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Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Delayed pairwise coupling in Kuramoto oscillators is equivalent to three-body coupling to second order, and the three-body term explains delay-induced bistability.

desk verdict A clean O(ε²) reduction from delayed pairwise to three-body Kuramoto coupling, but the claimed τ<1/ε regime is too broad; the numerics run where ετ is not small. read the letter →

arxiv 2512.16193 v2 pith:4CQUKDR2 submitted 2025-12-18 nlin.AO cond-mat.stat-mechmath-phmath.DSmath.MPnlin.PS

Narumi Fujii , Keisuke Taga , Riccardo Muolo , Bob Rink , Hiroya Nakao This is my paper · ORCID
classification nlin.AOcond-mat.stat-mechmath-phmath.DSmath.MPnlin.PS MSC 34C1534K1834K2037N25 PACS 05.45.Xt89.75.-k
keywords higher-orderinteractionsKuramotomodeltimedelaythree-bodycouplingorder-parameterreductionbistabilitysynchronizationtransitionphaseoscillators
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 argues that time-delayed pairwise interactions in Kuramoto oscillators are not merely a phase lag: when expanded to second order in the coupling strength, the delayed system becomes a delay-free higher-order Kuramoto model with both pairwise phase-lagged coupling and a three-body (2,-1,-1) interaction whose strength grows as the product of coupling and delay. This three-body term is what produces the bistability between incoherence and full synchrony seen in the original delayed model. If true, delays provide a concrete mechanistic origin for higher-order interactions, and delay-induced phenomena become tractable with standard oscillator-analysis tools.

What carries the argument

The central device is the second-order Taylor expansion of the delayed sine coupling in powers of the coupling strength, equivalently in powers of the product epsilon*tau. Iterated substitution of the equations of motion converts delayed derivatives into combinations of phase differences, and reorganizing terms by powers of epsilon yields the delay-free higher-order model. The Ott-Antonsen ansatz, which assumes the phase distribution's Fourier coefficients decay geometrically, then collapses the infinite-dimensional system to a closed equation for the synchronization degree R, from which the stability conditions for incoherent and fully synchronized states are read off.

What would settle it

Simulate the original delayed model and the reduced higher-order model for a parameter point where epsilon*tau is not small, e.g., epsilon = 0.5 and tau = 4 (product 2, violating tau < 1/epsilon), and compare synchronization transition curves: if the reduced model fails to reproduce bistable regions or transition thresholds, the claimed equivalence is limited to the small-parameter regime.

Watch

Extended reading notes

Core claim

For identical oscillators with small coupling epsilon and delay tau, the time-delayed Kuramoto model reduces to an ordinary differential equation: each pair experiences sine coupling with phase lag omega0 tau, plus a three-body (2,-1,-1) harmonic with amplitude proportional to epsilon^2 tau. The derivation expands the delayed argument in a Taylor series and repeatedly substitutes the equations of motion to eliminate delay derivatives, keeping terms through O(epsilon^2). Numerical simulations and the standard Ott-Antonsen order-parameter reduction show that this reduced model reproduces the synchronization transitions and, crucially, the bistable region of the original delayed system, provide

Load-bearing premise

The approximation is valid only when the product of coupling strength and delay, epsilon*tau, is small enough that truncating the Taylor expansion at second order is safe; the paper assumes tau < 1/epsilon, and if this fails the residual delay and four-body terms can change the dynamics.

Editorial extensions

If this is right

  • Bistability between incoherence and full synchrony in the delayed Kuramoto model is caused by the three-body term; the pairwise phase-lagged part alone cannot produce hysteresis.
  • The effective higher-order model is a system of ordinary differential equations, so all standard analysis and control methods for higher-order Kuramoto networks become applicable to time-delayed systems.
  • Stability boundaries of the incoherent and synchronized states reduce to simple trigonometric inequalities in epsilon and tau, which match the independently known diagram for the delayed model in the small-parameter regime.
  • The reduction offers a systematic route to derive many-body effective couplings for other delay-coupled oscillator systems beyond Kuramoto dynamics.

