pith. sign in

arxiv: 2605.22299 · v1 · pith:4L7SACM2new · submitted 2026-05-21 · 🧮 math.DS · nlin.CD

Data-Driven Reduced Modeling of Delayed Dynamical Systems via Spectral Submanifolds

Giacomo Abbasciano , Gergely Buza , George Haller This is my paper

Pith reviewed 2026-05-22 02:31 UTC · model grok-4.3

classification 🧮 math.DS nlin.CD
keywords spectral submanifoldsdelay differential equationsdata-driven modelingmodel reductionnonlinear dynamicschaotic systemsbifurcationsexperimental data
0
0 comments X

The pith

Spectral submanifolds enable data-driven reduction of nonlinear delay differential equations to predictive low-dimensional ODEs without any knowledge of the delays.

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

The paper shows that an extension of spectral submanifold theory to delay differential equations supports purely data-driven model reduction for nonlinear systems containing delays. The method produces finite-dimensional, delay-free ODE models directly from observed data and requires no information on the number, size, or functional form of the delays. These reduced models stay accurate for chaotic regimes and can also track bifurcations under parameter variation for both discrete and distributed delays. The same framework applies to experimental time series from a control system that includes feedback delay and quantization. The approach therefore converts infinite-dimensional delayed dynamics into tractable low-dimensional descriptions from data alone.

Core claim

Using the extension of spectral submanifold theory to DDEs, one can obtain purely data-driven, SSM-reduced, delay-free ODE models for nonlinear delayed systems. These models remain predictive even for chaotic dynamics and do not require any information about the form of the DDE or the delays it contains. The method also enables parametric SSM-reduction to capture bifurcations and extends to non-autonomous systems with periodic delays, as shown on experimental data.

What carries the argument

Spectral submanifolds of the delayed system, reconstructed from finite data to serve as the invariant manifold carrying the reduced dynamics.

If this is right

  • Low-dimensional ODE models accurately reproduce chaotic dynamics of the original delayed system.
  • Parametric reductions capture bifurcations in systems with discrete or distributed delays.
  • Data-driven SSM reduction applies to experimental data from control systems with feedback delay and quantization.
  • The theoretical results extend to non-autonomous systems with periodic delays.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same data-driven procedure could simplify controller design for systems whose delay values are unknown or time-varying.
  • Reduced models obtained this way might serve as fast surrogates inside real-time optimization loops that currently cannot handle infinite-dimensional DDEs.
  • The reconstruction step could be tested on other memory-dependent systems such as integro-differential equations to check the breadth of the approach.

Load-bearing premise

The spectral submanifolds of the delayed system can be reliably reconstructed from finite noisy data without prior knowledge of the delay structure or the underlying vector field.

What would settle it

A direct comparison in which the data-driven SSM-reduced ODE fails to reproduce the long-term attractor or a known bifurcation point of the original delayed system under the same initial conditions and parameters.

Figures

Figures reproduced from arXiv: 2605.22299 by George Haller, Gergely Buza, Giacomo Abbasciano.

