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REVIEW 3 major objections 5 minor 48 references

Macroscopic dynamics of oscillator ensembles with communities, higher-order interactions, and phase lags

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One shared phase lag drives two oscillator communities into chaos.

desk verdict The new result, that homogeneous phase lags can induce chaos when higher-order interactions are present, is likely correct, but the evidence linking the reduced-system chaos to the actual oscillator ensemble needs strengthening before publication. read the letter →

arxiv 2507.07956 v1 pith:C5BSNJ6I submitted 2025-07-10 nlin.AO

Sabina Adhikari , Juan G. Restrepo , Per Sebastian Skardal This is my paper · ORCID
classification nlin.AO MSC 34C1534C2837D45 PACS 05.45.Xt
keywords phaseoscillatorshigher-orderinteractionsKuramoto-SakaguchimodelcommunitystructureOtt-Antonsenansatzchaoticsynchronizationmultistabilitylag
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether phase frustration alone—one shared phase lag $\gamma$ on every pairwise and triadic interaction—can produce nontrivial macroscopic dynamics in a network of two communities of coupled oscillators. The answer it argues is yes: using the Ott-Antonsen reduction, the authors obtain a three-dimensional system for the community order parameters and show that, as the lag grows, steady synchronized states give way to periodic synchronization, a period-doubling cascade, and chaotic synchronization. The same reduction also reveals windows of bistability in which a fixed point and a chaotic or periodic attractor coexist, so initial conditions decide the macroscopic state. The result matters because inhomogeneous phase lags were previously seen as the trigger for chaos in oscillator networks; this paper claims that a uniform lag suffices once higher-order interactions are present.

What carries the argument

The load-bearing machinery is the Ott-Antonsen ansatz, which represents the Fourier modes of each community's phase distribution as powers of a single complex function, $f_n^\sigma(\omega,t)=(b^\sigma(\omega,t))^n$. Evaluating $b$ at the pole of the Lorentzian frequency distribution turns the community order parameters $z_\sigma=r_\sigma e^{i\psi_\sigma}$ into closed ODEs, and rewriting them in terms of $r_1$, $r_2$, and the phase difference $\Phi=\psi_2-\psi_1$ yields the three-dimensional system (19)–(21) that carries the entire analysis. Those equations provide the bifurcation diagram in $\gamma$, the linear stability calculation of the anti-phase state, the Lyapunov spectra and Lyapunov dimension of the chaotic attractors, and the basin-of-attraction plot. The phase lag enters inside $\cos$ and $\sin$ arguments such as $\Phi\pm\gamma$ and $2\Phi\pm\gamma$, shifting the relative phase and breaking the time-reversal symmetry of the original model, which is how a single parameter value can both destabilize the anti-phase fixed point and support a chaotic orbit.

What would settle it

Simulate the full oscillator system (2) for $N=10{,}000$ or more oscillators with $K_1=K_2=10$, $\alpha=-0.5$, $\Delta=1$, and $\gamma=1.41$ from many random initial conditions, and estimate the largest Lyapunov exponent from the full system or test whether long-time order parameters stay irregular and non-periodic; if no trajectory shows sustained chaos, or if the attractor collapses once higher Fourier modes are included, the central claim fails.

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Extended reading notes

Core claim

The central claim is that phase-lagged higher-order interactions, together with community structure, are sufficient for oscillatory and chaotic macroscopic dynamics even when all phase lags are identical. For two communities with pairwise and triadic coupling in which every interaction shares the same phase lag $\gamma$, the reduced order-parameter system (19)–(21) exhibits periodic and chaotic attractors for $\gamma$ near $1.4$ when $K_1=K_2=10$, $\alpha=-0.5$, and $\Delta=1$; the chaotic band $\gamma\in[1.4068,1.4163]$ has a positive largest Lyapunov exponent, and direct simulation of $N=10{,}000$ oscillators reproduces the reduced-system time series. The paper also shows bistability in overlapping intervals of $\gamma$, where chaotic and almost-anti-phase states or periodic and skew-phase states coexist depending on initial conditions. For attractive inter-community coupling ($\alpha=0.5$) the same period-doubling route to chaos appears in $\gamma\in[1.373,1.4035]$, so the phenomenon does not require repulsive cross-community coupling.

Load-bearing premise

The argument rests on the Ott-Antonsen ansatz being exact for these oscillator populations, so the chaotic behavior is demonstrated in the reduced three-dimensional system; direct simulation of the full oscillator ensemble checks that reduction at only a few parameter values.