Reading between the lines

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

  • Because the three-body strength scales as epsilon^2 tau, delay-induced effects grow linearly with delay but quadratically with coupling; experiments or numerics that vary epsilon and tau separately could test this scaling directly.
  • The same Taylor-expansion route should generate four-body interactions at O(epsilon^3); these may become visible for delays near tau ~ 1/epsilon and could produce cluster states or other phenomena absent from the truncated model.
  • For non-identical frequencies the effective couplings acquire frequency-dependent phase lags, suggesting a frequency-filtered interaction that could matter for broad frequency distributions.
  • The equivalence between delay and higher-order interactions may extend to non-global network topologies, where the three-body term would couple triangles of oscillators and make delayed network dynamics amenable to hypergraph analyses.
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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 studies globally coupled Kuramoto oscillators with a uniform time delay τ in the pairwise coupling, Eq. (1). Under a weak-coupling assumption, it Taylor-expands the delayed phase and substitutes the equations of motion repeatedly to obtain a delay-free model, Eq. (6), consisting of a Kuramoto–Sakaguchi pairwise term and a (2,−1,−1) three-body harmonic whose strength is ε²τ. The paper demonstrates bistability for N=2, compares finite-N simulations of the delayed and reduced models, applies the Ott–Antonsen ansatz to derive the order-parameter dynamics, Eqs. (10)–(12), and derives stability conditions, Eqs. (14)–(15), which it benchmarks against the Yeung–Strogatz result. The central assertion is that, within the regime ε≪1 and τ<1/ε, the higher-order Kuramoto model quantitatively reproduces the synchronization transitions and bistability of the time-delayed model.

Significance. The conceptual message — that delayed pairwise interactions can be recast, through a controlled weak-coupling expansion, as higher-order interactions — is interesting and potentially useful, connecting two active fields. The derivation is explicit and self-contained, with no fitted parameters; the reduction to a low-dimensional Ott–Antonsen equation makes the reduced model analytically tractable; and the stability boundaries are checked against the independent Yeung–Strogatz diagram. These are genuine strengths. The main weakness is that the stated domain of validity is broader than the expansion actually supports: the accuracy is controlled by ετ, not by ε alone, and the numerical comparisons include parameter values with ετ close to or above 1.

major comments (3)
  1. [Appendix A1 and Eq. (6)] The derivation is an expansion in ετ, not in ε alone. In Appendix A1, θ_k(t−τ) is replaced by θ_k(t)−ω_kτ plus a remainder of order ετ. When this remainder is substituted into the O(ε) coupling, the first omitted correction to the retained O(ε²τ) three-body term is smaller by a factor O(ετ). The paper itself states that the accuracy is characterized by the smallness of ετ, but then asserts the regime τ<1/ε, which allows ετ arbitrarily close to 1. At ετ≈1 the neglected corrections are comparable to the retained three-body term, so the statement that Eq. (6) approximates Eq. (1) to O(ε²) is not valid on the claimed domain. The validity condition should be ετ≪1, and the phrase τ<1/ε in the main text and in Fig. 3 should be replaced or supplemented.
  2. [Fig. 2 and Fig. 3] The numerical evidence for the central 'quantitatively reproduces' claim lies largely outside the small-ετ regime. In Fig. 2(a), τ=2.8 and the reported bifurcations occur at ε=0.12 and ε=0.27, giving ετ≈0.34 and 0.76; if the plotted ε-range extends beyond 0.3, ετ exceeds 1. In Fig. 2(b), ε=0.3, so ετ>1 for any τ>3.3, which covers most of the displayed periodic sequence. Fig. 3 compares stability boundaries over the entire region below the line τ=1/ε, including ετ close to 1. These comparisons therefore do not cleanly test the asymptotic regime in which the reduction is derived. The authors should either restrict the numerical and analytical comparisons to ετ≪1, or provide an explicit error estimate showing that the agreement persists when ετ is not small.
  3. [Appendix A3 and Eqs. (10)–(15)] The reduction to the identical-oscillator limit takes γ→0 after applying the Ott–Antonsen ansatz. This is a singular limit, and the replacement of f1 and f2 by τ and 0 uses (ω_l−ω_k)τ=O(ετ), which again requires ετ≪1. The paper does not justify that the N→∞ and γ→0 limits commute, nor does it discuss the fact that the Lorentzian becomes a δ-distribution. Since the stability diagram is a central quantitative output, the authors should either derive the identical-oscillator case directly or verify the γ→0 limit against finite-N simulations of Eq. (6) in the ετ≪1 regime.
minor comments (5)
  1. [Abstract and §5] The abstract says the reduced model exhibits 'qualitatively consistent' transitions, while the main text and conclusion say it 'quantitatively reproduces' the phenomena. These statements should be aligned to the same strength, especially after the validity regime is corrected.
  2. [Eq. (6)] The sums over k and l allow l=j, so the 'three-body' term includes O(1/N) pairwise and constant contributions. In the thermodynamic limit they vanish, but the paper should either exclude l=j explicitly or state that these contributions are negligible for large N.
  3. [Eqs. (10) and (11)] The phrase 'low-dimensional mode' should read 'low-dimensional model'. Also, the redefinition of z1 in the γ→0 limit is stated but the notation could be cleaner.
  4. [Fig. 2 caption] Please specify the full parameter ranges, the number of oscillators, initial conditions, and how the stable and unstable branches were obtained (e.g., forward/backward sweeps). It would also be helpful to mark the ετ≪1 validity boundary or indicate the ετ values at the reported transition points.
  5. [Appendix A1] The sentence 'The case with ω_l=ω_k is trivial' should be expanded: the limiting values f1=τ and f2=0 are used in the central Eq. (6), so they deserve an explicit derivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the delayed-to-higher-order reduction is a self-contained Taylor expansion checked against the independent Yeung–Strogatz result.