Figure 1
Figure 1. Figure 1: Eigenvalues of the linearization of the full system (7) at the fixed point at the origin (black [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a): Comparison between a full-system trajectory (black) of system (7) and the SSM-reduced [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a): Comparison between a trajectory of system (7) and the corresponding equation-driven [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between true dominant eigenvalues of the data-driven 2D SSM-reduced dynamics [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between two test trajectories of the full system (7) (black) and their data-driven [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: Comparison between a full system (7) test trajectory (black) together with its prediction [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Correlation dimension estimation and spectrum validation for the Mackey–Glass system (11). [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Short-time prediction performance of the SSM- reduced model of system (11). Left: training [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Analysis of the chaotic attractor of system (11) and its SSM-reduced model. Left: probability [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 3D projection of the delay embedding space showing convergence of the test trajectory to the [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: Comparison between eigenvalues of the data-driven reduced dynamics (green) and those [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Data-driven SSM reduction for system (12). Left: 3D projection of the delay embedding [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Left: Correlation dimension estimation for the attractor of system (13) from data in the [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Left: Training and test trajectories for the SSM-reduction of system (13) in the observable [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Analysis of the attractor of system (13). Left: Probability density functions of the reduced [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: 3D projection of the delay embedding space showing the chaotic attractor of system (13) and [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: For system (14), 3D projections of the delay embedding space showing the learned SSM (semi [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Predicted and verified portraits of the parametric SSM-reduced dynamics of system (15) at [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Illustration of the codimension one Poincar´e section viewed in the delay embedding space [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Validation of the parametric SSM model for system (15) at unseen delay values. The model [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Left: experimental rig used to control the Furuta pendulum. Right: schematic of the experi [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Illustration of the periodic-delay mechanism induced by the ZOH for a scalar DDE of the form [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Left: correlation dimension of the chaotic attractor, yielding a value of approximately 4 [PITH_FULL_IMAGE:figures/full_fig_p032_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: SSM-based model predictions for the Furuta pendulum experiments on previously unseen [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Comparison of the manifold coefficients obtained with the two SSM-reduction approaches, [PITH_FULL_IMAGE:figures/full_fig_p044_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Polynomial 6D ODE SSM-reduced model for the fulll system (11), with coefficients rounded [PITH_FULL_IMAGE:figures/full_fig_p049_26.png] view at source ↗
read the original abstract

We show how the recent extension of spectral submanifold (SSM) theory to delay differential equations (DDEs) enables data-driven model reduction of nonlinear delay systems. First, using a scalar DDE with a single discrete delay, we compare equation-based and data-driven SSM reductions, to illustrate the need for the latter. We then use the same algorithm to obtain purely data-driven, SSM-reduced, delay-free ODE models for several nonlinear delayed systems. Our approach requires no information about the form of the underlying DDE, or about the number and magnitude of the delays it contains. Our SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics. We also illustrate the use of parametric SSM-reduction to capture bifurcations in systems with both distributed and discrete delays. Finally we extend the theoretical underpinning of delayed SSM-reductions to non-autonomous systems with periodic delays, and apply these results to experimental data from a control system with feedback delay and quantization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper develops a data-driven approach to reduced-order modeling of nonlinear delay differential equations (DDEs) by extending spectral submanifold (SSM) theory. It first compares equation-based and data-driven SSM reductions on a scalar DDE with a single discrete delay, then applies the same data-driven algorithm to obtain delay-free low-dimensional ODE models for multiple nonlinear delayed systems. The central claims are that the method requires no information on the form of the DDE or the number/magnitude of delays, that the reduced models remain predictive even for chaotic dynamics, that parametric SSM reduction captures bifurcations for both distributed and discrete delays, and that the theory extends to non-autonomous systems with periodic delays, with a final application to experimental control-system data.

Significance. If the central claims are substantiated, the work would provide a practical route to finite-dimensional predictive models for infinite-dimensional delayed systems directly from time series, without explicit delay knowledge. This could impact data-driven modeling in control, fluid dynamics, and biological systems where delays are common. The inclusion of chaotic examples, bifurcation capture, and experimental validation adds to its potential utility, though the absence of quantitative metrics in the presented claims limits immediate assessment of impact.

major comments (3)
  1. [§3] §3 (data-driven algorithm description): The procedure for selecting embedding dimension, lags, or lifted features is not specified in a manner that demonstrates independence from the unknown delay scales; if these choices are tuned to unfold the attractor, the claim of requiring 'no information about the number and magnitude of the delays' is not yet load-bearing.
  2. [§5] §5 (chaotic dynamics examples): The assertion that 'SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics' lacks reported quantitative metrics such as normalized prediction error over multiple Lyapunov times, attractor reconstruction error, or comparison against full DDE integration; without these, the claim cannot be evaluated.
  3. [experimental validation section] Final experimental section: Validation on the control-system data with feedback delay and quantization does not include details on sampling rate relative to delay, noise characteristics, or how the SSM is identified from quantized observations without prior delay knowledge, which is central to the no-prior-information guarantee.
minor comments (2)
  1. Notation for the delay operator and spectral subspaces should be unified between the theoretical extension and the data-driven sections to avoid ambiguity.
  2. Figure captions for the bifurcation diagrams should explicitly state the parameter ranges and the number of delay values tested.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight important aspects that will improve the clarity and substantiation of our claims. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [§3] §3 (data-driven algorithm description): The procedure for selecting embedding dimension, lags, or lifted features is not specified in a manner that demonstrates independence from the unknown delay scales; if these choices are tuned to unfold the attractor, the claim of requiring 'no information about the number and magnitude of the delays' is not yet load-bearing.

