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Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles

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

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Pith's one-line read A k-nearest-neighbor forecast-error slope, evaluated at period-matched horizons, estimates the dominant negative Lyapunov exponent directly from short trajectory ensembles without equations or a Jacobian.

desk verdict A plausible equation-free negative-LLE estimator built on an asserted contraction premise; deserves a rigorous referee, not a desk reject. read the letter →

arxiv 2608.05522 v1 pith:6XPPXVXU submitted 2026-08-06 nlin.CD cs.LGphysics.data-an

Andrei Velichko , N'Gbo N'Gbo , Viet-Thanh Pham This is my paper · ORCID
classification nlin.CDcs.LGphysics.data-an MSC 37D2537M10
keywords negativeLyapunovexponentforecast-errorcontractionk-nearest-neighborpredictionshorttrajectoryensemblesperiodsynchronizationstableperiodicdynamicsequation-freeestimation
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 contraction, not just divergence, can be measured directly from data: in a stable regime, the geometric-mean error of a k-nearest-neighbor predictor decays exponentially with forecast horizon, and the decay rate is the dominant negative Lyapunov exponent. The proposed estimator trains the predictor on ensembles of short scalar trajectories, evaluates errors at horizons synchronized with the detected orbit period, and fits the logarithmic error profile. It accepts a slope only when several transient lengths agree, which guards against mistaking early transient decay for the asymptotic contraction rate. On the logistic map it recovers 92 of 112 negative-exponent parameter values with mean absolute error 0.0253 and $R^2=0.886$; on a two-dimensional no-fixed-point map, three independently processed scalar observables give MAE 0.00879–0.01145 and $R^2=0.983$–$0.986$. The significance, if true, is that negative Lyapunov exponents become available in settings where only repeated recovery responses are observed and no equations or Jacobian are known.

What carries the argument

The central object is the forecast-error contraction profile $\log E(h) \simeq C + \lambda_1 h$, where $E(h)$ is the geometric-mean absolute error of a k-nearest-neighbor predictor at horizon $h$. The mechanism has three parts: (i) an ensemble of short post-transient trajectories supplies many comparable initial conditions; (ii) horizons are evaluated at multiples of the detected period $p$ so that the same phase of the periodic orbit is compared, turning an oscillating multibranch profile into a single contraction envelope; and (iii) a stable-transient consensus rule keeps only slopes that agree across several transient lengths, rejecting isolated linear fits that may represent transient decay rather than the asymptotic exponent. This machinery turns prediction-error decay into a scalar Lyapunov-exponent estimate without tangent-space reconstruction.

What would settle it

Take the logistic map at a strongly contracting parameter, for example $r=3.52$, and run the estimator with increasing ensemble sizes $N=100, 500, 2000, 5000$. If the consensus slope converges to the reference $\lambda_1=\langle \log|r(1-2x_n)|\rangle$ as $N$ grows, the identification in Eq. (4) is supported; if the slope plateaus at a less negative value while the linear profile persists, the fitted slope is an artifact of finite-ensemble or numerical-floor effects rather than the Lyapunov exponent.

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Extended reading notes

Core claim

The central claim is that a dominant negative Lyapunov exponent can be estimated without reconstructing the governing equations or Jacobian, by fitting the geometric-mean forecast error of a kNN predictor. In a locally contracting regime, $\log E(h) \simeq C + \lambda_1 h$, so the slope of the logarithmic error profile is the exponent. Two design choices make this work for stable orbits: forecast horizons are synchronized to the detected period, comparing the same phase of the orbit, and candidate slopes are accepted only if they form a consensus across several transient lengths. The paper demonstrates the claim on two benchmarks: the logistic map, where 92 of 112 negative-exponent reference values are recovered with MAE 0.0253 and $R^2=0.886$, and a two-dimensional no-fixed-point map, where independent scalar pipelines for $x_n$, $y_n$, and $\sqrt{x_n^2+y_n^2}$ give MAE 0.00879–0.01145 and $R^2=0.983$–$0.986$.