full rationale

The central chain is Eq.(1) -> Taylor expansion Eq.(2) -> Eq.(3) -> Eq.(6) -> Ott–Antonsen reduction Eq.(10)-(12) -> stability conditions Eq.(14)-(15). No parameter is fitted to the quantities later called predictions: the three-body coupling strength ε²τ/2 is fixed by the expansion itself, not calibrated to simulation data. The OA closure uses the standard external Ott–Antonsen ansatz [45], and the stability diagram is compared with the independent Yeung–Strogatz [40] time-delay result, which is an external benchmark. Self-citations [48,52,53] are contextual or future-work references and do not carry any load-bearing step in the derivation. The only substantive caveat is a validity-regime concern, not circularity: the paper says the approximation is controlled by the smallness of ετ, yet also states the broader condition τ<1/ε, and some numerical points (e.g., Fig. 2, where ετ can reach about 0.76 or exceed 1) may lie outside the genuinely small-ετ regime. That is a correctness/scope issue, not a reduction of the claimed result to its inputs.

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

The paper does not fit any parameters to data; epsilon, tau, omega0, and gamma are model inputs. The main assumptions are the small-parameter Taylor truncation, the OA ansatz, and the frequency-width ordering gamma = O(epsilon). No new physical entities are introduced; the 'three-body interaction' is a mathematical term in the reduced model.

assumptions (4)
  • domain assumption The original dynamics are the all-to-all delayed Kuramoto model, Eq. (1), with uniform delay tau and frequency distribution g(omega).
    This is the baseline model from Yeung-Strogatz [40]; all results are about this specific system.
  • domain assumption The delay is Taylor-expanded and the equation is truncated at O(epsilon^2), with epsilon*tau < 1 assumed.
    Used to obtain Eq. (3) and Eq. (6); the paper explicitly states the convergence assumption but provides no rigorous error bound.
  • domain assumption The Ott-Antonsen ansatz P_m(omega,t) = alpha(omega,t)^m is valid for the Lorentzian frequency distribution.
    This is a standard invariant-manifold ansatz for globally coupled phase oscillators; it is not justified for all initial conditions in this paper.
  • ad hoc to paper gamma = O(epsilon) and (omega_l - omega_k)*tau small, so that f1 and f2 can be replaced by -i tau and 0; then gamma -> 0 for identical oscillators.
    Appendix 2 uses this to derive Eq. (8) and hence the OA equation. The order of limits (gamma -> 0 after OA) may affect the stability thresholds and is not rigorously controlled.