    Authors: We agree that the selection procedure for embedding parameters requires explicit clarification to support the no-prior-delay-information claim. In the current manuscript, these choices follow standard data-driven criteria (false nearest neighbors for dimension and mutual information for lags) that operate solely on the observed time series. To make this independence from delay scales load-bearing, we will revise §3 to include a dedicated subsection describing the automated, delay-agnostic selection algorithm and provide a numerical demonstration that the same procedure succeeds across different unknown delay values without retuning. revision: yes

  2. Referee: [§5] §5 (chaotic dynamics examples): The assertion that 'SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics' lacks reported quantitative metrics such as normalized prediction error over multiple Lyapunov times, attractor reconstruction error, or comparison against full DDE integration; without these, the claim cannot be evaluated.

    Authors: We concur that quantitative metrics are necessary to rigorously evaluate the predictive performance for chaotic dynamics. In the revised version we will augment §5 with normalized root-mean-square prediction errors computed over multiple Lyapunov times, attractor reconstruction errors (e.g., via Hausdorff distance between reconstructed and true attractors), and direct trajectory comparisons against numerical integration of the full DDE for each chaotic example. revision: yes

  3. Referee: [experimental validation section] Final experimental section: Validation on the control-system data with feedback delay and quantization does not include details on sampling rate relative to delay, noise characteristics, or how the SSM is identified from quantized observations without prior delay knowledge, which is central to the no-prior-information guarantee.

    Authors: We acknowledge that additional experimental details are needed to substantiate the no-prior-information guarantee. We will expand the final experimental section to report the sampling rate relative to the feedback delay, the noise characteristics of the measured signals, and a step-by-step account of the SSM identification procedure applied directly to the quantized observations, confirming that no delay value or DDE structure was used at any stage. revision: yes

Circularity Check

0 steps flagged

No significant circularity in data-driven SSM reduction for delayed systems

full rationale

The paper claims a data-driven algorithm that reconstructs spectral submanifolds from raw time series to produce delay-free ODE models without any input on the DDE form, number, or magnitude of delays. No load-bearing step in the abstract or described derivation reduces a prediction to a fitted quantity by construction, nor does it rely on self-citations for uniqueness or ansatz smuggling. The central result is presented as an application of extended SSM theory to data, remaining independent of the target outputs. This matches the default expectation for non-circular papers and the reader's assessment of minimal circularity risk.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the recent SSM extension to DDEs and on the assumption that data alone suffices to identify the reduced model for the systems tested.

axioms (1)
  • domain assumption The recent extension of spectral submanifold theory to delay differential equations is valid for the nonlinear systems considered.
    The abstract states that the method uses this extension as its foundation.

pith-pipeline@v0.9.0 · 5699 in / 1181 out tokens · 44597 ms · 2026-05-22T02:31:38.396531+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    A Time-Delayed Model for the Spread of COVID-19 with Vaccination

    Al-Tuwairqi, S. M. & Al-Harbi, S. K. “A Time-Delayed Model for the Spread of COVID-19 with Vaccination”.Sci. Rep.12(2022), 19435

    work page 2022

  2. [2]

    Reduction to Spectral Submanifolds in Guided Car-Following with Time Delay

    Szaksz, B., Orosz, G. & St´ ep´ an, G. “Reduction to Spectral Submanifolds in Guided Car-Following with Time Delay”.IFAC-Pap. OnLine58(2024), 67–72

    work page 2024

  3. [3]

    Discretization Based Spectral Submanifolds in Guided Car- Following with Time Delay

    Szaksz, B., Orosz, G. & St´ ep´ an, G. “Discretization Based Spectral Submanifolds in Guided Car- Following with Time Delay”.IFAC-Pap. OnLine59(2025), 165–170

    work page 2025

  4. [4]

