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REVIEW 3 major objections 4 minor 39 references

Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Replacing the ordinary Jacobian by a q-deformed one, this paper finds that coordinate contraction lowers positive Lyapunov exponents for chaotic orbits, dilatation raises them, and both tend to destabilize regular KAM-tori orbits.

desk verdict The analytical bifurcation curves are a real new result, but the headline q-trend in the Lyapunov exponents is mostly built into the definition, and the physical interpretation needs a lot more support. read the letter →

arxiv 2411.18691 v1 pith:P2ZGAZBV submitted 2024-11-27 nlin.CD

Tulio Meneghelli de Oliveira , Vinicius Wiggers , Eduardo Scafi , Silvio Zanin , Cesar Manchein , Marcus Werner Beims This is my paper · ORCID
classification nlin.CD MSC 37D4537C7537J40 PACS 05.45.-a05.45.Pq
keywords q-deformedLyapunovexponentq-Jacobianq-derivativeHénonmapHénon–HeilessystemdiamagneticKeplerproblemKAMtoristability
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 tries to establish that Lyapunov stability responds systematically to coordinate $q$-contraction ($q<1$) and $q$-dilatation ($q>1$) when stability is measured through a $q$-deformed Jacobian built from the $q$-derivative. In the dissipative Hénon map and the conservative Hénon–Heiles and diamagnetic Kepler problems, chaotic trajectories with positive Lyapunov exponents at $q=1$ show lower positive $q$-Lyapunov exponents under contraction and higher ones under dilatation, while regular orbits near KAM tori tend to be destabilized by both. The paper also derives analytical bifurcation curves for low-period orbits of the Hénon map and finds that the curves $P_{12}$ and $P_{1'2'}$, which coincide at $q=1$, split apart for $q\neq 1$, exposing a degeneracy hidden in the standard analysis. A reader would care because the results propose a concrete way to quantify how nonlinear stability responds to coordinate distortions in both dissipative maps and generic nonintegrable Hamiltonian flows.

What carries the argument

The carrier of the argument is the $q$-derivative, defined by $\partial^{(q)}_x f(x) = (f(qx)-f(x))/((q-1)x)$, which measures change under a $q$-dilatation of the variable rather than a translation. For a function that is homogeneous of degree $\psi$ in a coordinate, this derivative reduces to $[\psi]_q f(x)/x$ with $[\psi]_q = (q^{\psi}-1)/(q-1)$, a $q$-number that equals $\psi$ at $q=1$. Applying this derivative componentwise to the equations of motion gives the $q$-Jacobian $J^{(q)}$, whose products or eigenvalues replace the ordinary Jacobian in the Lyapunov exponent formula, yielding $\lambda^{(q)}$. Since $[1]_q=1$, the $q$-deformation acts only through nonlinear terms, and the homogeneity degrees of those terms determine the direction and size of the effect; this is what makes the Hénon map's quadratic term, the Hénon–Heiles cubic terms, and the diamagnetic Kepler quartic terms respond differently under contraction versus dilatation.

What would settle it

Take a chaotic orbit of the Hénon map, rescale the coordinates literally by $q$ in the equations of motion ($x \to qx$, $y \to qy$), and compute the standard maximal Lyapunov exponent of the rescaled map; smooth coordinate changes leave Lyapunov exponents invariant, so if the standard exponent stays equal to the $q=1$ value while the $q$-Jacobian $\lambda^{(q)}$ changes, the interpretation of $\lambda^{(q)}$ as the Lyapunov stability under coordinate $q$-dilatation is refuted.

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Extended reading notes

Core claim

The central claim is that the $q$-deformed Jacobian, obtained by replacing ordinary derivatives with the $q$-derivative in the linearized equations, yields a $q$-dependent Lyapunov exponent $\lambda^{(q)}$ that tracks whether coordinates are contracted or dilated. For $q>1$ the positive $\lambda^{(q)}$ of chaotic orbits increases relative to $q=1$; for $q<1$ it decreases; and for regular orbits both transformations tend to push the $q$-Lyapunov exponent upward, signalling destabilization. In the Hénon map the analytical stability curves $P_{12}$, $P_{1'2'}$, and $P_{24}$, which mark the period-1 to 2, period-1' to 2', and period-2 to 4 bifurcations, vary with $q$, and the two curves that are degenerate at $q=1$ become distinct at $q\neq 1$, so the $q$-deformation resolves a degeneracy invisible in the standard limit. In the conservative systems the same qualitative rule holds for chaotic initial conditions, with exceptions clustered around specific KAM tori, and the maximal $q$-Lyapunov exponent follows a power law in energy in the Hénon–Heiles system and a linear law in the diamagnetic Kepler problem.