Load-bearing premise

The load-bearing premise is that the shrinkage of forecast errors measured from repeated short trajectories equals the true contraction rate of nearby states, so the slope of the error profile is the dominant negative Lyapunov exponent.

Editorial extensions

If this is right

  • Negative Lyapunov exponents become estimable from data sets too short for classical divergence-based methods, because the estimator reads contraction before the signal hits the numerical floor.
  • The same estimator can be applied to repeated relaxation records, such as structural ring-down or event-triggered sensor responses, without a mechanistic model.
  • The reliability decision is part of the output: parameter values that fail the transient-consensus test are rejected rather than reported as high-confidence fits.
  • Independent scalar observables of a higher-dimensional system yield nearly the same contraction rate, so the observable does not need to be a full state vector.
  • Because the period-aware horizon is a scalar analogue of one-period Floquet comparison, the method may estimate transverse or Floquet decay rates for limit cycles when implemented in a phase-aligned way.

Reading between the lines

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

  • Inference: if the method is applied to a continuous-time limit cycle, the fitted slope should be interpreted as a transverse Lyapunov or Floquet decay exponent, not the largest exponent, which is zero along the phase direction; the paper gestures at this distinction but leaves its implementation for future work.
  • Inference: the observed conservative bias toward zero on strongly contracting logistic windows suggests a general identifiability limit: contraction rates faster than the noise floor divided by the available horizon cannot be recovered by any forecast-error estimator, no matter how the transient is chosen.
  • Inference: a direct practical test would be to compare the forecast-error contraction rate with conventional logarithmic decrement or damping estimates on the same bridge ring-down records; agreement is not guaranteed because a Lyapunov exponent is an observation-dependent nonlinear return rate rather than a modal damping ratio.
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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 introduces an equation-free estimator of the dominant negative largest Lyapunov exponent (LLE) from ensembles of short scalar trajectories. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at period-synchronized horizons, and the slope of its logarithm is taken to estimate a negative LLE. A transient-slope consensus rule is used to accept or reject candidate estimates. The method is validated on the logistic map and on a two-dimensional no-fixed-point map, with reference exponents computed independently from the governing equations. The central claim is that, in a locally contracting regime, the log-error profile is approximately C + λ1 h, so the fitted slope estimates the dominant negative exponent without reconstructing a Jacobian or a surrogate model.

Significance. If the central premise holds, the method would be a practically useful tool for extracting contraction rates from repeated relaxation measurements, complementing positive-LLE estimators and avoiding the need for explicit tangent-space reconstruction. The paper is honest about limitations, and it explicitly distinguishes the target quantity as a 'dominant observable contraction rate' that may equal a negative LLE in some settings. The estimation stage does not use reference exponents during fitting or selection, and the benchmark systems have known reference values, which is a credit. However, the central relation in Eq. (4) is asserted rather than derived, and the reported accuracy metrics are computed only on accepted parameter points. The lack of sensitivity analysis and of all-point metrics means the headline numbers may overstate the method's reliability; the manuscript needs strengthening on these points before the claim is fully supported.

