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REVIEW 3 major objections 5 minor 93 references

Learning a quantitative criterion for distinguishing chaos from noise

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single cross-prediction score separates chaotic time series from noisy ones.

desk verdict A useful heuristic for chaos/noise in the tested regimes, but the claimed quantitative threshold does not survive a simple AR(1) counterexample; worth reviewing, needs major revision. read the letter →

arxiv 2608.07109 v1 pith:GST64RZH submitted 2026-08-07 nlin.CD cond-mat.stat-mech

Jaesung Choi , Athokpam Langlen Chanu , Jong-Min Park This is my paper · ORCID
classification nlin.CDcond-mat.stat-mech PACS 05.45.-a
keywords chaosvsnoisereservoircomputingechostatenetworkcross-predictionsquaredPearsoncorrelationtimeseriesclassificationdeterministicstochasticrobustnessanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that one number computed from the data, the squared Pearson correlation $R^2$ between the true future change of a variable and the future change predicted by a reservoir computer, cleanly separates deterministic chaos from stochastic noise. The task is a cross-prediction: the model receives the current value $x_t$ and must predict the next difference $\Delta x_{t+1}$, combining short-term predictability with a test of whether a smooth deterministic rule connects consecutive observations. The authors show that for seven chaotic systems and six noise processes, $R^2$ is close to one for chaos and below roughly 0.1 for noise, and they adopt $R^2 = 0.5$ as an operational threshold. If the claim holds, it gives a purely data-driven classification that needs no embedding parameters, no Lyapunov estimates, and no prior assumptions about the generating system.

What carries the argument

The load-bearing element is the reservoir-computing cross-prediction task with a value-to-difference scheme. An echo state network with a fixed recurrent weight matrix and fixed random input weights evolves a reservoir state $r_t = (1-a_{\rm leak})\,r_{t-1} + a_{\rm leak}\,\tanh(W_{\rm res} r_{t-1} + W_{\rm in} x_t)$; only the linear readout $\hat{y}_t = W_{\rm out}\,[r_t;1]$ is trained by ridge regression, with target $y_t = \Delta x_{t+1} = x_{t+2} - x_{t+1}$. Because the reservoir keeps memory of past inputs, it builds an implicit delay-coordinate representation without explicit embedding, and because the readout is linear and the reservoir fixed, the model has a restricted inductive bias against memorizing stochastic fluctuations. The squared Pearson correlation $R^2$ between the true and predicted differences, averaged over five contiguous validation folds, is the resulting classifier statistic.

What would settle it

Run the cross-prediction pipeline on a long realization of a stochastic process deliberately built to be linearly readable from a fixed recurrent reservoir, such as $x_{t+1} = f(x_t) + \eta_t$ where $f$ is drawn from the reservoir's own function class and $\eta_t$ is i.i.d.; if the resulting $R^2$ exceeds 0.5, the binary threshold misclassifies noise. A less elaborate check is to apply the method to phase-randomized or shuffled surrogates of a known chaotic series: if the surrogates keep $R^2 > 0.5$, the criterion fails to detect the loss of determinism.

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Extended reading notes

Core claim

The central discovery is that the cross-prediction score, not the prediction itself, is the discriminator. Chaotic dynamics have enough learnable deterministic structure that a fixed random recurrent reservoir with only a linear readout can map the current state to the next increment; noise processes, including strongly correlated flicker noise, fractional Gaussian noise, and infinite-variance Cauchy noise, cannot be fit this way. The squared Pearson correlation between predicted and true first differences therefore lands near unity for chaos and near zero for noise, with a clean gap. The authors validate the criterion on empirical records as well: voice, laser pulsation, squid giant axon membrane potential, and an experimental Chua circuit fall above the threshold, while the North Atlantic Oscillation index, sunspot number, and RR-interval series fall below. Parkinsonian tremor, whose classification has been debated, sits just on the chaos side with an intermediate score, which the authors interpret as mixed stochastic-deterministic dynamics.