Load-bearing premise

The load-bearing premise is that the $q$-deformed Jacobian measures stability under coordinate contraction and dilatation: unlike the ordinary Jacobian, it is not the derivative of any physical perturbation of the trajectory, so the claim that $\lambda^{(q)}$ quantifies coordinate-deformation stability rests on interpreting the algebraic rescaling $[\psi]_q$ as a genuine coordinate effect.

Editorial extensions

If this is right

  • In the Hénon map, the analytical curves $P_{12}$ and $P_{1'2'}$ coincide at $q=1$ and split for $q\neq 1$, resolving a degeneracy in the period-doubling structure that the standard analysis cannot see.
  • Bifurcation points that are stable at $q=1$ acquire positive $q$-Lyapunov exponents under $q>1$ dilatation, meaning coordinate dilatation destabilizes them while contraction stabilizes them.
  • For chaotic initial conditions in both conservative systems, contraction lowers the positive $q$-Lyapunov exponent and dilatation raises it, making $\lambda^{(q)}$ a monotone stability indicator along the $q$ direction for chaos.
  • Regular orbits near KAM tori do not follow the simple rule: both contraction and dilatation can increase the $q$-Lyapunov exponent, so the $q$-deformation acts as a generic perturbation for tori, consistent with KAM stability theory.
  • The energy dependence of the maximal $q$-Lyapunov exponent remains a power law in the Hénon–Heiles system and a linear law in the diamagnetic Kepler problem, with $q$ changing the fitted exponents or the slope and intercept.

Reading between the lines

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

  • A direct testable extension is to literally rescale coordinates in the equations of motion, $x \to qx$, compute the standard Lyapunov exponent of the rescaled system, and compare it with $\lambda^{(q)}$; the paper's interpretation predicts a match only if the $q$-derivative genuinely captures coordinate dilatation, and a mismatch would isolate the algebraic part of the effect.
  • The $q=1$ degeneracy splitting suggests that $q$ can be used as a symmetry-breaking parameter to unfold other hidden coincidences in bifurcation diagrams, potentially serving as a numerical continuation tool for stability curves in maps with higher-degree terms.
  • Since $[\psi]_q$ is increasing in $\psi$ for $q>1$ and decreasing for $q<1$, a heuristic prediction is that chaotic orbits whose instability is dominated by higher-degree homogeneous terms show the strongest response to dilatation; classifying orbits by the dominant degree in the $q$-Jacobian could turn the observed trend into a per-term prediction.
  • The exceptions near KAM tori may reflect that tori with different rotation numbers respond differently to the algebraic weighting of the $q$-Jacobian; connecting the size of the $q$-effect to the torus's winding number would be a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript introduces a q-deformed Lyapunov exponent obtained by replacing ordinary derivatives in Jacobians with q-derivatives, and applies it to the Hénon map, the Hénon-Heiles system, and the diamagnetic Kepler problem. The central reported findings are that q-contraction (q<1) decreases positive Lyapunov exponents for chaotic trajectories, q-dilatation (q>1) increases them, and both transformations tend to increase the exponents of regular orbits. For the Hénon map the authors also derive analytical bifurcation curves P1, P12, P1'2', and P24, which split at q≠1 and are compared with numerically computed maximal q-Lyapunov exponents. The paper claims that these results demonstrate Lyapunov stability under coordinate contraction and dilatation and reveal a hidden degeneracy in the usual q=1 bifurcation structure.

Significance. If the proposed interpretation were established, the work would provide a general formalism for assessing stability under nonlinear coordinate deformations and would extend q-deformation methods to Lyapunov analysis in both dissipative and conservative systems. The analytical Hénon bifurcation curves and their numerical verification are self-contained and potentially useful, as is the observation that the q=1 degeneracy of P12 and P1'2' is lifted for q≠1. However, the central physical claim is not currently supported: the q-Jacobian is an algebraic rescaling of the ordinary Jacobian evaluated on the same q=1 trajectory, not the linearization of any defined q-deformed dynamics, and the main monotonic trend follows directly from the definition of [d]_q. The manuscript also contains a concrete formal error in the Hénon sum rule. The numerical results may be reproducible, but the interpretation needs either a dynamical derivation or an explicit reframing.