major comments (4)
  1. [Sec. 2.1, Eq. (4)] The central relation log E(h) ≈ C + λ1 h is asserted, not derived. It does not follow trivially from local contraction: for a deterministic noiseless stable system, the kNN predictor is consistent as N→∞ and E(h)→0 for any stable system regardless of λ1, while for finite N the error is controlled by neighbor-gap statistics, K, history length Y, embedding delay, and the numerical floor. The paper should provide either an analytic derivation under explicit assumptions (e.g., large N, low-dimensional observable, phase-aligned trajectories) or a sensitivity study showing that the fitted slope is invariant under variation of K, N, and Y. All reported results depend on Eq. (4), so this gap is load-bearing.
  2. [Sec. 3.2, Table 1] The headline metrics (MAE, RMSE, R²) are computed only on accepted parameter points, after applying the consensus and acceptance rules. For the logistic negative branch, 92 of 112 reference points are accepted (82.14% coverage), so the reported MAE of 0.0253 and R² of 0.886 do not characterize the whole negative-exponent region. Moreover, the text says six isolated single-transient fits were rejected but the retained set contains 92 points; 112 − 6 = 106, so the fate of the remaining 14 points is unclear. Please report coverage-weighted or all-point metrics and clarify the missing-point accounting.
  3. [Sec. 2.3] The acceptance criteria are not specified precisely: 'sufficient linearity', 'predominantly decreasing log-error values', 'limited contamination by exact-zero or numerical-floor errors', and the period-selection rule 'close to the minimum' in Eq. (5) are all qualitative. Because these rules determine which points enter the reported metrics, the paper should give exact thresholds and test the sensitivity of the results to those thresholds. Without this, the coverage and accuracy numbers are not reproducible.
  4. [Sec. 3.1] The fixed-point example at r = 2.7 is exact because, in one dimension, the kNN forecast error reduces to neighbor-gap contraction; it therefore does not validate the general higher-dimensional scalar-observable premise. The two-dimensional no-fixed-point benchmark is a more appropriate test, but the reported agreement across x_n, y_n, and the radial observable is not accompanied by any uncertainty quantification. A bootstrap or repeated-split analysis showing the distribution of estimated exponents would strengthen the claim that the slope is robust and not an artifact of a particular train/test partition.
minor comments (6)
  1. [Eq. (3)] The floor ϵ in the geometric-mean error is never defined; please state its value and how it is chosen, because it directly affects the log-error profile for strongly contracting systems.
  2. [Sec. 2.2, Eq. (5)] The period-detection rule 'smallest period close to the minimum of Q_p' and the phrase 'confirmed in a shifted window' are not formalized; a precise criterion is needed for reproducibility.
  3. [Sec. 3.2] The positive-branch context row in Table 1 is described as a prior-work context calculation, but it is given the same prominence as the new negative-LLE results; please state explicitly that this row is not a new contribution of the present method.
  4. [Discussion] The paper nicely qualifies the target as a 'dominant observable contraction rate', but the title and abstract repeatedly say 'negative largest Lyapunov exponent'; please harmonize the terminology so that readers do not over-interpret the estimator as a full Lyapunov-spectrum method.
  5. [References] Several references are incomplete (e.g., [22], [24], [34] lack complete volume/page information); please check the journal's reference style.
  6. [Figure 2] Figure 2(b) should distinguish accepted points from rejected or missing points with different markers, so that coverage is visually transparent rather than only summarized in Table 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the estimator is validated against independent reference exponents and no fitted input is renamed as a prediction.

full rationale

The negative-LLE estimator does not fit, select, or post-process using reference exponents. The central relation, Eq. (4) ('log E(h) ≈ C + λ1 h, λ1 < 0'), is a stated finite-sample linearization premise, but λ1 is not defined in terms of E(h); reference values are computed independently via Eq. (9) for the logistic map and via QR accumulation of the Jacobian for the 2-D map (Sec. 2.4). The forecast error E(h) is constructed only from observed trajectories through Eqs. (1)-(3), and the fitted slope is then compared externally to the reference exponents. Period detection, the transient-consensus rule, and the accept/reject classification (Sec. 2.3) do not use reference LLE values, and the paper explicitly notes that rejecting a point is preferable to reporting an unstable slope. The only self-citation, Ref. [9], is used for the positive-branch context experiment and to position the method; it is not load-bearing for the negative-branch claim. The paper itself flags Eq. (4) as a modeling assumption and discusses identifiability limits for strongly contracting cases, which is a correctness/robustness concern rather than a circular reduction. No equation or fitted parameter is shown to be equivalent to its own input by construction, so no circular step can be exhibited.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The paper contributes an estimator rather than a derivation. The main burden falls on the unproved Eq. (4) and on hand-chosen hyperparameters. No new physical entities are postulated.