    & Moln´ ar, T

    Orosz, G. & Moln´ ar, T. G. Dynamics and Control of Connected Vehicles, Surveys and Tutorials in the Applied Mathematical Sciences, (Springer, Cham, 2025) https://doi.org/10.1007/978-3-031-94555-7

    work page doi:10.1007/978-3-031-94555-7 2025

  5. [5]

    Quantization Improves Stabilization of Dynamical Sys- tems with Delayed Feedback

    St´ ep´ an, G., Milton, J. G. & Insperger, T. “Quantization Improves Stabilization of Dynamical Sys- tems with Delayed Feedback”.Chaos27(2017), 114306

    work page 2017

  6. [6]

    & St´ ep´ an, G

    Insperger, T. & St´ ep´ an, G. Semi-Discretization for Time-Delay Systems, (Springer, New York, 2011) https://doi.org/10.1007/978-1-4614-0335-7

    work page doi:10.1007/978-1-4614-0335-7 2011

  7. [7]

    Existence of Spectral Submanifolds in Time Delay Systems

    Buza, G. & Haller, G. “Existence of Spectral Submanifolds in Time Delay Systems”.arXiv(2025), arXiv:2512.22062

    work page arXiv 2025

  8. [8]

    Oscillation and Chaos in Physiological Control Systems

    Mackey, M. C. & Glass, L. “Oscillation and Chaos in Physiological Control Systems”.Science197 (1977), 287–289

    work page 1977

  9. [10]

    Nonlinear Normal Modes and Spectral Submanifolds: Existence, Unique- ness and Use in Model Reduction

    Haller, G. & Ponsioen, S. “Nonlinear Normal Modes and Spectral Submanifolds: Existence, Unique- ness and Use in Model Reduction”.Nonlinear Dyn.86(2016), 1493–1534

    work page 2016

  10. [11]

    Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds

    Cenedese, M., Ax˚ as, J., B¨ auerlein, B., Avila, K. & Haller, G. “Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds”.Nat. Commun.13(2022), 5514. 49

    work page 2022

  11. [12]

    Capturing the Edge of Chaos as a Spectral Submanifold in Pipe Flows

    Kasz´ as, B. & Haller, G. “Capturing the Edge of Chaos as a Spectral Submanifold in Pipe Flows”. J. Fluid Mech.979(2024), A48

    work page 2024

  12. [13]

    Joint Reduced Model for the Laminar and Chaotic Attractors in Plane Couette Flow

    Kasz´ as, B. & Haller, G. “Joint Reduced Model for the Laminar and Chaotic Attractors in Plane Couette Flow”. Journal of Fluid Mechanics1023(2025), A1

    work page 2025

  13. [14]

    Data-Driven Modeling of Subharmonic Forced Response Due to Nonlinear Resonance

    Ax˚ as, J., B¨ auerlein, B., Avila, K. & Haller, G. “Data-Driven Modeling of Subharmonic Forced Response Due to Nonlinear Resonance”.Sci. Rep.14(2024), 25991

    work page 2024

  14. [15]

    Model Reduction to Spectral Submanifolds in Piecewise Smooth Dynamical Systems

    Bettini, L., Cenedese, M. & Haller, G. “Model Reduction to Spectral Submanifolds in Piecewise Smooth Dynamical Systems”.Int. J. Non-Linear Mech.163(2024), 104753

    work page 2024

  15. [16]

    Data-Driven Modeling of Global Bifurcations and Chaos in a Mechanical System under Delayed and Quantized Control

    Abbasciano, G., Endr´ esz, B., St´ ep´ an, G. & Haller, G. “Data-Driven Modeling of Global Bifurcations and Chaos in a Mechanical System under Delayed and Quantized Control”.J. Sound Vib.628 (2026), 119660

    work page 2026

  16. [17]

    Data-Driven Spectral Sub- manifold Reduction for Nonlinear Optimal Control of High-Dimensional Robots