major comments (3)
  1. [Section II, Eq. (9)] The definition of the q-Lyapunov exponent is not derived from a variational equation. The text states that 'it is possible to construct the corresponding linearized equations dy(q)/dt = J(q) y(q)' using the q-Jacobian, but no q-deformed map or vector field is defined whose tangent dynamics would produce this equation. The trajectories entering J(q) are still generated by the original q=1 equations, and J(q) is obtained by pointwise rescaling of the ordinary Jacobian entries. Consequently, the claim that λ(q) measures 'Lyapunov stability under dilatation and contraction of coordinates' is an interpretation without derivation. The manuscript should either define a concrete q-deformed dynamics whose variational equation is dy(q)/dt = J(q)y(q), or clearly state that λ(q) is a diagnostic of the rescaled Jacobian along unchanged trajectories.
  2. [Section III A, text after Eq. (11)] The statement 'we have two q-LEs, λ1(q) > λ2(q), which satisfy λ1(q) + λ2(q) = -b' is incorrect. The determinant of J(q) in Eq. (11) is det J(q) = -b, independent of q, so the sum of the Lyapunov exponents is ln|b|, not -b. This is a concrete formal error in a central example and should be corrected; it also indicates that the q-LE is being treated as if it were an ordinary Lyapunov exponent of the matrix product, for which the standard sum rule would apply.
  3. [Sections III B 1, III B 2, and IV] The main qualitative trend—contraction lowers positive q-LEs and dilatation raises them—is essentially a consequence of the definition. Each nonlinear monomial derivative is multiplied by [d]_q, and [d]_q is strictly increasing in q for every d≥2, e.g. [2]_q = q+1. Since the trajectory is unchanged and the q-LE is a time average of log norms of these rescaled matrices, the stated monotonicity is expected from the rescaling alone and does not by itself establish a dynamical effect. The paper acknowledges exceptions for KAM tori in the abstract and conclusion, further showing that the trend is not universal. To make the central claim load-bearing, the authors should provide a quantitative argument separating the rescaling effect from any genuinely new stability property, or reframe the contribution as an analysis of rescaled Jacobian spectra.
minor comments (4)
  1. [Figure 3 caption] The caption labels the bottom row as 'q = 0.5', but the text and the subsequent discussion indicate that the bottom row should be q = 1.5; this typo should be fixed.
  2. [Eq. (8) and Eq. (9)] The notation |J(t)| and |Ji| is ambiguous for matrices; the authors should specify the matrix norm used in the numerical computations, for example the induced 2-norm or singular value norm.
  3. [Reference [8]] The author name 'Ayse' appears incomplete or incorrect; the reference should be verified and completed, since it is the source of the q-derivative definition in Eq. (1).
  4. [Section III B 1, Fig. 5] The power-law fits report many significant digits without uncertainties or goodness-of-fit measures, which would be helpful for assessing the claim that the exponent changes with q.

Circularity Check

1 steps flagged · score 6.0 of 10

The main q-LE stability trend is a rescaling artifact of the q-Jacobian definition; the analytical bifurcation curves remain self-contained.

  1. other [Abstract; Section II Eq. (9); Section III A Eq. (11)]
    "Simulations show that q-contraction (q-dilatation) generally decreases (increases) positive Lyapunov exponents relative to the q = 1 case... J_n^(q) = ... -[ψ_x^(2)]_q x_n ... b; 1; 0, [ψ_x^(2)]_q = (q^2-1)/(q-1), and λ^(q) = lim_{n→∞} (1/n) ln(∏_i |J_i^(q)|)."

    For the Hénon map, [2]_q = q+1, so J^(q) is the ordinary Jacobian with the (1,1) entry multiplied by (q+1)/2, evaluated on the same q=1 trajectory {x_n}. More generally, the q-Jacobian is the standard Jacobian with each monomial derivative multiplied by the q-number [d]_q, which is strictly increasing in q for every degree d≥2. Since λ^(q) in Eq. (9) is a time-average of logarithms of norms of these pointwise rescaled matrices, the reported trend that contraction decreases and dilatation increases positive Lyapunov exponents follows by construction from the monotonicity of [d]_q. No q-deformed evolution law is defined: the orbit feeding J^(q) is still generated by the ordinary q=1 equations. Thus the central numerical claim is equivalent to the definitional rescaling in Eqs.

full rationale

Most of the paper is self-contained numerical computation: the analytical bifurcation curves P1, P12, P1'2', P24 are derived from the q-Jacobian determinant and match the zero-LE contours, and these mathematical consequences are not circular. The central interpretive claim, however, is the q-dependence of positive Lyapunov exponents. That claim reduces by construction to the definition of the q-derivative: for the Hénon map, [2]_q = q+1, so J^(q) is the ordinary Jacobian with the (1,1) entry multiplied by (q+1)/2, evaluated on the same q=1 trajectory; more generally, every monomial derivative in the q-Jacobian is multiplied by [d]_q, a strictly increasing function of q. Because λ^(q) is the time-average of log norms of these pointwise rescaled matrices, the stated 'contraction decreases, dilatation increases' pattern for chaotic orbits follows immediately from that monotonicity rather than from an emergent property of the systems. No q-deformed tangent-space dynamics is defined, so the paper's assertion that this measures Lyapunov stability under coordinate transformations is an interpretation imported from Ref. [12] (same author group) and from the q-derivative formalism, not a derived result. The analytical content and the PSS simulations retain independent value, but the main stability conclusion is a rescaling artifact. The inconsistent sum rule λ1^(q)+λ2^(q) = -b (whereas det J^(q) = -b would give ln|b| for two-dimensional maps) is a correctness issue rather than a circularity, but it reinforces that the q-LE is treated formally. Overall score 6: the central 'prediction' for positive Lyapunov exponents reduces by construction, while the analytical curves and the regular-orbit exceptions are not circular.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central quantity λ(q) is defined through an ad hoc q-derivative and a non-physical q-Jacobian; the paper does not provide a derivation from the underlying dynamics. The main 'discoveries' (contraction lowers LEs, dilatation raises them) follow from the definition's monotonic scaling of Jacobian entries.