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Cite this review

Pith. "Pith review of Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay." pith.science (2026). https://pith.science/paper/4CQUKDR2

@misc{pith2026251216193,
  author       = {Pith},
  title        = {Pith review of: Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CQUKDR2}},
  note         = {Machine review of arXiv:2512.16193}
}
read the original abstract

We show that higher-order interactions naturally emerge from time-delayed pairwise coupling in Kuramoto oscillators. By expanding the delayed pairwise coupling to the second order, we derive a delay-free Kuramoto model possessing both pairwise and three-body interactions. Numerical simulations and stability analysis demonstrate that the three-body Kuramoto model and the time-delayed pairwise Kuramoto model exhibit qualitatively consistent synchronization transitions under appropriate conditions. In particular, the bistability arising in the time-delayed Kuramoto model is accounted for by the three-body interactions. Our findings reveal that time delays can be recast effectively as higher-order interactions, providing an insight into how coupling delays shape collective dynamics.

Figures

Figures reproduced from arXiv: 2512.16193 by the authors.

Figure 1
Figure 1. Synchronization transitions in the case of 2 oscil [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Synchronization transitions in the case of (a) fixed time delay [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Stability diagram of the incoherent state and the fully synchronized state obtained from (a) the order parameter [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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  2. Impact of Channel Dynamics on Higher-order Interactions of Oscillators

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  3. On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model

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Reference graph

Works this paper leans on

56 extracted references · 2 linked inside Pith · cited by 3 Pith papers

  1. [1]

    Battiston, G

    F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lu- cas, A. Patania, J.-G. Young, and G. Petri. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep., 874:1–92, 2020

    2020

  2. [2]

    Bianconi.Higher-Order Networks: An introduction to simplicial complexes

    G. Bianconi.Higher-Order Networks: An introduction to simplicial complexes. Cambridge University Press, 2021

    2021

  3. [3]

    Battiston, E

    F. Battiston, E. Amico, A. Barrat, G. Bianconi, G.F. de Arruda, B. Franceschiello, I. Iacopini, S. K´ efi, V. La- tora, Y. Moreno, M.M. Murray, T.P. Peixoto, F. Vac- carino, and G. Petri. The physics of higher-order inter- actions in complex systems.Nat. Phys., 17:1093–1098, 2021

    2021

  4. [4]

    Majhi, M

    S. Majhi, M. Perc, and D. Ghosh. Dynamics on higher- order networks: a review.Journal of The Royal Society Interface, 19(188):20220043, 2022

    2022

  5. [5]

    Bick, Elizabeth Gross, Heather A Harrington, and Michael T Schaub

    C. Bick, Elizabeth Gross, Heather A Harrington, and Michael T Schaub. What are higher-order networks? SIAM Rev., 65(3):686–731, 2023

    2023

  6. [6]

    Boccaletti, P

    S. Boccaletti, P. De Lellis, C.I. Del Genio, K. Alfaro- Bittner, R. Criado, S. Jalan, and M. Romance. The structure and dynamics of networks with higher order interactions.Phys. Rep., 1018:1–64, 2023

    2023

  7. [7]

    Muolo, L

    R. Muolo, L. Giambagli, H. Nakao, D. Fanelli, and T. Carletti. Turing patterns on discrete topologies: from networks to higher-order structures.Proceedings of the Royal Society A, 480(2302):20240235, 2024

    2024

  8. [8]

    Mill´ an, H

    A.P. Mill´ an, H. Sun, L. Giambagli, R. Muolo, T. Carletti, J.J. Torres, F. Radicchi, J. Kurths, and G. Bianconi. Topology shapes dynamics of higher-order networks.Na- ture Physics, 21:353–361, 2025. 6

    2025

Show all 56 references
  1. [9]

    Battiston, C

    F. Battiston, C. Bick, M. Lucas, A.P. Mill´ an, P.S. Skardal, and Y. Zhang. Collective dynamics on higher- order networks.arXiv preprint arXiv:2510.05253, 2025

    2025

  2. [10]

    Mill´ an, J.J

    A.P. Mill´ an, J.J. Torres, and G. Bianconi. Explo- sive higher-order Kuramoto dynamics on simplicial com- plexes.Phys. Rev. Lett., 124(21):218301, 2020

    2020

  3. [11]

    Di Patti, L

    L.V Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Ro- mance, R. Criado, M. Frasca, V. Latora, and S. Boc- caletti. Stability of synchronization in simplicial com- plexes.Nat. Comm., 12(1):1–13, 2021

    2021

  4. [12]

