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REVIEW 4 major objections 6 minor 58 references

The symbolic partition with generalized Koopman analysis

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Zero-eigenvalue Koopman left eigenfunctions mark the symbolic partition boundary of chaotic maps, and a generalized Koopman analysis refines that boundary to match published partitions.

desk verdict A heuristic but promising Koopman-based route to symbolic partitions: the zero-mode eigenfunction idea is new and the numerical matches to published boundaries are real, but the central mechanism is a conjecture and the pipeline depends on hand-tuning. read the letter →

arxiv 2506.06973 v1 pith:CTATGNRW submitted 2025-06-08 nlin.CD

Haipeng Li , Pengfei Guo , Yueheng Lan This is my paper
classification nlin.CD MSC 37B1037D4537M10 PACS 05.45.-a05.45.Tp
keywords symbolicdynamicsKoopmanoperatorpartitionboundarylefteigenfunctionsEDMDchaotictimeseriesHénonmaphyperchaos
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 claims that the symbolic partition boundary of a chaotic map—the pre-image of its folding point—can be located from data by computing left eigenfunctions of an approximate Koopman operator and looking at those with eigenvalue zero whose oscillation is confined to an interior subregion (VLEZ). It argues that because a zero-eigenvalue density distribution vanishes after one step, the two sides of such an oscillation must overlap after evolution, which is exactly what happens across a folding point; the zero-crossing or the oscillation subregion therefore marks the boundary. To turn that coarse marker into a precise boundary, the paper introduces Generalized Koopman Analysis (GKA), which applies a local affine transformation so that a small region and its evolved image lie in the same coordinate frame and a one-dimensional folding analysis becomes valid. The claim is verified by matching previously published boundary positions for the Hénon map (two parameter sets), the Duffing return map, and a three-dimensional hyperchaotic map, including noisy versions.

What carries the argument

The central object is the Valid Left Eigenfunction with Zero (VLEZ): a left eigenvector $u$ of the finite-dimensional Koopman approximation matrix with eigenvalue $\lambda \approx 0$ (within $10^{-5}$) whose associated left eigenfunction $Ku$ oscillates only in an interior subregion. The argument rests on the Perron–Frobenius picture: left eigenvectors describe density evolution, and $\lambda=0$ means the density vanishes after one iteration, so the oscillation's two lobes must map onto the same region—a folding event—and the pre-image of the folding point lies between them. For maps too complex to yield VLEZ directly, the paper uses Characteristic Left Eigenfunctions (CLE, eigenvalues close to but not equal to zero) to redistribute basis functions and then applies the Generalized Koopman Analysis (GKA), which fits a curve to a localized subregion and uses an affine transformation to put the region and its evolved image on the same footing.

What would settle it

Construct a one-dimensional unimodal map whose folding pre-image falls exactly on the edge between two basis cells and whose computed VLEZ oscillation zero-crossing shifts away from that pre-image as the basis is refined; if such a map exists, the claimed localization mechanism fails. Concretely, one can take the logistic map at $\alpha=3.75$ with a deliberately shifted or non-uniform basis and check whether the zero-crossing points of VLEZ still converge to the critical point's pre-image; a mismatch would falsify the paper's central mechanism.

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Extended reading notes

Core claim

The central claim is that the pre-image of a folding point—the symbolic partition boundary—is carried by a specific spectral object: a Koopman left eigenfunction with eigenvalue zero whose oscillation is confined to an interior subregion. The paper calls this object the Valid Left Eigenfunction with Zero (VLEZ) and argues that because left eigenfunctions evolve densities and a zero eigenvalue makes the density vanish after one iteration, the two lobes of the oscillation must overlap after one step; that overlap is the folding event, so the boundary between the lobes is where the symbolic partition must cut. For maps with multiple or incomplete foldings, where VLEZ does not appear at low resolution, the paper first obtains a Characteristic Left Eigenfunction (CLE) with a small nonzero eigenvalue, uses its oscillation to redistribute the basis functions, and then obtains VLEZ. The refined boundary is produced by Generalized Koopman Analysis (GKA), which fits a curve to a localized attractor region and applies an affine transformation so that the region and its evolved image live in the same coordinate frame; the reported boundaries agree with the published partitions for the Hénon map ($\alpha=1.4, \beta=0.3$ and $\alpha=1.0, \beta=0.54$), the Duffing return map, and a three-dimensional hyperchaotic map, with weak Gaussian noise shifting the positions only within the stated tolerances.