Load-bearing premise

The method assumes that a noise process cannot be mimicked by a linear readout of a fixed recurrent network; if some stochastic process with memory can be approximated that way, its cross-prediction score would wrongly land in the chaos regime.

Editorial extensions

If this is right

  • A single scalar time series can be classified as chaos or noise by computing one cross-prediction $R^2$ and comparing it with 0.5, with no embedding dimension or delay chosen by hand.
  • The criterion tolerates finite levels of measurement noise: $R^2$ declines continuously with the noise-mixing coefficient $\alpha$, so the classification changes only when deterministic predictability is substantially destroyed.
  • Short records of a few hundred samples usually suffice: synthetic chaos and noise are classified consistently at all tested lengths, and empirical records stabilize for $L_{\rm given} \ge 2000$.
  • The score can serve as a continuous measure of how strongly observed dynamics support a deterministic description, not only as a binary label.
  • The cross-prediction strategy can be transplanted to other recurrent neural network architectures, since only the readout is trained.

Reading between the lines

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

  • A natural but untested extension is to apply the same scalar $R^2$ threshold to regime-switching or intermittently chaotic data, where the score would presumably interpolate between the deterministic and stochastic regimes.
  • Because the reservoir implicitly builds an embedding, the method may be sensitive to sampling rate; the authors' observation that downsampling SGAMP improved $R^2$ suggests temporal resolution itself can shift the classification, which deserves systematic testing.
  • The threshold 0.5 is operational rather than derived from first principles; the wide gap seen in synthetic data (near unity versus below 0.1) suggests a more conservative threshold could be more robust on mixed empirical records.
  • A practical tool could combine the cross-prediction $R^2$ with surrogate-data tests: if phase-randomized or shuffled surrogates of the same series also exceed the threshold, the deterministic label would be suspect.
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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 / 5 minor

Summary. The paper proposes a reservoir-computing cross-prediction scheme for distinguishing deterministic chaos from stochastic noise from scalar time series. An echo state network is trained to predict the first difference Δx_{t+1}=x_{t+2}-x_{t+1} from the current value x_t, and the squared Pearson correlation R² between true and predicted differences is proposed as a classifier: R²>0.5 is labeled chaos and R²<0.5 noise. The method is applied to seven synthetic chaotic systems, six synthetic noise processes, and several empirical datasets, with additional robustness experiments varying additive noise, time-series length, and prediction lag.

Significance. The cross-prediction idea is attractive: it combines short-term predictability with a deterministic-flow test, and the paper presents a broad, clearly described validation across synthetic and empirical data. The reservoir implementation is transparent and the pseudo-code in Algorithm 1 makes the procedure reproducible. The robustness analyses in Sec. IV (noise mixing, finite length, prediction lag) are a strength and give useful practical information. If the universal R²>0.5 threshold were valid, this would be a practical advance over embedding-based methods. However, the threshold is not universal: a simple AR(1) noise process violates it, and the reported R² values are inflated by per-dataset hyperparameter optimization and by selecting the top 100 of 200 reservoir realizations. The paper is better read as a promising heuristic with benchmark-specific separation than as a quantitative law.