    Gallo, R

    L. Gallo, R. Muolo, L.V. Gambuzza, V. Latora, M. Frasca, and T. Carletti. Synchronization induced by directed higher-order interactions.Comm. Phys., 5:236, 2022

    2022

  5. [13]

    Von Der Gracht, E

    S. Von Der Gracht, E. Nijholt, and B. Rink. Hyper- networks: cluster synchronization is a higher-order ef- fect.SIAM Journal on Applied Mathematics, 83(6):2329– 2353, 2023

    2023

  6. [14]

    von der Gracht, E

    S. von der Gracht, E. Nijholt, and B. Rink. Higher- order interactions lead to ‘reluctant’ synchrony breaking. Proceedings A of the Royal Society, 480(2301):20230945, 2024

    2024

  7. [15]

    Zhang, P.S

    Y. Zhang, P.S. Skardal, F. Battiston, G. Petri, and M. Lucas. Deeper but smaller: Higher-order interactions increase linear stability but shrink basins.Science Ad- vances, 10(40):eado8049, 2024

    2024

  8. [16]

    Z. Wang, J. Zhu, and X. Liu. Network stochastic res- onance under higher-order interactions.arXiv preprint arXiv:2509.14796, 2025

    2025

  9. [17]

    Muolo, L

    R. Muolo, L. Gallo, V. Latora, M. Frasca, and T. Car- letti. Turing patterns in systems with high-order inter- actions.Chaos, Solitons & Fractals, 166:112912, 2023

    2023

  10. [18]

    Carletti, D

    T. Carletti, D. Fanelli, and S. Nicoletti. Dynamical sys- tems on hypergraphs.J. Phys. Complex., 1(3):035006, 2020

    2020

  11. [19]

    Iacopini, G

    I. Iacopini, G. Petri, A. Barrat, and V. Latora. Simpli- cial models of social contagion.Nat. Comm., 10(1):2485, 2019

    2019

  12. [20]

    Neuh¨ auser, A

    L. Neuh¨ auser, A. Mellor, and R. Lambiotte. Multibody interactions and nonlinear consensus dynamics on net- worked systems.Phys. Rev. E, 101:032310, Mar 2020

    2020

  13. [21]

    De Lellis, F

    P. De Lellis, F. Della Rossa, F. Lo Iudice, and D. Li- uzza. Pinning control of hypergraphs.IEEE Control Syst. Lett., 7:691–696, 2022

    2022

  14. [22]

    Della Rossa, D

    F. Della Rossa, D. Liuzza, F. Lo Iudice, and P. De Lellis. Emergence and control of synchronization in networks with directed many-body interactions.Phys. Rev. Lett., 131(20):207401, 2023

    2023

  15. [23]

    Xia and L

    R. Xia and L. Xiang. Pinning control of simplicial com- plexes.European Journal of Control, 77:100994, 2024

    2024

  16. [24]

    Kuramoto

    Y. Kuramoto. Self-entrainment of a population of cou- pled non-linear oscillators. In H. Araki, editor,Inter- national Symposium on Mathematical Problems in Theo- retical Physics, pages 420–422, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg

    1975

  17. [25]

    Kuramoto.Chemical Oscillations, Waves, and Tur- bulence

    Y. Kuramoto.Chemical Oscillations, Waves, and Tur- bulence. Springer, Berlin, 1984

    1984

  18. [26]

    Acebr´ on, L.L

    J.A. Acebr´ on, L.L. Bonilla, C.J. P´ erez Vicente, F. Ri- tort, and R. Spigler. The Kuramoto model: A simple paradigm for synchronization phenomena.Reviews of modern physics, 77(1):137–185, 2005

    2005

  19. [27]

    Tanaka and T

    T. Tanaka and T. Aoyagi. Multistable attractors in a network of phase oscillators with three-body interactions. Phys. Rev. Lett., 106(22):224101, 2011

    2011

  20. [28]

    Skardal and A

    P.S. Skardal and A. Arenas. Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching.Communications Physics, 3(1):218, Nov 2020

    2020

  21. [29]

    Lucas, G

    M. Lucas, G. Cencetti, and F. Battiston. Multiorder Laplacian for synchronization in higher-order networks. Physical Review Research, 2(3):033410, 2020

    2020

  22. [30]