Load-bearing premise

The load-bearing premise is that the oscillation pattern of a zero-eigenvalue Koopman left eigenfunction—demonstrated for the tent map—localizes the pre-image of a folding event for general chaotic maps and basis choices; no proof is given that EDMD left eigenvectors at eigenvalue zero must oscillate exactly across that pre-image.

Editorial extensions

If this is right

  • If the VLEZ mechanism is correct, symbolic partitions can be estimated directly from scalar or vector chaotic time series without knowing the dynamical equations.
  • The GKA refinement yields boundary positions that match published generating partitions for the Hénon map, the Duffing return map, and a 3D hyperchaotic map, so the method is a viable alternative when stable/unstable manifold or periodic-orbit data are unavailable.
  • The method tolerates weak Gaussian observational noise, meaning the same pipeline can be applied to experimental data where generating partitions are formally ill-defined.
  • Since the required building blocks are generic (basis functions, EDMD, spectral decomposition), the approach extends to higher-dimensional and continuous-time systems by discretization and Poincaré sections.

Reading between the lines

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

  • If the VLEZ mechanism holds beyond the tested maps, the method offers a data-only route to generating partitions for high-dimensional and hyperchaotic systems, where unstable-manifold and periodic-orbit constructions are impractical.
  • The thresholds used to define VLEZ (eigenvalue within $10^{-5}$ of zero, oscillation-to-zero ratio bounds) are empirical; a systematic characterization of when these thresholds fail—for example, the eigenvalue gap closing at higher dimensions—would sharpen the method's applicability.
  • Because GKA relies on fitting curves or surfaces and checking $R^2 \ge 0.9$, its success for a new system depends on local geometric regularity of the attractor; a testable extension would be applying GKA to a system whose attractor has cusps or non-smooth folds, where curve fitting would break down.
  • The paper's observation that an invalid LEZ (oscillation at the edge) indicates one side of the boundary suggests a possible iterative refinement strategy during data collection, turning VLEZ localization into an online detector for streaming time series.
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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

4 major / 6 minor

Summary. The paper proposes a data-driven method for constructing symbolic partitions of chaotic time series using left eigenfunctions of a finite-dimensional approximation of the Koopman operator (the KA method). The authors introduce 'Valid Left Eigenfunctions with Zero' (VLEZ), whose interior oscillations are claimed to localize the coarse symbolic boundary, and a 'Generalized Koopman Analysis' (GKA) that refines the boundary via affine transformations and localized basis functions. The method is demonstrated on the logistic and tent maps, a multimodal quartic map, two Hénon maps, the return map of a Duffing oscillator, and a three-dimensional hyperchaotic map, including noisy versions, and the results are compared with previously published symbolic boundaries in Table 1.

Significance. If the central claim holds, the paper offers a new operator-theoretic route to symbolic partitioning that extends the authors' earlier work (Ref. [49]) and is applicable to multivariate and noisy data. The main strength is the empirical agreement with independent literature values: the boundaries in Table 1 match Grassberger–Kantz, Hansen, and Giovannini–Politi results to within about 0.001–0.01, which is a meaningful quantitative validation. The method's reliance on Koopman eigenfunctions rather than unstable periodic orbits or stable/unstable manifolds is a refreshing alternative. However, the work does not provide machine-checked proofs, reproducible code, or a parameter-free derivation; the central mechanism is supported by a single worked example and a set of hand-set thresholds. The significance is therefore conditional on the generality of the VLEZ phenomenon and on the method's reproducibility as a fully specified algorithm.