major comments (3)
  1. [Sec. IIA, Eq. (3), Fig. 6] The proposed criterion R²>0.5 ⇒ chaos is violated by a stationary AR(1) process x_{t+1}=φx_t+ε_t with negative autocorrelation. The conditional expectation of Δx_{t+1}=x_{t+2}-x_{t+1} given x_t is φ(φ-1)x_t, and the squared Pearson correlation between Δx_{t+1} and this optimal predictor is R²=φ²(1-φ)/2. For φ=-0.8 this gives R²≈0.576>0.5, and for φ=-0.9 it gives ≈0.77. This is a purely stochastic, stationary linear process with no deterministic flow, so the theoretical optimum of the cross-prediction task exceeds the paper's chaos threshold. The noise set in Sec. IIIA and Fig. 6 contains no negatively autocorrelated process, so the observed separation does not establish the claimed generality. Since the reservoir state can represent x_t (e.g., in the linear regime of tanh) and the readout is linear, an ESN can in principle realize this optimal predictor; the paper's hyperparameter optimization and realization selection would only push R² upward. The abstract's statement that R² provides a quantitative criterion for distinguishing chaos from noise is therefore not supported as stated.
  2. [Sec. VI, Algorithm 1, lines 16-18] The reported R² is computed after choosing the best hyperparameters on each dataset using the same CV5 folds, and the final score is the mean over the top 100 of 200 reservoir realizations ranked by validation RMSE. Selecting realizations by validation RMSE is a form of peeking at the validation set; it biases R² upward, most strongly for noise processes where many realizations have near-zero skill and only a few appear predictive by chance. The threshold R²=0.5 is then chosen after observing the distributions in Figs. 6 and 9. Thus the 'quantitative criterion' is partly a post-hoc fitted decision boundary applied to an optimized score, not a fixed, parameter-free law. The authors should report R² without the top-100 selection, or use nested cross-validation, and should describe how the threshold was selected before seeing the data.
  3. [Sec. VIB, Tables IV-VI] All six reservoir hyperparameters are optimized separately for each dataset, prediction scheme, and lag on the same data subsequently used for evaluation, with no independent test set. Because the per-dataset optimization can exploit chance regularities in each realization, the reported R² values are not directly comparable across datasets, and the separation in Fig. 6 may be optimistically biased. To support a universal quantitative criterion, the authors should validate with a fixed or lightly tuned hyperparameter set, or with nested cross-validation that leaves untouched data for final evaluation. This is a load-bearing issue because the paper's main claim is a single threshold that holds across all datasets, not merely a demonstration that some tuning can separate benchmark examples.
minor comments (5)
  1. [Eq. (3)] The notation R² for the squared Pearson correlation is easily confused with the coefficient of determination; please add an explicit sentence distinguishing these quantities.
  2. [Sec. IIA] The statement that the 'restricted inductive bias' of a fixed reservoir and linear readout reduces memorization of stochastic fluctuations is an empirical assumption, not a theorem; the AR(1) counterexample in the major comments shows that this bias does not automatically exclude stochastic processes from high R².
  3. [Fig. 6] The synthetic noise set includes fBm, which is non-stationary, alongside stationary processes; the paper should state whether the proposed criterion is intended to apply to non-stationary noise, since fBm has different properties from the other noise benchmarks.
  4. [Table I] The final column '≥2000' summarizes the results for L_given=2000, 5000, and 10000, but the text does not specify whether the hyperparameters were re-optimized at each length; please clarify.
  5. [Appendix B] The optimized hyperparameter tables would be more useful if accompanied by the corresponding achieved R² values, so readers can see the variability across the 200 reservoir realizations rather than only the top-100 summary.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity: R^2 is a genuinely out-of-sample cross-validation score, and the 0.5 threshold is an explicit operational rule rather than a derived or fitted parameter.

full rationale

The paper's central quantity, R^2, is computed on held-out validation folds after training the linear readout on the remaining folds, so it is not equal to a training objective by construction. The threshold 0.5 is introduced explicitly as an operational classification rule ('we operationally classify time series with R^2 > 0.5 as chaos and those with R^2 < 0.5 as noise'), not as a theorem derived from first principles. The fixed-reservoir/linear-readout restriction is stated as an architectural assumption, and the empirical separation between chaotic systems and noise processes is presented as a benchmark demonstration rather than as a mathematically forced consequence. The only self-citations in the paper (Refs. [61] and [62], both by the first author) appear in the introduction as background on reservoir computing recovering chaotic attractors under noise; they are not used as load-bearing justification for the chaos/noise criterion. The skeptical AR(1) counterexample is a potential false-positive limitation of the empirical generalization, but it does not show that the paper's R^2 values reduce to its inputs by construction. Hyperparameter optimization on the same datasets and top-100 realization selection may inflate reported R^2 values, but this is a statistical optimism concern, not a definitional circularity. No equation in the paper defines the criterion in terms of itself, and no fitted parameter is renamed as a prediction. Hence no significant circularity is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the inductive-bias assumption of the reservoir, the representational sufficiency of the recurrent state, the representativeness of the tested noise set, and the hand-chosen threshold. No new entities are introduced.