    Le´ on, R

    I. Le´ on, R. Muolo, S. Hata, and H. Nakao. Higher-order interactions induce anomalous transitions to synchrony. Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence, 34(1):013105, 01 2024

    2024

  23. [31]

    Muolo, S

    I Le´ on, R. Muolo, S. Hata, and H. Nakao. Theory of phase reduction from hypergraphs to simplicial com- plexes: A general route to higher-order Kuramoto mod- els.Physica D: Nonlinear Phenomena, 482:134858, 2025

    2025

  24. [32]

    H. Nakao. Phase reduction approach to synchronisa- tion of nonlinear oscillators.Contemporary Physics, 57(2):188–214, Oct. 2016

    2016

  25. [33]

    Monga, D

    B. Monga, D. Wilson, T. Matchen, and J. Moehlis. Phase reduction and phase-based optimal control for biological systems: a tutorial.Biological cybernetics, 113(1):11–46, 2019

    2019

  26. [34]

    Pietras and A

    B. Pietras and A. Daffertshofer. Network dynamics of coupled oscillators and phase reduction techniques. Physics Reports, 819:1–105, 2019

    2019

  27. [35]

    Le´ on and D

    I. Le´ on and D. Paz´ o. Phase reduction beyond the first or- der: The case of the mean-field complex ginzburg-landau equation.Phys. Rev. E, 100:012211, Jul 2019

    2019

  28. [36]

    C. Bick, T. B¨ ohle, and C. Kuehn. Higher-order network interactions through phase reduction for oscillators with phase-dependent amplitude.Journal of Nonlinear Sci- ence, 34(4):77, 2024

    2024

  29. [37]

    Schwartz, Luis Mier-y Teran Romero, Christoffer R

    Klementyna Szwaykowska, Ira B. Schwartz, Luis Mier-y Teran Romero, Christoffer R. Heckman, Dan Mox, and M. Ani Hsieh. Collective motion patterns of swarms with delay coupling: Theory and experiment.Phys. Rev. E, 93:032307, Mar 2016

    2016

  30. [38]

    Lag synchronization of coupled time- delayed fitzhugh–nagumo neural networks via feedback control.Scientific Reports, 11(1):3884, 2021

    Malik Muhammad Ibrahim, Muhammad Ahmad Kam- ran, Malik Muhammad Naeem Mannan, Il Hyo Jung, and Sangil Kim. Lag synchronization of coupled time- delayed fitzhugh–nagumo neural networks via feedback control.Scientific Reports, 11(1):3884, 2021

    2021

  31. [39]

    Pulse-coupled oscillator synchroniza- tion: Bridging theory and experiments with electronic firefly networks.Chaos, Solitons & Fractals, 196:116377, 2025

    Mois´ es Santill´ an. Pulse-coupled oscillator synchroniza- tion: Bridging theory and experiments with electronic firefly networks.Chaos, Solitons & Fractals, 196:116377, 2025

    2025

  32. [40]

    M. K. Stephen Yeung and Steven H. Strogatz. Time delay in the Kuramoto model of coupled oscillators.Phys. Rev. Lett., 82:648–651, Jan 1999

    1999

  33. [41]

    Kozyreff, A

    G. Kozyreff, A. G. Vladimirov, and Paul Mandel. Global coupling with time delay in an array of semiconductor lasers.Phys. Rev. Lett., 85:3809–3812, Oct 2000

    2000

  34. [42]

    Time delay effect in a living coupled oscillator system with the plasmodium of physarum polycephalum.Phys

    Atsuko Takamatsu, Teruo Fujii, and Isao Endo. Time delay effect in a living coupled oscillator system with the plasmodium of physarum polycephalum.Phys. Rev. Lett., 85:2026–2029, Aug 2000

    2026

  35. [43]

    En- hancing power grid synchronization and stability through time-delayed feedback control.Phys

    Halgurd Taher, Simona Olmi, and Eckehard Sch¨ oll. En- hancing power grid synchronization and stability through time-delayed feedback control.Phys. Rev. E, 100:062306, Dec 2019

    2019

  36. [44]

    B¨ ottcher, Andreas Otto, Stefan Kettemann, and Carsten Agert

    Philipp C. B¨ ottcher, Andreas Otto, Stefan Kettemann, and Carsten Agert. Time delay effects in the control of synchronous electricity grids.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 30(1):013122, 01 7 2020