major comments (4)
  1. [§2.2.2, Eq. (12)] The central claim that VLEZ oscillations localize the pre-image of the folding point is not established. The only derivation is a hand-computed M=4 tent map example (Eq. (12)), from which the authors infer a general mechanism. For the EDMD matrix in Eq. (7), a zero-eigenvalue left eigenvector satisfies a cancellation condition on the empirical transition matrix; it is not shown why such a signed measure must cancel exactly across the pre-image of the critical point, nor that a VLEZ with the required single interior oscillation exists at finite M for a given map. This is load-bearing because the entire KA/GKA method rests on this equivalence. The authors should either supply a proof for a class of piecewise-linear or smooth unimodal maps, or provide a systematic numerical study across many maps, basis sizes, and initial conditions showing that VLEZ consistently appears and localizes the boundary. Without this, the mechanism remains a conjecture.
  2. [§2.2.2, threshold definitions] The criteria for VLEZ selection are ad hoc: the eigenvalue tolerance |λ|<10^{-5}, the zero-valued-region tolerance of 1/10 of the oscillation extrema, and the tiny-zero-subregion threshold of 1/5 of the oscillation size are all hand-set. No sensitivity analysis is given to show that the results are stable with respect to these choices, and no principled reason is offered for why these particular values were selected. Since these thresholds are used to decide whether a given left eigenfunction qualifies as a VLEZ, they may be doing a significant amount of the work in matching the known boundaries. I ask the authors to report how the output boundaries in Table 1 vary when each threshold is changed by, say, a factor of two, and to justify the choices or replace them with an automatic selection rule.
  3. [§2.1, Eq. (9)] Equation (9), written as 'Ku Ũ = λ Ku', is dimensionally inconsistent: K is n×M, u is an M-dimensional left eigenvector, so Ku is n×1 while Ũ is M×M, and the product Ku Ũ is undefined. The intended relation is presumably K(u Ũ) = λ K u, but as written it suggests a confusion between left and right eigenvector conventions. This is not a mere typographical nuisance because it obscures the derivation of the left eigenfunction and should be corrected before publication.
  4. [§3.1 and §3.2] The method as described requires many manual interventions that are not specified as an algorithm: the text repeatedly states 'we find', 'we set', 'we choose', 'we shrink', and 'we separate into curved-type subregions', and the decisions rely on visual inspection of figures (e.g., 'we find that the continuity of the fractal region cannot be broken when M_x are reset to 1'). The choice of R²>0.9 for a 'curved-type region', the decision to use interpolation with n=100 points, and the selection of 'the complete folding region Part1S3' are not automated. This makes the method difficult to reproduce and weakens the claim that the authors 'propose a practical algorithm'. I strongly recommend that the authors provide a pseudocode or a step-by-step algorithmic description that removes (or explicitly parameterizes) these human choices, so that a reader could apply the method to a new dataset without reference to the expected answer.
minor comments (6)
  1. [Global] There are numerous typographical and grammatical errors, such as 'Egienfuction' in the Section 2.1 heading, 'Althougth' in the Introduction, and incomplete sentences in the abstract ('The limitations of these methods are heuristic and empirical for the partition the multivariate chaotic state space'). The paper needs a thorough language editing pass.
  2. [§3.2.2] The parameter is given as 'β=5.4' for the Hénon II map, but the standard value in the literature is β=0.54 (and Eq. (16) uses β=0.3 for Hénon I). This appears to be a typo that makes the comparison in Table 1 ambiguous; please correct it.
  3. [Table 1] The column structure of Table 1 is confusing: the headers list 'practical boundary' and 'theoretical boundary', but each of these is further split into two rows for the noiseless and noisy cases, and the reader must infer which entries correspond to which noise level. Please redesign the table with explicit row/column labels.
  4. [§3.2.1 and Fig. 5(c)] The notation 'Part1S1', 'Part1S2', 'Part1S3' for higher iterates is introduced without definition; the reader must deduce that S denotes the one-step evolved region and the number counts iterations. Please define this notation at first use.
  5. [References] Reference [20] is cited as 'Misha Chai', but the correct author name is M. S. Chai; also, the citation label in the text appears to be 'Chai MS, Lan YH'. Please standardize the citation format.
  6. [§3.2.4] In the discussion of the three-dimensional hyperchaotic map, the sentence 'the eigenvalue contains an imaginary part, indicating that a significant rotation occurred during the folding process' is not explained; an eigenvalue with an imaginary part is generic for a real matrix, so this observation does not by itself justify the interpretation. Please clarify or remove the speculation.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the central claim is validated against external benchmarks, and the self-citations are not load-bearing.