free parameters (3)
  • Classification threshold R^2 = 0.5 = 0.5
    Chosen as a round decision boundary between observed chaos (R^2 near 1) and noise (R^2 < 0.1). The paper states 'we operationally classify time series with R^2 > 0.5 as chaos' (Sec. IIA).
  • Reservoir hyperparameters (n, rho, sigma_in, a_leak, beta, k) = Varies per dataset; see Tables IV-VI
    Tuned via surrogate-based optimization on CV5 RMSE for each dataset, so they are fitted to the same data used for evaluation.
  • Number of top reservoir realizations reported = 100 out of 200
    R^2 is reported from the top 100 realizations ranked by validation RMSE, which is a selection rule that biases scores upward (Sec. VI.B).
assumptions (4)
  • domain assumption The reservoir state provides a sufficient representation of the underlying dynamics from scalar input without explicit delay-coordinate embedding.
    Invoked in Sec. IIA: 'the reservoir naturally retains information from previous inputs, enabling it to construct an effective representation of the hidden dynamics without explicit phase-space reconstruction.'
  • domain assumption The fixed reservoir and linear readout impose a restricted inductive bias that reduces memorization of stochastic fluctuations.
    Sec. IIA: this is the stated mechanism for why noise yields low R^2; not proven.
  • domain assumption The tested noise processes (Gaussian, uniform, Cauchy, flicker, fBm, fGn) are representative of 'noise' for the proposed criterion.
    Sec. IIIA: the claim that noise stays in a low-correlation regime is based on these six processes.
  • standard math Standard ridge regression and Pearson correlation are appropriate tools for the prediction and scoring tasks.
    Used in Sec. VI; uncontroversial.

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Cite this review

Pith. "Pith review of Learning a quantitative criterion for distinguishing chaos from noise." pith.science (2026). https://pith.science/paper/GST64RZH

@misc{pith2026260807109,
  author       = {Pith},
  title        = {Pith review of: Learning a quantitative criterion for distinguishing chaos from noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GST64RZH}},
  note         = {Machine review of arXiv:2608.07109}
}
read the original abstract

Distinguishing chaos from noise using time-series data is fundamentally challenging because both exhibit irregular fluctuations and share many statistical and dynamical characteristics. Existing methods face two key limitations: temporally correlated noise can yield spurious signatures of chaos, and analyses of scalar time series often require explicit choices of embedding parameters. Here, we propose a purely data-driven method for distinguishing chaos and noise based on a reservoir-computing framework with a cross-prediction scheme. In the proposed approach, the model is trained to predict the future change of a variable from its current value, thereby combining a short-term predictability test with a test of the smoothness of deterministic flows. The recurrent structure of reservoir computing enables effective prediction of high-dimensional chaotic dynamics even from scalar time series without explicit delay-coordinate reconstruction, while the cross-prediction framework strongly suppresses spurious predictive correlations arising from noise. We apply the proposed method to diverse synthetic and empirical time series. Chaotic systems consistently yield strong correlations between the true and predicted future changes, whereas noise processes remain clearly separated in a low-correlation regime. The method also exhibits substantial robustness to practical limitations in empirical data, including measurement noise, limited data length, and increasing prediction lag. These results demonstrate that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.

Figures

Figures reproduced from arXiv: 2608.07109 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Scatter plot of the true and predicted values obtained [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scatter plots obtained using the conventional [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of ∆ [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 9
Figure 9. Figure 9: presents R2 values for empirical time series. The obtained R2 values demonstrate a consistent overall tendency, albeit with a less clear separation than in the model-generated time series, which is likely attributable to measurement noise, finite resolution, and other …
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: shows R2 as a function of Lgiven for all dataset categories. For synthetic chaos in panel (a), R2 remains close to unity across nearly all time series lengths. Only at Lgiven = 100 do a few systems start lower, around 0.8, before recovering to near unity by Lgiven = 2…
Figure 12
Figure 12. Figure 12: FIG. 12. Comparison of the first 1 [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

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Reference graph

Works this paper leans on

93 extracted references · 79 canonical work pages

  1. [1]

    The syn- thetic chaotic datasets comprise both discrete-time and continuous-time systems

    Synthetic chaotic systems All synthetic time series used in our analysis have a total length ofL given = 10,000 samples. The syn- thetic chaotic datasets comprise both discrete-time and continuous-time systems. Discrete-time systems are it- erated directly. Continuous-time systems are integrated numerically using a fourth-order Runge–Kutta method with a f...