    Alora, J. I., Cenedese, M., Schmerling, E., Haller, G. & Pavone, M. “Data-Driven Spectral Sub- manifold Reduction for Nonlinear Optimal Control of High-Dimensional Robots”.arXiv(2022)

    work page 2022

  17. [18]

    Practical Deployment of Spectral Submanifold Reduction for Optimal Control of High-Dimensional Systems

    Alora, J. I., Cenedese, M., Schmerling, E., Haller, G. & Pavone, M. “Practical Deployment of Spectral Submanifold Reduction for Optimal Control of High-Dimensional Systems”.IFAC-Pap. OnLine56(2023), 4074–4081

    work page 2023

  18. [19]

    Nonlinear Spectral Modeling and Control of Soft-Robotic Muscles from Data

    Bettini, L., Kazemipour, A., Katzschmann, R. K. & Haller, G. “Nonlinear Spectral Modeling and Control of Soft-Robotic Muscles from Data”.arXiv(2026)

    work page 2026

  19. [20]

    Parametric Spectral Submanifolds across Hopf Bifurcations with Applications to Fluid Dynamics

    King, J., Kasz´ as, B., Buza, G., Jussiau, W. & Haller, G. “Parametric Spectral Submanifolds across Hopf Bifurcations with Applications to Fluid Dynamics”.arXiv(2026)

    work page 2026

  20. [21]

    Quasiperiodic Oscillations in Robot Dynamics

    St´ ep´ an, G. & Haller, G. “Quasiperiodic Oscillations in Robot Dynamics”.Nonlinear Dyn.8(1995), 513–528

    work page 1995

  21. [22]

    M., Gils, S

    Diekmann, O., Verduyn Lunel, S. M., Gils, S. A. van & Walther, H.-O. Delay Equations: Functional- , Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, vol. 110, (Springer, New York, NY, 1995) https://doi.org/10.1007/978-1-4612-4206-2

    work page doi:10.1007/978-1-4612-4206-2 1995

  22. [23]

    Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems

    Brunton, S. L., Proctor, J. L. & Kutz, J. N. “Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems”.Proc. Natl. Acad. Sci. U.S.A.113(2016), 3932– 3937

    work page 2016

  23. [24]

    Learning Time Delay Systems with Neural Ordinary Differential Equations

    Ji, X. A. & Orosz, G. “Learning Time Delay Systems with Neural Ordinary Differential Equations”. IFAC-Pap. OnLine55(2022), 79–84

    work page 2022

  24. [25]

    Learn from One and Predict All: Single Trajectory Learning for Time Delay Systems

    Ji, X. A. & Orosz, G. “Learn from One and Predict All: Single Trajectory Learning for Time Delay Systems”.Nonlinear Dyn.112(2024), 3505–3518

    work page 2024

  25. [26]

    Data-Driven Methods for Delay Differential Equa- tions

    Breda, D., Ji, X. A., Orosz, G. & Tanveer, M. “Data-Driven Methods for Delay Differential Equa- tions”.arXiv(2025), arXiv:2512.04894

    work page arXiv 2025

  26. [27]

    Data-Driven Discovery of Delay Differ- ential Equations with Discrete Delays

    Pecile, A., Demo, N., Tezzele, M., Rozza, G. & Breda, D. “Data-Driven Discovery of Delay Differ- ential Equations with Discrete Delays”.J. Comput. Appl. Math.461(2025), 116439

    work page 2025

  27. [28]

    SINDy for Delay-Differential Equations: Application to Model Bacterial Zinc Response

    Sandoz, A., Ducret, V., Gottwald, G. A., Vilmart, G. & Perron, K. “SINDy for Delay-Differential Equations: Application to Model Bacterial Zinc Response”.Proc. R. Soc. A479(2023), 20220556

    work page 2023

  28. [29]

    Semigroups of linear operators and applications to partial differential equa- tions, volume 44 of Applied Mathematical Sciences

    Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, (Springer-Verlag, New York, 1983) https://doi.org/10.1007/978-1-4612-5561-1

    work page doi:10.1007/978-1-4612-5561-1 1983

  29. [30]

    Invariant Spectral Foliations with Applications to Model Order Reduction and Synthe- sis