full rationale

The paper's central chain, from the EDMD left eigenvectors to the VLEZ oscillation and then to the symbolic boundary, is not circular by construction. The VLEZ selection criteria in Section 2.2.2 are operational (lambda approximately zero, oscillation in an inner subregion, tolerance thresholds) and do not use the known boundary values to select the eigenfunction; the claimed correspondence between the oscillation subregion and the pre-image of the folding point is presented as a conjecture inferred from the hand-computed tent map, not as a definition of the boundary. The final partitions are checked against independent published results (Grassberger and Kantz, Hansen, Giovannini and Politi) that are never used to set any parameter of the algorithm, so the comparisons are genuine external tests. The self-citations to the same group's earlier work [49] and [20] motivate the approach and provide a comparison, but the central mechanism is re-derived in the paper from the tent map example and does not reduce to those citations. I also noted that Eq. (9) has a dimension mismatch and that the threshold choices in Section 2.2.2 are ad hoc; these are correctness and reproducibility concerns, not circularity, and therefore do not raise the circularity score beyond a minor self-citation adjustment.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The method relies on several unproven or hand-set choices: the EDMD approximation of the Koopman operator, the heuristic interpretation of zero-eigenvalue left eigenfunctions as folding indicators, the ad hoc thresholds for VLEZ classification, and the affine-transformation assumption that local regions become unimodal. These are documented in Section 2 and Appendix B. No new physical entities are introduced; VLEZ and CLE are internal algorithmic constructs with no independent empirical handle.

free parameters (6)
  • eigenvalue tolerance for VLEZ = 1e-5
    Section 2.2.2: the module of eigenvalue of VLEZ can be permitted approximately equal to zero with an error within 10^-5 but ensure one of the three minimum eigenvalues.
  • zero-valued region tolerance = 1/10 of oscillation max/min
    Section 2.2.2: function value of zero-valued region can be permitted in scope of 1/10 of the maximum and minimum of oscillation subregion.
  • tiny zero-valued subregion size threshold = 1/5 of oscillation region
    Section 2.2.2: if the coordinate range of a tiny zero-valued subregion is less than one-fifth of that of the oscillation region, it may be allowed to appear within the oscillation subregion.
  • minimum state point count n = 100
    Section 2.2.2: 'we set the condition n >= 100' to ensure validity of basis functions and eigenfunctions.
  • maximum basis count M = 5n
    Section 2.2.2: 'we set the condition M <= 5n' to keep the number of basis functions smaller than the number of state points.
  • coefficient of determination threshold for curved regions = 0.9
    Appendix B: 'we determine whether the original region can be a curved region, with the constraint that the points of the original region and the fitted curve points have a coefficient of determination R^2 >= 0.9'.
assumptions (6)
  • domain assumption A finite-dimensional matrix U~ built from basis functions approximates the Koopman operator on the chaotic attractor.
    Section 2.1, Eq. (7): K U~ = L. This is the standard EDMD assumption; convergence for chaotic systems is not established here.
  • domain assumption Left eigenvectors of U~ approximate eigenfunctions of the Perron-Frobenius operator, and eigenvalue zero corresponds to density evolution that vanishes after one step.
    Section 2.1 Eqs. (8)-(9) and Section 2.2.2 discussion.
  • standard math The pre-image of the folding point is the symbolic partition boundary for 1D non-invertible maps.
    Section 2.2.1, citing ref. [16].
  • ad hoc to paper An oscillation (one positive and one negative subregion) in a zero-eigenvalue left eigenfunction indicates a folding event and locates the coarse symbolic boundary.
    Section 2.2.2: inferred from the tent map example, not proven.
  • domain assumption Affine transformation Eq. (13) preserves the folding structure so that KA applies to the localized transformed region.
    Section 2.2.3 and Appendix B.
  • domain assumption For multivariate maps, complete folding is achieved over multiple steps, and the one-step folding is 'primary' with local foldings as interference.
    Section 3.2, opening paragraphs.
invented entities (2)
  • Valid Left Eigenfunction with Zero (VLEZ)
    purpose: Identify the coarse symbolic boundary subregion from its localized oscillation.
    Defined and identified by the paper's own criteria (eigenvalue near zero, interior oscillation); no external falsifiable prediction.
  • Characteristic Left Eigenfunction (CLE)
    purpose: Estimate symbolic boundary when eigenvalue is near zero but not zero, enabling basis redistribution.
    Introduced as a practical fallback for multimodal and multivariate maps; no independent evidence.