  2. [2]

    •Gaussian noise (Gauss):An i.i.d

    Synthetic noise processes We generate time series from the following noise pro- cesses with total lengthL given = 10,000 samples. •Gaussian noise (Gauss):An i.i.d. sequence drawn from the standard normal distribution,Xt ∼ N(0,1). •Uniform noise (Uniform):An i.i.d. sequence drawn from a uniform distribution on (0,1),X t ∼ U(0,1) with a flat probability den...

  3. [3]

    Unless otherwise stated below, the empirical datasets are analyzed at their full available lengths, with the normalization in Eq

    Empirical datasets We evaluate the proposed method on a collection of empirical time series data spanning biomedical, physical, and geophysical systems. Unless otherwise stated below, the empirical datasets are analyzed at their full available lengths, with the normalization in Eq. (5) as the only preprocessing step. The specific time series lengthL given...

    1937

  4. [4]

    Bradley and H

    E. Bradley and H. Kantz, Chaos25, 097610 (2015)

    2015

  5. [5]

    Kantz and T

    H. Kantz and T. Schreiber,Nonlinear time series analysis (Cambridge university press, 2003)

    2003

  6. [6]

    M. B. Kennel and S. Isabelle, Phys. Rev. A46, 3111 (1992)

    1992

  7. [7]

    O. A. Rosso, H. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Phys. Rev. Lett.99, 154102 (2007)

    2007

  8. [8]

    D. J. Wales, Nature350, 485 (1991)

    1991

Show all 93 references
  1. [9]

    Grassberger and I

    P. Grassberger and I. Procaccia, Phys. Rev. Lett.50, 346 (1983)

    1983

  2. [10]

    Grassberger and I

    P. Grassberger and I. Procaccia, Physica D9, 189 (1983)

    1983

  3. [11]

    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D16, 285 (1985)

    1985

  4. [12]

    Theiler, Phys

    J. Theiler, Phys. Lett. A155, 480 (1991)

    1991

  5. [13]

    Grassberger and I

    P. Grassberger and I. Procaccia, Phys. Rev. A28, 2591 (1983)

    1983

  6. [14]

    Eckmann and D

    J.-P. Eckmann and D. Ruelle, Physica D56, 185 (1992)

    1992

  7. [15]

    A. R. Osborne and A. Provenzale, Physica D35, 357 (1989)

    1989

  8. [16]

    Provenzale, A

    A. Provenzale, A. R. Osborne, and R. Soj, Physica D47, 361 (1991)

    1991

  9. [17]

    D¨ ammig and F

    M. D¨ ammig and F. Mitschke, Phys. Lett. A178, 385 (1993)

    1993

  10. [18]

    Tanaka, K

    T. Tanaka, K. Aihara, and M. Taki, Phys. Rev. E54, 2122 (1996)

    1996

  11. [19]

    Ikeguchi and K

    T. Ikeguchi and K. Aihara, Int. J. Bifurcation Chaos7, 1267 (1997)

    1997

  12. [20]

    Luque, L

    B. Luque, L. Lacasa, F. J. Ballesteros, and A. Robledo, Chaos22, 013109 (2012)

    2012

  13. [21]

    Barahona and C.-S

    M. Barahona and C.-S. Poon, Nature381, 215 (1996)

    1996

  14. [22]

    Lacasa and R

    L. Lacasa and R. Toral, Phys. Rev. E82, 036120 (2010)

    2010

  15. [23]

