64²–1024²); controls approximation quality but is not fitted to external observations.
  • Newton tolerance tol / tol_c
    User-chosen stopping thresholds (e.g. 10^{-16}, 10^{-50}); ordinary numerical parameters.
  • Continuation step-size adaptation factors (1.1 / 0.8)
    Heuristic multipliers for pseudo-arclength step control; not data-fitted.
axioms (4)
  • domain assumption ω is Diophantine (small-divisor condition with constants ν, τ)
    Required for solvability of the cohomological equations in Fourier space (§2.1).
  • domain assumption Normal bundle is reducible: linearized normal dynamics conjugate to a constant hyperbolic matrix Λ_N (A1–A4)
    Central structural hypothesis that decouples the mode-by-mode solves (§2.2).
  • standard math Vector field F is C² (or smoother) so that second-order Taylor remainders are well-defined
    Used throughout the Newton linearizations (§2–3).
  • domain assumption A sufficiently accurate initial guess (K0,N0) exists so that Newton converges
    Stated as a practical prerequisite in §5; not constructed by the algorithm.
invented entities (1)
  • Unfolding / pseudo-arclength scalar ς = ⟨K, v_c⟩ no independent evidence
    purpose: Provides a regular coordinate that remains transversal through the saddle-node fold of the torus branch.
    Standard device in numerical continuation; not a new physical object. independent_evidence is false because it is a computational coordinate defined inside the algorithm.

pith-pipeline@v1.1.0-grok45 · 42111 in / 2666 out tokens · 24830 ms · 2026-07-12T02:04:03.138985+00:00 · methodology

0 comments
read the original abstract

We present a method for computing reducible, normally hyperbolic, invariant tori with internal quasiperiodic dynamics in autonomous ordinary differential equation systems. The approach is based on the parameterization method of KAM theory; thus, it is a Newton scheme with small divisors. Since the inner dynamics of the torus is prescribed, the corresponding system parameters for which such a torus exists are simultaneously determined. The method is amenable to a form of pseudo-arclength continuation, enabling the traversal and computation of saddle-node bifurcations. We give explicit algorithms for the methods and demonstrate their applicability with two numerical examples.

Figures

Figures reproduced from arXiv: 2607.03498 by Jeremy P. Parker, Joan Gimeno, Jordi-Llu\'is Figueras.

Figure 2
Figure 2. Figure 2: Using 211-digits arithmetic and a model (41), residual errors in different meshes of pθ1, θ2q of the torus E and the distinguished direction E c v. The last panel shows raw CPU time with no parallelization -80 -70 -60 -50 -40 -30 -20 -10 0 0 1 2 3 4 5 6 7 8 log10 |EK| Newton Iteration 10242 5122 2562 1282 642 -80 -70 -60 -50 -40 -30 -20 -10 0 0 1 2 3 4 5 6 7 8 log10 |Ev| Newton Iteration 10242 5122 2562 12… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the torus solution (39) using Algorithm 2 0.998 0.999 1 1.001 1.002 0.997 0.998 0.999 1 1.001 1.002 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 θ1=0 θ2=0 r1,2 r3,4 h -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 ε=0.01 [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Continuation w.r.t. ε using Algorithm 2. (1,1): Illustration of the solution at the last continuation solution; (1,2): Evolution of eigenvalues λ2,3 (λ1 stays close to ´3); (1,3): Evolution of the parameters µ1,2; and (1,4): Illustration of the first and last solution and evolution sectioned on tθ2 “ 0u 0.96 0.98 1 1.02 1.04 0.94 0.96 0.98 1 1.02 1.04 1.06 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 θ1=0 θ2=0 r1,2 r3… view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

32 extracted references · 4 canonical work pages

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    doi: 10 . 1088 / 0951 - 7715 / 18 / 4 / 018. url: https : / / doi . org / 10 . 1088 / 0951 - 7715/18/4/018. [Bro+97] Broer et al. “Algorithms for computing normally hyperbolic invariant manifolds”. In: Zeitschrift f¨ ur angewandte Mathematik und Physik48.3 (1997), p. 480. issn: 0044-2275. doi: 10.1007/ s000330050044. url: https://dx.doi.org/10.1007/s00033...

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    work page doi:10.1016/j.jde.2025.113681 2003

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    A tutorial on KAM theory

    doi: 10 . 1016 / 0167 - 2789(91 ) 90227 - Z. url: https : / / doi . org / 10 . 1016 / 0167 - 2789(91)90227-Z. [De +01] Rafael De la Llave et al. “A tutorial on KAM theory”. In: Proceedings of Symposia in Pure Mathematics. Vol. 69. Providence, RI; American Mathematical Society; 1998. 2001, pp. 175– 296. [Doo+22] Patrick Doohan et al. “The state space and t...