    2020

  37. [45]

    Ott and T.M

    E. Ott and T.M. Antonsen. Low dimensional behavior of large systems of globally coupled oscillators.Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(3), September 2008

    2008

  38. [46]

    Antonsen

    Wai Shing Lee, Edward Ott, and Thomas M. Antonsen. Large coupled oscillator systems with heterogeneous in- teraction delays.Phys. Rev. Lett., 103:044101, Jul 2009

    2009

  39. [47]

    A solu- ble active rotater model showing phase transitions via mutual entertainment.Progress of Theoretical Physics, 76(3):576–581, 09 1986

    Hidetsugu Sakaguchi and Yoshiki Kuramoto. A solu- ble active rotater model showing phase transitions via mutual entertainment.Progress of Theoretical Physics, 76(3):576–581, 09 1986

    1986

  40. [48]

    Namura, R

    N. Namura, R. Muolo, and H. Nakao. Optimal inter- action functions realizing higher-order kuramoto dynam- ics with arbitrary limit-cycle oscillators.arXiv preprint arXiv:2510.14501, 2025

    2025

  41. [49]

    Slow switching in a population of delayed pulse-coupled oscillators.Phys

    Hiroshi Kori. Slow switching in a population of delayed pulse-coupled oscillators.Phys. Rev. E, 68:021919, Aug 2003

    2003

  42. [50]

    D’Souza, M

    R.M. D’Souza, M. di Bernardo, and Y.-Y. Liu. Con- trolling complex networks with complex nodes.Nature Reviews Physics, 5(4):250–262, 2023

    2023

  43. [51]

    von der Gracht, E

    S. von der Gracht, E. Nijholt, and B. Rink. A parametri- sation method for high-order phase reduction in coupled oscillator networks.arXiv preprint arXiv:2306.03320, 2023

    2023 arXiv

  44. [52]

    C. Bick, B. Rink, and B.A.J. de Wolff. When time de- lays and phase lags are not the same: higher-order phase reduction unravels delay-induced synchronization in os- cillator networks.arXiv preprint arXiv:2404.11340, 2024

    2024 arXiv

  45. [53]

    C. Bick, B. Rink, and B.A.J. de Wolff. Higher- order phase reduction for delay-coupled oscillators be- yond the phase-shift approximation.arXiv preprint arXiv:2510.27524, 2025. Appendix A: Supplementary

    2025

  46. [54]

    (3), from the time-delay Kuramoto model, Eq

    Derivation of the higher-order interaction from time-delayed pairwise coupling We show the details of the calculation to derive the higher-order Kuramoto model, Eq. (3), from the time-delay Kuramoto model, Eq. (1), under the assumption of smallϵ. Eq. (1) and its Taylor expansi...

  47. [55]

    (8) from Eq

    Derivation of Eq.(8) We derive Eq. (8) from Eq. (3) under the assumptionγ=O(ϵ) andN≫1. From Eq. (3), ˙θj(t) =ω j + ϵ N NX k=1 k̸=j ei(θk(t)−θj (t)−ωkτ) −e −i(θk(t)−θj (t)−ωkτ) 2i + ϵ2 N 2 NX k=1 k̸=j NX l=1 l̸=k ei(θk(t)−θj (t)−ωkτ) +e −i(θk(t)−θj (t)−ωkτ) 2 ei(θl(t)−θk(t)−ωlτ...

  48. [56]

    (9) is redefined as zm(t) = Z 2π 0 Z ∞ −∞ eim(θ′−ω′τ) P(θ ′, ω′, t)g(ω′)dω′dθ′,(A10) and the time evolution ofP(θ, ω, t) is given from Eq

    Ott-Antonsen ansatz In theN→ ∞limit, the general complex order parameter in Eq. (9) is redefined as zm(t) = Z 2π 0 Z ∞ −∞ eim(θ′−ω′τ) P(θ ′, ω′, t)g(ω′)dω′dθ′,(A10) and the time evolution ofP(θ, ω, t) is given from Eq. (8) as ∂P ∂t =− ∂ ∂θ ˙θP =− ∂ ∂θ ω+ 1 2i e−iθH−e iθH ∗ P ,...

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Reviewed August 3, 2026 · model on record in the stance chip above.