    Luque, L

    B. Luque, L. Lacasa, F. J. Ballesteros, and A. Robledo, PLoS One6, e22411 (2011)

    2011

  16. [24]

    Zhang and M

    J. Zhang and M. Small, Phys. Rev. Lett.96, 238701 (2006)

    2006

  17. [25]

    M. G. Ravetti, L. C. Carpi, B. A. Gon¸ calves, A. C. Frery, and O. A. Rosso, PLoS One9, e108004 (2014)

    2014

  18. [26]

    Poon and M

    C.-S. Poon and M. Barahona, Proc. Natl. Acad. Sci. U.S.A.98, 7107 (2001)

    2001

  19. [27]

    Zhang, X

    J. Zhang, X. Luo, and M. Small, Phys. Rev. E73, 016216 (2006)

    2006

  20. [28]

    Zanin, Commun

    M. Zanin, Commun. Nonlinear Sci. Numer. Simul.114, 106708 (2022)

    2022

  21. [29]

    D. M. Wolpert and R. Miall, Proc. R. Soc. London, Ser. B242, 82 (1990)

    1990

  22. [30]

    J. B. Borges, H. S. Ramos, R. A. Mini, O. A. Rosso, A. C. Frery, and A. A. Loureiro, Appl. Math. Comput. 362, 124554 (2019)

    2019

  23. [31]

    Boaretto, R

    B. Boaretto, R. C. Budzinski, K. L. Rossi, T. L. Prado, S. R. Lopes, and C. Masoller, Sci. Rep.11, 15789 (2021)

    2021

  24. [32]

    Zunino, M

    L. Zunino, M. C. Soriano, and O. A. Rosso, Phys. Rev. E86, 046210 (2012)

    2012

  25. [33]

    T. L. Prado, B. R. R. Boaretto, G. Corso, G. Z. dos Santos Lima, J. Kurths, and S. R. Lopes, New J. Phys. 24, 033027 (2022)

    2022

  26. [34]

    J. V. V. Flauzino, T. L. Prado, N. Marwan, J. Kurths, and S. R. Lopes, Phys. Rev. Lett.135, 097401 (2025)

    2025

  27. [35]

    Olivares, A

    F. Olivares, A. Plastino, and O. A. Rosso, Phys. Lett. A 376, 1577 (2012)

    2012

  28. [36]

    J. M. Amig´ o, S. Zambrano, and M. A. Sanju´ an, EPL79, 50001 (2007)

    2007

  29. [37]

    Quintero-Quiroz, S

    C. Quintero-Quiroz, S. Pigolotti, M. Torrent, and C. Ma- soller, New J. Phys.17, 093002 (2015)

    2015

  30. [38]

    Zanin and F

    M. Zanin and F. Olivares, Commun. Phys.4, 190 (2021)

    2021

  31. [39]

    J. M. Amig´ o, L. Kocarev, and J. Szczepanski, Phys. Lett. A355, 27 (2006)

    2006

  32. [40]

    Kulp and L

    C. Kulp and L. Zunino, Chaos24, 033116 (2014)

    2014

  33. [41]

    J. M. Amig´ o, S. Zambrano, and M. A. Sanju´ an, EPL83, 60005 (2008)

    2008

  34. [42]

    L. C. Carpi, P. M. Saco, and O. A. Rosso, Physica A 389, 2020 (2010)

    2010

  35. [43]

    O. A. Rosso, L. C. Carpi, P. M. Saco, M. G. Ravetti, A. Plastino, and H. A. Larrondo, Physica A391, 42 (2012)

    2012

  36. [44]

    Tsonis and J

    A. Tsonis and J. Elsner, Nature358, 217 (1992)

    1992

  37. [45]

    U. S. Freitas, C. Letellier, and L. A. Aguirre, Phys. Rev. E79, 035201 (2009)

    2009

  38. [46]

    J. Gao, J. Hu, X. Mao, and W.-w. Tung, Chaos Solitons Fractals45, 213 (2012)

    2012

  39. [47]