    work page doi:10.1016/j.physd.2024.134127 1998

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    On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems

    issn: 0022-0396. doi: 10.1016/j.jde.2018.08.003 . url: https://dx.doi.org/10. 1016/j.jde.2018.08.003. [HM05] H. Hanßmann and J. C. van der Meer. “On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems”. In: EQUADIFF 2003. World Sci. Publ., Hackensack, NJ, 2005, pp. 476–481. isbn: 981-256-169-2. doi: 10.1142/9789812702067\_0077 . url: https://doi....

    work page doi:10.1016/j.jde.2018.08.003 2018

  5. [5]

    Epθq Ð LωrK0spθq ` F pK0pθq; µ0q ▷E : Td Ñ Rn

  6. [6]

    P0pθq Ð ` DK0pθq N0pθq ˘ ▷P0 : Td Ñ Rpnˆpp`1q`nˆpn´dqq

  7. [7]

    pηL0 , ηN0 q Ð P0pθq´1Epθq ▷ηL0 : Td Ñ Rd and ηN0 : Td Ñ Rn´d

  8. [8]

    pbL0 , bN0 q Ð P0pθq´1DµF pK0pθq; µ0q ▷pbL0 , bN0 q: Td Ñ Rppdq`pn´dqqˆpdq

  9. [9]

    Fourier step to solve ∆µ, ∆ω0, and ξL0 pθq: for all k P Zd pξL0 0 “ 0, normalization condition, pηL0 0 ` pbL0 0 ∆µ “ 0, for |k| “ 0, ∆µ is solved, ´ipk ¨ ωqpξL0 k ` pηL0 k ` pbL0 k ∆µ “ 0, for |k| ‰ 0, pξL0 k is solved 30 Quasiperiodic Reducible Saddle-Node Bifurcations

  10. [10]

    Fourier step to solve ξN0 pθq: for all k P Zd, pΛN0 ´ ipk ¨ ωqqpξN0 k ` pηN0 k ` pbN0 k ∆µ “ 0

  11. [11]

    K0pθq Ð K0pθq ` P0pθq ˆ ξL0 pθq ξN0 pθq ˙ and µ0 Ð µ0 ` ∆µ

  12. [12]

    Eredpθq Ð LωrN0spθq ` DzF pK0pθq; µ0qN0pθq ´ N0pθqΛN0 ▷Ered : Td Ñ Rnˆpn´dq

  13. [13]

    ˆ ηL0 redpθq ηN0 redpθq ˙ Ð P0pθq´1Eredpθq ▷ηL0 red : Td Ñ Rdˆpn´dq and ηN0 red : Td Ñ Rpn´dqˆpn´dq

    P0pθq Ð ` DK0pθq N0pθq ˘ ▷ updated frame 10. ˆ ηL0 redpθq ηN0 redpθq ˙ Ð P0pθq´1Eredpθq ▷ηL0 red : Td Ñ Rdˆpn´dq and ηN0 red : Td Ñ Rpn´dqˆpn´dq

  14. [14]

    ηN0 red “ ˆ ηss red ηsu red ηus red ηuu red ˙ ▷ block view of

  15. [15]

    Solve p pQss i,jqk p pQuu i,j qk p pQsu i,jqk, and p pQus i,jqk, for all k P Zd, p pQss i,iq0 “ p pQuu i,i q0 “ 0 normalization condition pλs i ´ λs jqp pQss i,jq0 “ ´pp pηss redqi,jq0 |k| “ 0, i ‰ j pλu i ´ λu j qp pQuu i,j q0 “ ´pp pηuu redqi,jq0 |k| “ 0, i ‰ j pλs i ´ λs j ´ ik ¨ ωqp pQss i,jqk “ ´pp pηss redqi,jqk |k| ‰ 0 pλu i ´ λu j ´ ik ¨ ωqp pQuu ...

  16. [16]

    Fourier step to solve p pQL0 i,j qk for all k P Zd, p´λj ´ ik ¨ ωqp pQL0 i,j qk “ ´pp pηL0 redqi,jqk

  17. [17]

    ΛN0 Ð ˆ λs i ` pppηss redqi,iq0 0 0 λu i ` pppηuu redqi,iq0 ˙ and N0pθq Ð N0pθq`DK0pθqQL0 pθq`N0pθq ˆ Qsspθq Qsupθq Quspθq Quupθq ˙

  18. [18]

    ‹ Input: ODE like (1), ergodic frequency ω “ pω0, ω1q P RˆRd´1

    Iterate from step 1 until convergence of E and Ered ♢ Algorithm 3 (Steps to correct pK0, µ0, ω0q and pN0, ΛN0q). ‹ Input: ODE like (1), ergodic frequency ω “ pω0, ω1q P RˆRd´1. Initial guesses ω0 P R, embedding K0 : Td Ñ Rn, µ0 P Rd´1, normal bundle N0 : Td Ñ Rnˆpn´dq, and matrix ΛN0 “ diagpΛs, Λuq P Rpn´dqˆpn´dq with Λs “ diagpλs i q P Rnsˆns and Λu “ di...