free parameters (7)
  • N (ensemble size) = 5000 trajectories
    Chosen by hand for all sweeps; affects forecast-error floor and coverage, and no sensitivity analysis is reported.
  • K (nearest neighbors) = 3
    Fixed in the main sweeps; no sensitivity analysis is reported.
  • Y (history length) = 1 (logistic); 5 (fixed-point example); not stated for 2D map
    History-vector length is central to the predictor and differs across experiments without a stated selection rule.
  • tau (history step) = 1
    Embedding step for history vectors; fixed by hand.
  • H (profile length range) = 5 to 10
    Number of horizons considered for slope fits; range chosen by hand.
  • pmax (candidate period range) = 16
    Upper bound for period detection through Q_p.
  • Transient lengths T_j and consensus thresholds = unspecified
    Several candidate transient lengths and consistent-group thresholds are stated qualitatively, but exact values are not given.
assumptions (5)
  • ad hoc to paper log E(h) is approximately C + lambda_1 h in a locally contracting regime
    Eq. (4) is asserted without derivation; the entire estimator rests on kNN geometric-mean forecast errors contracting at the dominant negative Lyapunov exponent.
  • domain assumption kNN forecast errors behave like infinitesimal perturbations at finite K and short H
    No finite-sample or observability theory connects nearest-neighbor forecast error to tangent-space contraction, especially for scalar observations of a higher-dimensional map.
  • domain assumption Q_p recurrence statistic identifies the true orbit period
    Eq. (5) selects the smallest period near the minimum, confirmed in a shifted window; no proof is given that this works for all periodic windows or observables.
  • ad hoc to paper Transient-slope consensus across several T_j recovers the asymptotic rate
    Section 2.3 heuristic; no theoretical guarantee that agreement across transient lengths removes finite-time bias.
  • domain assumption Scalar observable contraction rate equals the dominant negative LLE of the full state
    Validated empirically by agreement across x_n, y_n, and z_n in the 2D map, but no observability argument is given.

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Cite this review

Pith. "Pith review of Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles." pith.science (2026). https://pith.science/paper/6XPPXVXU

@misc{pith2026260805522,
  author       = {Pith},
  title        = {Pith review of: Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XPPXVXU}},
  note         = {Machine review of arXiv:2608.05522}
}
abstract

Estimating positive largest Lyapunov exponents from data is comparatively natural because neighboring trajectories separate, whereas stable dynamics require resolving contraction before measurement noise or finite precision erases the signal. We introduce a period-aware forecast-error contraction procedure for estimating a dominant negative Lyapunov exponent from ensembles of short scalar trajectories without using governing equations or an analytical Jacobian. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at phase-consistent horizons, and the exponent is obtained from the slope of the logarithmic error profile. Unlike data-driven approaches that reconstruct local evolution matrices or differentiate a learned surrogate, the proposed method extracts the contraction rate directly from out-of-sample forecast errors. Two adaptations are essential: the forecast step is synchronized with the detected orbit period, and candidate slopes are accepted only when they form a stable consensus across several transient lengths. On the logistic map, the method recovers 92 of 112 negative-exponent parameter values with a mean absolute error of 0.0253 and $R^2=0.886$. On a two-dimensional map without fixed points, independent scalar pipelines based on the three observables $x_n$, $y_n$, and $z_n$ give mean absolute errors of 0.00879--0.01145 and $R^2=0.983$--$0.986$. Because the estimation stage uses only observed trajectories, the framework provides a basis for repeated-relaxation experiments in which short sensor responses are available but the governing equations and analytical Jacobian are unknown. Experimental validation remains a subject of future work.

Figures

Figures reproduced from arXiv: 2608.05522 by the authors.