    Sugihara and R

    G. Sugihara and R. M. May, Nature344, 734 (1990)

    1990

  40. [48]

    L. W. Salvino and R. Cawley, Phys. Rev. Lett.73, 1091 (1994)

    1994

  41. [49]

    D. T. Kaplan and L. Glass, Phys. Rev. Lett.68, 427 (1992)

    1992

  42. [50]

    Wayland, D

    R. Wayland, D. Bromley, D. Pickett, and A. Passamante, Phys. Rev. Lett.70, 580 (1993)

    1993

  43. [51]

    D. T. Kaplan and L. Glass, Physica D64, 431 (1993)

    1993

  44. [52]

    Takens, inDynamical Systems and Turbulence, War- wick 1980, Lecture Notes in Mathematics, Vol

    F. Takens, inDynamical Systems and Turbulence, War- wick 1980, Lecture Notes in Mathematics, Vol. 898, edited by D. Rand and L.-S. Young (Springer, Berlin, Heidelberg, 1981) pp. 366–381

    1980

  45. [53]

    G. J. Ortega and E. Louis, Phys. Rev. Lett.81, 4345 (1998)

    1998

  46. [54]

    Ikeguchi and K

    T. Ikeguchi and K. Aihara, Phys. Rev. E55, 2530 (1997)

    1997

  47. [55]

    Jeong, M

    J. Jeong, M. S. Kim, and S. Y. Kim, Phys. Rev. E60, 831 (1999)

    1999

  48. [56]

    Z. Lu, B. R. Hunt, and E. Ott, Chaos28, 061104 (2018)

    2018

  49. [57]

    Jaeger and H

    H. Jaeger and H. Haas, Science304, 78 (2004)

    2004

  50. [58]

    Pathak, Z

    J. Pathak, Z. Lu, B. R. Hunt, M. Girvan, and E. Ott, Chaos27, 121102 (2017)

    2017

  51. [59]

    Pathak, B

    J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, Phys. Rev. Lett.120, 024102 (2018)

    2018

  52. [60]

    Zhai, L.-W

    Z.-M. Zhai, L.-W. Kong, and Y.-C. Lai, Phys. Rev. Res. 5, 033127 (2023)

    2023

  53. [61]

    Z. Lu, J. Pathak, B. Hunt, M. Girvan, R. Brockett, and E. Ott, Chaos27, 041102 (2017)

    2017

  54. [62]

    Margazoglou and L

    G. Margazoglou and L. Magri, Nonlinear Dyn.111, 8799 (2023)

    2023

  55. [63]

    H. Fan, J. Jiang, C. Zhang, X. Wang, and Y.-C. Lai, Phys. Rev. Res.2, 012080 (2020)

    2020

  56. [64]

    Gantert, J

    C. Gantert, J. Honerkamp, and J. Timmer, Biol. Cybern. 66, 479 (1992)

    1992

  57. [65]

    Choi and P

    J. Choi and P. Kim, Chaos35, 083136 (2025)

    2025

  58. [66]

    Choi and P

    J. Choi and P. Kim, Knowl.-Based Syst.330, 114574 (2025)

    2025

  59. [67]

    Cazelles and R

    B. Cazelles and R. Ferriere, Nature355, 25 (1992)

    1992

  60. [68]

    Langarica and F

    S. Langarica and F. N´ u˜ nez, Eng. Appl. Artif. Intell.120, 105838 (2023)

    2023

  61. [69]

    SadeghiRazlighi, A

    M. SadeghiRazlighi, A. H. Jafari, S. M. Firoozabadi, and G. A. Shahidi, Australas. Phys. Eng. Sci. Med.35, 25 (2012)

    2012

  62. [70]

    Gao and W.-w

    J. Gao and W.-w. Tung, Biol. Cybern.86, 263 (2002)

    2002

  63. [71]

    Sarbaz and H

    Y. Sarbaz and H. Pourakbari, Biomed. Signal Process. Control61, 102040 (2020). 18

    2020

  64. [72]

    J. R. M. Hosking, Water Resour. Res.20, 1898 (1984)

    1984

  65. [73]