    Szalai, R. “Invariant Spectral Foliations with Applications to Model Order Reduction and Synthe- sis”.Nonlinear Dyn.101(2020), 2645–2669. 50

    work page 2020

  30. [31]

    Haller, G. Modeling Nonlinear Dynamics from Equations and Data — with Applications to Solids, Fluids, and Controls, (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2025) https://doi.org/10.1137/1.9781611978353

    work page doi:10.1137/1.9781611978353 2025

  31. [32]

    Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection

    Bettini, L., Kasz´ as, B., Zybach, B., Dual, J. & Haller, G. “Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection”.Chaos35(2025), 043135

    work page 2025

  32. [33]

    General Oblique Projections for Model Reduction via Spectral Submanifolds

    Bettini, L., Kazemipour, A., Katzschmann, R. K. & Haller, G. “General Oblique Projections for Model Reduction via Spectral Submanifolds”.Chaos36(2026), 053122

    work page 2026

  33. [34]

    Detecting Strange Attractors in Turbulence

    Takens, F. “Detecting Strange Attractors in Turbulence”.Dyn. Syst. Turbulence898(1981), 366– 381

    work page 1981

  34. [35]

    Measuring the Strangeness of Strange Attractors

    Grassberger, P. & Procaccia, I. “Measuring the Strangeness of Strange Attractors”.Phys. D: Non- linear Phenom.9(1983), 189–208

    work page 1983

  35. [36]

    Embedology

    Sauer, T., Yorke, J. A. & Casdagli, M. “Embedology”.J. Stat. Phys.65(1991), 579–616

    work page 1991

  36. [37]

    Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations

    Nastac, G., Labahn, J. W., Magri, L. & Ihme, M. “Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations”. Physical Review Fluids2(2017), 094606

    work page 2017

  37. [38]

    Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds

    Liu, A., Ax˚ as, J. & Haller, G. “Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds”.Chaos34(2024), 033140

    work page 2024

  38. [39]

    Data-Driven Modelling of the Regular and Chaotic Dynamics of an Inverted Flag from Experiments

    Xu, Z., Kasz´ as, B., Cenedese, M., Berti, G., Coletti, F. & Haller, G. “Data-Driven Modelling of the Regular and Chaotic Dynamics of an Inverted Flag from Experiments”.J. Fluid Mech.987(2024), R7

    work page 2024

  39. [40]

    Modeling nonlinear dynamics from videos Rights / license: Creative Commons Attribution 4.0 International Modeling nonlinear dynamics from videos

    Haller, G., Yang, A., Ax˚ as, J., K´ ad´ ar, F. & St´ ep´ an, G. “Modeling nonlinear dynamics from videos Rights / license: Creative Commons Attribution 4.0 International Modeling nonlinear dynamics from videos”.113(2025), 10881–10909

    work page 2025

  40. [41]

    Circular Causal Systems in Ecology

    Hutchinson, G. E. “Circular Causal Systems in Ecology”.Ann. N. Y. Acad. Sci.50(1948), 221– 246

    work page 1948

  41. [42]

    Spectral Submanifolds in Time Delay Systems

    Szaksz, B., Orosz, G. & St´ ep´ an, G. “Spectral Submanifolds in Time Delay Systems”.Nonlinear Dyn.113(2025), 14265–14286

    work page 2025

  42. [43]

    TDS-CONTROL: a MATLAB Package for the Analysis and Controller-Design of Time-Delay Systems

    Appeltans, P., Silm, H. & Michiels, W. “TDS-CONTROL: a MATLAB Package for the Analysis and Controller-Design of Time-Delay Systems”.IFAC-Pap. OnLine55(2022), 272–277

    work page 2022

  43. [44]

    Analysis and Controller-Design of Time-Delay Systems Using TDS- CONTROL: A Tutorial and Manual

    Appeltans, P. & Michiels, W. “Analysis and Controller-Design of Time-Delay Systems Using TDS- CONTROL: A Tutorial and Manual”.arXiv(2023)

    work page 2023

  44. [45]

    Time Delays in Single Species Growth Models

    Cushing, J. M. “Time Delays in Single Species Growth Models”.J. Math. Biol.4(1977), 257–264

    work page 1977

  45. [46]