Figure 1
Figure 1. Forecast-error contraction for a representative stable logistic-map point, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Logistic-map results. (a) Reference LLE over 3 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Negative largest Lyapunov exponent for the two-dimensional map without fixed points. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Works this paper leans on

37 extracted references · 23 canonical work pages

  1. [1]

    Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 1: Theory,

    G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 1: Theory,” Meccanica15, 9–20 (1980). doi:10.1007/BF02128236

    doi:10.1007/bf02128236 1980

  2. [2]

    Determining Lyapunov exponents from a time series,

    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D16, 285–317 (1985). doi:10.1016/0167-2789(85)90011-9 10

    doi:10.1016/0167-2789(85)90011-9 1985

  3. [3]

    Measurement of the Lyapunov spectrum from a chaotic time series,

    M. Sano and Y. Sawada, “Measurement of the Lyapunov spectrum from a chaotic time series,” Phys. Rev. Lett.55, 1082–1085 (1985). doi:10.1103/PhysRevLett.55.1082

    work page doi:10.1103/physrevlett.55.1082 1985

  4. [4]

    Lyapunov exponents from time series,

    J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, and S. Ciliberto, “Lyapunov exponents from time series,” Phys. Rev. A34, 4971–4979 (1986). doi:10.1103/PhysRevA.34.4971

    doi:10.1103/physreva.34.4971 1986

  5. [5]

    A practical method for calculating largest Lyapunov exponents from small data sets,

    M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D65, 117–134 (1993). doi:10.1016/0167- 2789(93)90009-P

    doi:10.1016/0167- 1993

  6. [6]

    A robust method to estimate the maximal Lyapunov exponent of a time series,

    H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time series,” Phys. Lett. A185, 77–87 (1994). doi:10.1016/0375-9601(94)90991-1

    doi:10.1016/0375-9601(94)90991-1 1994

  7. [7]

    Simple mathematical models with very complicated dynamics,

    R. M. May, “Simple mathematical models with very complicated dynamics,” Nature261, 459–467 (1976). doi:10.1038/261459a0

    doi:10.1038/261459a0 1976

  8. [8]

    Predictability: A way to characterize complexity,

    G. Boffetta, M. Cencini, M. Falcovich, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep.356, 367–474 (2002). doi:10.1016/S0370-1573(01)00025-4

    work page doi:10.1016/s0370-1573(01)00025-4 2002

Show all 37 references
  1. [9]

    A novel approach for estimating largest Lyapunov exponents in one-dimensional chaotic time series using machine learning,

    A. Velichko, M. Belyaev, and P. Boriskov, “A novel approach for estimating largest Lyapunov exponents in one-dimensional chaotic time series using machine learning,” Chaos35, 101101 (2025). doi:10.1063/5.0289352

    2025 doi

  2. [10]

    Chaotic map with no fixed points: Entropy, implementation and control,

    V. V. Huynh, A. Ouannas, X. Wang, V.-T. Pham, X. Q. Nguyen, and F. E. Alsaadi, “Chaotic map with no fixed points: Entropy, implementation and control,” Entropy21, 279 (2019). doi:10.3390/e21030279

    2019 doi

  3. [11]

    A robust method on estimation of Lyapunov exponents from a noisy time series,

    C. Yang and C. Wu, “A robust method on estimation of Lyapunov exponents from a noisy time series,” Nonlinear Dyn.64, 279–292 (2011). doi:10.1007/s11071-010-9860-x

    2011 doi

  4. [12]

    Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping,

    C. Yang, C. Wu, and P. Zhang, “Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping,” Nonlinear Dyn.69, 1493–1507 (2012). doi:10.1007/s11071-012-0364-8

    2012 doi

  5. [13]

    A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability,

    Y. Sun and C. Wu, “A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability,” Nonlinear Dyn. 70, 1689–1708 (2012). doi:10.1007/s11071-012-0567-z

    2012 doi

  6. [14]

    Improved assessment of orbital stability of rhythmic motion with noise,

    J. Ahn and N. Hogan, “Improved assessment of orbital stability of rhythmic motion with noise,” PLoS ONE10, e0119596 (2015). doi:10.1371/journal.pone.0119596