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Cite this review

Pith. "Pith review of The symbolic partition with generalized Koopman analysis." pith.science (2026). https://pith.science/paper/CTATGNRW

@misc{pith2026250606973,
  author       = {Pith},
  title        = {Pith review of: The symbolic partition with generalized Koopman analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CTATGNRW}},
  note         = {Machine review of arXiv:2506.06973}
}
read the original abstract

Symbolic dynamics serves as a crucial tool in the study of chaotic systems, prompting extensive research into various methods for symbolic partitioning. The limitations of these methods are heuristic and empirical for the partition the multivariate chaotic state space. Notably, the use of operator theory in partitioning the multivariable chaotic series into precise symbolic cells has been underexplored. In this paper, we propose a novel symbolic partition method, referred to as Koopman Analysis(KA) method, exploiting Koopman operator theory to address the symbolic partition, especially multivariate chaotic time series. We map the chaotic time series into the basis functions to obtain the approximate representation of the Koopman operator.Then we transpose the Koopman approximate matrix and subsequently perform spectral decomposition to obtain the Koopman left eigenfunctions. We apply KA method to one-dimensional unimodal chaotic map to obtain Koopman left eigenfunctions. Then we find some particular eigenfunctions whose eigenvalues are zero, some of which can be used to identify the symbolic boundary of region composed of chaotic series in that the oscillation coincides with the subregion where the theoretical symbolic boundary is located. We refer to the function as the Valid Left Eigenfunction with Zero(VLEZ). Based on the number of oscillations, we further classify VLEZ into two categories. Then, we modify the KA method applicable to chaotic localized subregion and further propose the Generalized Koopman Analysis (GKA) method. The KA method can be also applied to the multimodal maps, multivariate chaotic maps and hyperchaotic maps and their noisy version. The current work can be well further expand to higher dimensional and more complex time series due to its interpretability and availability.

Figures

Figures reproduced from arXiv: 2506.06973 by the authors.

Figure 1
Figure 1. The graph relating 𝐱 to 𝐱p of chaotic unimodal map Eq.(10) with 𝛼=3.75 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. The four basis functions 𝐾 and evolved functions 𝐿 and three typical left eignfunctions of the Koopman operator 𝑈 from the tent map Eq.(11). Haipeng Li et al.: Preprint submitted to Elsevier Page 15 of 14 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. The KA and GKA methods for the chaotic unimodal map. (a) One MVLEZ in the whole region via KA method. (b) The UVLEZ in the localized region via GKA method. 0.4 0.6 0.8 x n 0.2 0.4 0.6 0.8 1 x n+1 (a) =-1.7539e-17 M=11 n=10000 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 critical point 0.4 0.6 0.8 -0.5 0 0.5 =0.012915 M=10 (b) =-5.1254e-18 M=30 n=10000 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 critical point (c) (d) [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The basis functions of the chaotic map governed by the Eq.(15) and KA method for the chaotic multimodal map. (a) The graph relating 𝐱 to 𝐱p of chaotic map Eq.(15) (b) The UVLEZ to obtain the first symbolic boundary. (c) The MVLEZ to obtain two symbolic boundaries. (d) …
Figure 5
Figure 5. Figure 5: The KA method on the series governed by the Hénon I map (a)The CLE for primary symbolic boundary (b)The VLEZ for primary symbolic boundary (c)The multiple-step evolution of Part1 (d)The eventually symbolic boundary and symbolic subregions Haipeng Li et al.: Preprint su…
Figure 6
Figure 6. Figure 6: The KA method for the series governed by the Hénon II map (a) The CLE via KA method for primary symbolic boundary. (b) The VLEZ via KA method for primary symbolic boundary. (c) The three modified subregions based on the coarse boundary. (d)The eventually refined symbol…
Figure 7
Figure 7. Figure 7: The symbolic partition of the series from the Duffing oscillator return map Eq.(18) via KA method. (a) The CLE via KA method for primary symbolic boundary. (b)-(c) The VLEZ via KA method for different two primary symbolic boundaries. (d) Eventual refined symbolic bound…
Figure 8
Figure 8. Figure 8: The symbolic partition of the 3-dimensional series from the map Eq.(19) (a) The CLE via KA method for primary symbolic boundary. (b)-(c) The VLEZ to obtain different two primary symbolic boundaries (d) Eventual refined symbolic boundary 0.84 0.86 0.88 0.9 0.92 0.94 x n…

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