    Digital Stability of the Furuta Pendulum Based on Angle Detection

    Vizi, M. B. & St´ ep´ an, G. “Digital Stability of the Furuta Pendulum Based on Angle Detection”. J. Vib. Control30(2024), 588–597

    work page 2024

  46. [47]

    Semi-Discretization and the Time-Delayed PDA Feedback Control of Human Balance

    Insperger, T., Milton, J. & St´ ep´ an, G. “Semi-Discretization and the Time-Delayed PDA Feedback Control of Human Balance”.IFAC-Pap. OnLine48(2015), 93–98

    work page 2015

  47. [48]

    Micro-Chaos in Digital Control

    Haller, G. & St´ ep´ an, G. “Micro-Chaos in Digital Control”.J. Nonlinear Sci.6(1996), 415–448

    work page 1996

  48. [49]

    doi: 10.1007/BFb0084432

    Hino, Y., Murakami, S. & Naito, T. Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, (Springer-Verlag, New York, 1991) https://doi.org/10.1007/BFb0084432

    work page doi:10.1007/bfb0084432 1991

  49. [50]

    Fast Data-Driven Model Reduction for Nonlinear Dynamical Systems

    Ax˚ as, J., Cenedese, M. & Haller, G. “Fast Data-Driven Model Reduction for Nonlinear Dynamical Systems”.Nonlinear Dyn.111(2023), 7941–7957

    work page 2023

  50. [51]

    Periodic Center Manifolds for DDEs in the Light of Suns and Stars

    Lentjes, B., Spek, L., Bosschaert, M. M. & Kuznetsov, Y. A. “Periodic Center Manifolds for DDEs in the Light of Suns and Stars”.J. Dyn. Differ. Equ.37(2025), 815–858. 51

    work page 2025

  51. [52]

    Data-Driven Reduction of the Finite-Element Model of a Tribomechadynamics Benchmark Problem

    Morsy, A. A., Xu, Z., Tiso, P. & Haller, G. “Data-Driven Reduction of the Finite-Element Model of a Tribomechadynamics Benchmark Problem”.Nonlinear Dyn.113(2025), 24539–24555

    work page 2025

  52. [53]

    Discovering Dominant Dynamics for Nonlinear Continuum Robot Control

    Alora, J. I., Cenedese, M., Haller, G. & Pavone, M. “Discovering Dominant Dynamics for Nonlinear Continuum Robot Control”.npj Robot.3(2025), 5

    work page 2025

  53. [54]

    Data-Driven Soft Robot Control via Adiabatic Spectral Submanifolds

    Kaundinya, R. S., Alora, J. I., Matt, J. G., Pabon, L. A., Pavone, M. & Haller, G. “Data-Driven Soft Robot Control via Adiabatic Spectral Submanifolds”.arXiv(2025)

    work page 2025

  54. [55]

    Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, vol

    Kato, T. Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, vol. 132, (Springer, Berlin, Heidelberg, 2013) https://doi.org/10.1007/978-3-662-12678-3

    work page doi:10.1007/978-3-662-12678-3 2013

  55. [56]

    Invariant Foliations forC 1 Semigroups in Banach Spaces

    Chen, X.-Y., Hale, J. K. & Tan, B. “Invariant Foliations forC 1 Semigroups in Banach Spaces”.J. Differ. Equ.139(1997), 283–318

    work page 1997

  56. [57]

    & Salamon, D

    B¨ uhler, T. & Salamon, D. A. Functional Analysis, Graduate Studies in Mathematics, vol. 191, (American Mathematical Society, Providence, RI, 2018)

    work page 2018

  57. [58]

    A New Proof of the Pseudostable Manifold Theorem

    Irwin, M. C. “A New Proof of the Pseudostable Manifold Theorem”.J. Lond. Math. Soc.s2-21 (1980), 557–566

    work page 1980

  58. [59]

    Spectral Submanifolds of the Navier–Stokes Equations

    Buza, G. “Spectral Submanifolds of the Navier–Stokes Equations”.SIAM J. Appl. Dyn. Syst.23 (2024), 1052–1089. 52

    work page 2024