    2015 doi

  7. [15]

    Development of Floquet multiplier estimator to determine nonlinear oscillatory behavior in power system data measurement,

    N. Choi, H. Cho, and B. Lee, “Development of Floquet multiplier estimator to determine nonlinear oscillatory behavior in power system data measurement,” Energies12, 1824 (2019). doi:10.3390/en12101824

    2019 doi

  8. [16]

    Improving empirical characteristic multiplier estimation through a change of basis,

    J. A. Little, J. Turner, and B. P. Mann, “Improving empirical characteristic multiplier estimation through a change of basis,” J. Sound Vib.488, 115613 (2020). doi:10.1016/j.jsv.2020.115613

    2020

  9. [17]

    Poincar´ e maps for multi- scale physics discovery and nonlinear Floquet theory,

    J. J. Bramburger, J. E. Bramburger, and J. N. Kutz, “Poincar´ e maps for multi- scale physics discovery and nonlinear Floquet theory,” Physica D408, 132479 (2020). doi:10.1016/j.physd.2020.132479

    2020

  10. [18]

    A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems,

    D. Wilson, “A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems,” Chaos30, 013121 (2020). doi:10.1063/1.5126122 11

    2020 doi

  11. [19]

    Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time-series data,

    N. Namura, S. Takata, K. Yamaguchi, R. Kobayashi, and H. Nakao, “Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time-series data,” Phys. Rev. E 106, 014204 (2022). doi:10.1103/PhysRevE.106.014204

    2022 doi

  12. [20]

    A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,” J. Nonlinear Sci.25, 1307–1346 (2015). doi:10.1007/s00332-015-9258-5

    2015 doi

  13. [21]

    Global stability analysis using the eigenfunctions of the Koopman operator,

    A. Mauroy and I. Mezi´ c, “Global stability analysis using the eigenfunctions of the Koopman operator,” IEEE Trans. Autom. Control61, 3356–3369 (2016). doi:10.1109/TAC.2016.2518918

    2016

  14. [22]

    Data-driven transient sta- bility analysis using the Koopman operator,

    A. Matavalam, B. Hou, H. Choi, S. Bose, and U. Vaidya, “Data-driven transient sta- bility analysis using the Koopman operator,” Int. J. Electr. Power Energy Syst. (2024). doi:10.1016/j.ijepes.2024.110307

    2024

  15. [23]

    Big data-driven predictive control for nonlinear systems—A trajectory cluster-based contraction approach,

    S. Han, Y. Yan, J. Bao, and B. Huang, “Big data-driven predictive control for nonlinear systems—A trajectory cluster-based contraction approach,” J. Process Control, 103474 (2025). doi:10.1016/j.jprocont.2025.103474

    2025

  16. [24]

    Convex data-driven contraction with Riemannian metrics,

    A. Oliveira, J. Zheng, and M. Sznaier, “Convex data-driven contraction with Riemannian metrics,” IEEE Control Syst. Lett.9, 595–600 (2025). doi:10.1109/LCSYS.2025.3578275

    2025

  17. [25]

    Supervised machine learning to estimate instabilities in chaotic systems: Estimation of local Lyapunov exponents,

    D. Ayers, J. Lau, J. Amezcua, A. Carrassi, and V. Ojha, “Supervised machine learning to estimate instabilities in chaotic systems: Estimation of local Lyapunov exponents,” Q. J. R. Meteorol. Soc.149, 1236–1262 (2023). doi:10.1002/qj.4450

    2023 doi

  18. [26]

    Full Lyapunov exponents spectrum with deep learning from single-variable time series,

    C. Mayora-Cebollero, A. Mayora-Cebollero, A. Lozano, and R. Barrio, “Full Lyapunov exponents spectrum with deep learning from single-variable time series,” Physica D, 134510 (2024). doi:10.1016/j.physd.2024.134510

    2024

  19. [27]