  19. [19]

    E0pα, βq Ð Lpω0,ω1qrK0spα, βq ` F pK0pα, βq; µ0q ▷E0 : Td Ñ Rn

  20. [20]

    P0pα, βq Ð ` DK0pα, βq N0pα, βq ˘ ▷P0 : Td Ñ Rpnˆd`nˆpn´dqq

  21. [21]

    pηL0 , ηN0 q Ð P0pα, βq´1E0pα, βq ▷ηL0 : Td Ñ Rd and ηN0 : Td Ñ Rn´d

  22. [22]

    pbL0 , bN0 q Ð P0pα, βq´1 ` DµF pK0pα, βq; µ0q Dω0 F pK0pα, βq; µ0q ´ BαK0pα, βq ˘ ▷pbL0 , bN0 q: Td Ñ Rpd`pn´dqqˆd

  23. [23]

    Figueras, J

    Fourier step to solve ∆µ, ∆ω0, and ξLpα, βq: for all k P Zd pξL0 0 “ 0, normalization condition, pηL0 0 ` pbL0 0 ˆ ∆µ ∆ω0 ˙ “ 0, for |k| “ 0, p∆µ, ∆ω0q is solved, ´ipk ¨ pω0, ω1qqpξL0 k ` pηL0 k ` pbL0 k ˆ ∆µ ∆ω0 ˙ “ 0, for |k| ‰ 0, pξL0 k is solved J.Ll. Figueras, J. Gimeno, and J. Parker 31

  24. [24]

    Fourier step to solve ξN0 pα, βq: for all k P Zd, pΛN0 ´ ipk ¨ pω0, ω1qqqpξN0 k ` pηN0 k ` pbN0 k ˆ ∆µ ∆ω0 ˙ “ 0

  25. [25]

    K0pα, βq Ð K0pα, βq ` P0pα, βq ˆ ξL0 pα, βq ξN0 pα, βq ˙ , µ0 Ð µ0 ` ∆µ, and ω0 Ð ω0 ` ∆ω0

  26. [26]

    Ered 0 pα, βq Ð Lpω0,ω1qrN0spα, βq ` DzF pK0pα, βq; µ0qN0pα, βq ´ N0pα, βqΛN0 ▷Ered 0 : Td Ñ Rnˆpn´dq

  27. [27]

    ˆ ηL0 redpα, βq ηN0 redpα, βq ˙ Ð P0pα, βq´1Ered 0 pα, βq ▷ηL0 red : Td Ñ Rdˆpn´dq and ηN0 red : Td Ñ Rpn´dqˆpn´dq

    P0pα, βq Ð ` DK0pα, βq N0pα, βq ˘ ▷ updated frame 10. ˆ ηL0 redpα, βq ηN0 redpα, βq ˙ Ð P0pα, βq´1Ered 0 pα, βq ▷ηL0 red : Td Ñ Rdˆpn´dq and ηN0 red : Td Ñ Rpn´dqˆpn´dq

  28. [28]

    ηN0 red “ ˆ ηss red ηsu red ηus red ηuu red ˙ ▷ block view of ηN0 red

  29. [29]

    Solve p pQss i,jqk, p pQuu i,j qk, p pQsu i,jqk, and p pQus i,jqk, for all k P Zd, p pQss i,iq0 “ p pQuu i,i q0 “ 0 normalization condition pλs i ´ λs jqp pQss i,jq0 “ ´pp pηss redqi,jq0 |k| “ 0, i ‰ j pλu i ´ λu j qp pQuu i,j q0 “ ´pp pηuu redqi,jq0 |k| “ 0, i ‰ j pλs i ´ λs j ´ ik ¨ pω0, ω1qqp pQss i,jqk “ ´pp pηss redqi,jqk |k| ‰ 0 pλu i ´ λu j ´ ik ¨ ...

  30. [30]

    Fourier step to solve p pQL0 i,j qk for all k P Zd, p´λj ´ ik ¨ pω0, ω1qqp pQL0 i,j qk “ ´pp pηL0 redqi,jqk

  31. [31]

    ΛN0 Ð ˆλs i ` pppηss redqi,iq0 0 0 λu j ` pppηuu redqj,jq0 ˙ and N0pα, βq Ð N0pα, βq ` DK0pα, βqQL0 pα, βq ` N0pα, βq ˆ Qsspα, βq Qsupα, βq Quspα, βq Quupα, βq ˙

  32. [32]

    Iterate from step 1 until convergence ♢ A.1 Synthetic example To assess the accuracy and convergence of the proposed Algorithms 1 and 3, we consider a synthetic model similar to (39). 9h “ ´3h ` εpx1 ` x3q 9x1 “ ´7p1 ´ r1,2qx1 ` µ1x2˜ω1 ` ε cosphq ri,j def “ b x2 i ` x2 j 9x2 “ ´7p1 ´ r1,2qx2 ´ µ1x1˜ω1 ` ε sinphq 9x3 “ ´5p1 ´ r3,4qx3 ` µ2x4˜ω2 ` ε sinphq ...