    Y. Liu, Y. Zhou, K. Yang, and X. Wang, IEEE Internet Things J.10, 14285 (2023)

    2023

  66. [74]

    M. L. Heltberg, S. Krishna, and M. H. Jensen, Nat. Com- mun.10, 71 (2019)

    2019

  67. [75]

    Timmer and M

    J. Timmer and M. K¨ onig, Astron. Astrophys.300, 707 (1995)

    1995

  68. [76]

    Santa Fe time series competition, dataset a: Fluctuations in a far-infrared laser,https://www.comp-engine.org/ browse/category/real/physics/laser, (accessed June 2026)

    2026

  69. [77]

    Cesari, G

    U. Cesari, G. De Pietro, E. Marciano, C. Niri, G. San- nino, and L. Verde, Comput. Electr. Eng.68, 310 (2018)

    2018

  70. [78]

    Herzel, D

    H. Herzel, D. Berry, I. R. Titze, and I. Steinecke, Chaos 5, 30 (1995)

    1995

  71. [79]

    Tao and J

    C. Tao and J. J. Jiang, Phys. Rev. E77, 061922 (2008)

    2008

  72. [80]

    A. Mees, K. Aihara, M. Adachi, K. Judd, T. Ikeguchi, and G. Matsumoto, Phys. Lett. A169, 41 (1992)

    1992

  73. [81]

    H¨ ubner, N

    U. H¨ ubner, N. B. Abraham, and C. O. Weiss, Physical Review A40, 6354 (1989)

    1989

  74. [82]

    Paydarfar, D

    D. Paydarfar, D. B. Forger, and J. R. Clay, J. Neuro- physiol.96, 3338 (2006)

    2006

  75. [83]

    Aihara and G

    K. Aihara and G. Matsumoto, inChaos, edited by A. V. Holden (Manchester University Press and Princeton Uni- versity Press, Manchester and Princeton, NJ, 1986)

    1986

  76. [84]

    Timmer, S

    J. Timmer, S. H¨ außler, M. Lauk, and C.-H. L¨ ucking, Chaos10, 278 (2000)

    2000

  77. [85]

    T. L. Prado, V. S. Machado, G. Corso, G. Z. d. S. Lima, and S. R. Lopes, Parameter free determination of opti- mum time delay (2020), arXiv:2010.02951

    2020 arXiv

  78. [86]

    L. O. Chua, J. Circuits Syst. Comput.4, 117 (1994)

    1994

  79. [87]

    Tremor dataset,https://jeti.uni-freiburg.de/path_ tremor/readme, (accessed June 2026)

    2026

  80. [88]

    Glass, Chaos19, 028501 (2009)

    L. Glass, Chaos19, 028501 (2009)

    2009

  81. [89]

    gov/products/precip/CWlink/pna/nao.shtml, accessed June 2026

    NOAA climate prediction center: Daily North Atlantic Oscillation (NAO) index,https://www.cpc.ncep.noaa. gov/products/precip/CWlink/pna/nao.shtml, accessed June 2026

    2026

  82. [90]

    Toker, F

    D. Toker, F. T. Sommer, and M. D’Esposito, Commun. Biol.3, 11 (2020)

    2020

  83. [91]

    SILSO world data center: Sunspot number archive, http://www.sidc.be/silso/infosndtot, (accessed June 2026)

    2026

  84. [93]

    PhysioNet chaos challenge: Is the normal heart rate chaotic?,https://archive.physionet.org/challenge/ chaos/, (accessed June 2026)

    2026

  85. [1950]

    •Sunspot number (Sunspot):We use the daily total sunspot number (V2.0) from the Sunspot In- dex and Long-term Solar Observations (SILSO) repository [87]

    A previous study reported a stochastic origin for these data [86]. •Sunspot number (Sunspot):We use the daily total sunspot number (V2.0) from the Sunspot In- dex and Long-term Solar Observations (SILSO) repository [87]. To avoid missing values in the early records, we use the...

    1970

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Reviewed August 10, 2026 · model on record in the stance chip above.