free parameters (2)
  • power-law exponent β for Hénon-Heiles = β ≈ 2.99 (q=0.5), 3.45 (q=1.0), 2.38 (q=1.5)
    Exponents from log-log linear fit of λ vs energy in Fig. 5; no uncertainties given.
  • linear fit coefficients (a,b) for diamagnetic Kepler = q=0.5: (1.12334, -0.290927); q=1.0: (1.62469, -0.480846); q=1.5: (2.41462, -0.612911)
    Phenomenological linear fit of q-LE vs energy in Fig. 8; no uncertainties given.
assumptions (3)
  • ad hoc to paper The q-derivative defined in Eq. (1) and its homogeneous extension in Eq. (4) correctly represent the effect of coordinate dilatation/contraction on stability.
    This is a definition introduced by the authors (following Ref. [8]) without derivation from first principles; the interpretation as 'stability under coordinate dilatation' is assumed.
  • domain assumption The q-Lyapunov exponent computed from the q-Jacobian via Eq. (9) captures the stability of trajectories under the q-deformed perturbation.
    Standard Lyapunov theory applies to the linearized flow; here the q-Jacobian is not the linearization of the flow, so the ergodic meaning of λ(q) is not established.
  • domain assumption The standard numerical algorithm [14] for Lyapunov exponents remains valid when the Jacobian is replaced by the q-Jacobian.
    The Wolf algorithm is designed for the true Jacobian; its application to a rescaled, non-physical Jacobian is an assumption.

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Pith. "Pith review of Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates." pith.science (2026). https://pith.science/paper/P2ZGAZBV

@misc{pith2026241118691,
  author       = {Pith},
  title        = {Pith review of: Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2ZGAZBV}},
  note         = {Machine review of arXiv:2411.18691}
}
abstract

This study examines the Lyapunov stability under coordinate $q$-contraction and $q$-dilatation in three dynamical systems: the discrete-time dissipative H\'enon map, and the conservative, non-integrable, continuous-time H\'enon-Heiles and diamagnetic Kepler problems. The stability analysis uses the $q$-deformed Jacobian and $q$-derivative, with trajectory stability assessed for $q > 1$ (dilatation) and $q < 1$ (contraction). Analytical curves in the parameter space mark boundaries of distinct low-periodic motions in the H\'enon map. Numerical simulations compute the maximal Lyapunov exponent across the parameter space, in Poincar\'e surfaces of section, and as a function of total energy in the conservative systems. Simulations show that $q$-contraction ($q$-dilatation) generally decreases (increases) positive Lyapunov exponents relative to the $q = 1$ case, while both transformations tend to increase Lyapunov exponents for regular orbits. Some exceptions to this trend remain unexplained regarding Kolmogorov-Arnold-Moser (KAM) tori stability.

Figures

Figures reproduced from arXiv: 2411.18691 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. displays the maximal q-LE (see color bar) in the parameter space (a, q) for b = 0.3. For q = 1, we have the usual period-doubling bifurcation diagram as a function of a. The curves P12 (B) and P1 ′2 ′ (C) cross at q = 1 where the usual bifurcation from period 1 → 2 occurs. This crossing point is the origin of the chaotic motion for q > 1. Similar behavior is expected for P24 (D) at q = 1, where the usual bifurcation… view at source ↗
Figure 3
Figure 3. displays the PSS for three distinct energies EH = 1/8 (left column), 1/7 (middle column) and 1/6 (right column), and three different values of q = 0.5 (top row), q = 1.0 (middle row) and 1.5 (bottom row). The values of the q-LEs are provided by the colorbar. Lilac for vanishing q-LEs, and dark to light blue, yellow to red for increasing positive q-LEs. For q = 1, we have the usual case. Islands and tori have a negat… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: displays the positive q-LE as a function of the energy and for three distinct values of q, namely red points for q = 1.5, black points for q = 1.0 and blue points for q = 0.5. Roughly speaking, the q-LE increases as a power-law with the energy, whose exponent does not …
Figure 6
Figure 6. Figure 6: FIG. 6: Poincar´e surface of section ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Plot of the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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