    Ambient vibration re-testing and operational modal analysis of the Humber Bridge,

    J. M. W. Brownjohn, F. Magalh˜ aes, E. Caetano, and A. Cunha, “Ambient vibration re-testing and operational modal analysis of the Humber Bridge,” Eng. Struct.32, 2003–2018 (2010). doi:10.1016/j.engstruct.2010.02.034

    2010 doi

  20. [28]

    Ambient and free-vibration tests to improve the quantification and estimation of modal parameters in existing bridges,

    F. Lorenzoni, N. De Conto, F. da Porto, and C. Modena, “Ambient and free-vibration tests to improve the quantification and estimation of modal parameters in existing bridges,” J. Civ. Struct. Health Monit.9, 617–637 (2019). doi:10.1007/s13349-019-00357-4

    2019 doi

  21. [29]

    Influence of the modal damping ratio calculation method in the analysis of dynamic events obtained in structural health monitoring of bridges,

    J. L´ opez-Arag´ on, V. Puchol, and M. Astiz, “Influence of the modal damping ratio calculation method in the analysis of dynamic events obtained in structural health monitoring of bridges,” J. Civ. Struct. Health Monit.14, 1191–1213 (2024). doi:10.1007/s13349-023-00760-y

    2024 doi

  22. [30]

    Intelligent automatic operational modal analysis,

    M. M. Rosso, A. Aloisio, J. Parol, G. C. Marano, and G. Quaranta, “Intelligent automatic operational modal analysis,” Mech. Syst. Signal Process., 110669 (2023). doi:10.1016/j.ymssp.2023.110669

    2023

  23. [31]

    Quantitative analysis of non-equilibrium systems from short-time experimental data,

    S. K. Manikandan, S. Ghosh, A. Kundu, B. Das, V. Agrawal, D. Mitra, A. Banerjee, and S. Krishnamurthy, “Quantitative analysis of non-equilibrium systems from short-time experimental data,” Commun. Phys.4(2021). doi:10.1038/s42005-021-00766-2

    2021 doi

  24. [32]

    Modeling nonlinear dynamics from videos,

    A. Yang, J. Ax ˚ as, F. K´ ad´ ar, G. St´ ep´ an, and G. Haller, “Modeling nonlinear dynamics from videos,” Nonlinear Dyn. (2024). doi:10.1007/s11071-024-10687-8 12

    2024 doi

  25. [33]

    Condition monitoring of impact vibrations using observed signals and estimated principal Lyapunov exponents,

    J.-Y. Lee, “Condition monitoring of impact vibrations using observed signals and estimated principal Lyapunov exponents,” J. Vib. Control (2024). doi:10.1177/10775463241259131

    2024 doi

  26. [34]

    Data-driven method for identifying the expression of the Lyapunov exponent from discrete random data,

    X. Chen, X. Jin, and Z. Huang, “Data-driven method for identifying the expression of the Lyapunov exponent from discrete random data,” Int. J. Non-Linear Mech., 104268 (2022). doi:10.1016/j.ijnonlinmec.2022.104268

    2022

  27. [35]

    Data-driven spectral analysis of the Koopman operator,

    M. Korda, M. Putinar, and I. Mezi´ c, “Data-driven spectral analysis of the Koopman operator,” Appl. Comput. Harmon. Anal. (2018). doi:10.1016/j.acha.2018.08.002

    2018 doi

  28. [36]

    Modern Koopman theory for dynamical systems,

    S. L. Brunton, M. Budiˇ si´ c, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,” SIAM Rev.64, 229–340 (2022). doi:10.1137/21M1401243

    2022 doi

  29. [37]

    Rigorous data-driven computation of spectral proper- ties of Koopman operators for dynamical systems,

    M. J. Colbrook and A. Townsend, “Rigorous data-driven computation of spectral proper- ties of Koopman operators for dynamical systems,” Commun. Pure Appl. Math.77(2024). doi:10.1002/cpa.22125 13

    2024 doi

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