Editorial extensions

If this is right

  • Chaos in oscillator networks does not require heterogeneity in phase lags: a common lag on every pairwise and triadic edge suffices when higher-order interactions are present.
  • Near the chaos window the system is multistable, so for fixed parameters initial conditions decide whether the communities settle to a fixed point or a chaotic attractor; numerical studies should report this dependence.
  • The reduced three-dimensional system (19)–(21) tracks the full $N=10{,}000$ oscillator dynamics at the tested parameter values, supporting reduced-order models as a tool for locating chaos.
  • Attractive inter-community coupling ($\alpha>0$) also produces period-doubling and chaos, so the effect is not an artifact of repulsive cross-community interactions.
  • The anti-phase synchronized state is linearly unstable exactly where the chaotic attractors appear, and it stays stable on the symmetric $r_1=r_2$ manifold; this explains why adiabatic parameter sweeps can miss the chaotic branch.

Reading between the lines

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

  • Extension: with more than two communities the reduced system would have additional phase differences, but the same resonance structure ($\Phi\pm\gamma$ and $2\Phi\pm\gamma$) should still create instability windows, so chaos is plausible in larger community hypergraphs.
  • Extension: a quantitative check of whether the Ott-Antonsen ansatz remains exact on the chaotic attractor—monitoring the decay of higher Fourier modes in direct simulations—would settle whether the reported chaos belongs to the true thermodynamic limit or only to the reduced model.
  • Extension: systematic sweeps of $K_1$, $K_2$, and $\alpha$, which the paper leaves for future work, could test whether the positive-Lyapunov window widens as triadic coupling strengthens, which would sharpen the claim that higher-order interactions are the mechanism behind the uniform-lag chaos.
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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. This paper studies a two-community population of Kuramoto-Sakaguchi phase oscillators with dyadic and triadic (simplicial) coupling and a single homogeneous phase lag gamma. Using the Ott-Antonsen ansatz, the authors reduce the continuum limit to a three-dimensional system, Eqs. (19)-(21), for the community order parameters r1, r2 and phase difference Phi. Numerical bifurcation analysis of this reduced system for alpha = -0.5 and alpha = 0.5 with K1 = K2 = 10, Delta = 1 reveals, as gamma is varied, skew-phase and anti-phase synchronized fixed points, asymmetric and almost anti-phase states, periodic orbits, period-doubling cascades, a chaotic window with positive Lyapunov exponents and Lyapunov dimension between 1 and 2, and multistability with computed basins of attraction. Direct simulation of the original oscillator ensemble, Eq. (2), with N = 10,000 is reported for three gamma values and agrees visually with the reduced-system time series. The paper concludes that homogeneous phase lags, together with higher-order interactions and community structure, are sufficient to induce chaotic macroscopic dynamics in coupled oscillators.

Significance. If correct, the result broadens the known mechanisms for macroscopic chaos in oscillator populations: whereas chaotic dynamics had been reported for networks with heterogeneous phase lags, here a single uniform phase lag suffices once triadic interactions and community structure are present. The reduced three-dimensional system is explicit, and the bifurcation analysis is thorough: continuation sweeps in both directions, Lyapunov spectra, Lyapunov dimension, and basin calculations are all provided, and the code and data are promised on GitHub. The main reserve is the usual one for Ott-Antonsen reductions: the chaotic claim is established for the reduced system, and its transfer to the finite-N oscillator ensemble rests on the assumption that the Ott-Antonsen manifold is attracting, supported only by three visual time-series matches in Fig. 2. With an additional quantitative check and with the algebraic corrections requested below, this would be a solid contribution to the nonlinear dynamics of oscillator populations.

major comments (3)
  1. [II.B, Eq. (8)] The Ott-Antonsen equation (8) is not consistent with the reduced equation (13) that follows. Substituting z = b*(omega0 - i Delta) into Eq. (8) and complex-conjugating gives dot z = (i omega0 - Delta) z + (H* - H z^2)/2, whereas Eq. (13) contains (H - H* z^2)/2 on the right-hand side. Since the later derivation and all numerics are based on Eq. (13), Eq. (8) appears to have H and H* interchanged. Please correct Eq. (8) to read db/dt + i omega b - (H* - H b^2)/2 = 0, and revise the sentence describing the substitution.
  2. [III.A.1, Eq. (22)] The analytical anti-phase solution (22) does not solve the stated fixed-point equations. For r1 = r2 = r and Phi = pi in Eqs. (19)-(20) with Delta = 1, the fixed-point condition is 2 = (1 - r^2)[(1 - alpha)K1 cos(gamma) + (1 - alpha^2)K2 r^2 cos(gamma)], whose solution for r^2 is not Eq. (22). In addition, Eq. (22) does not go to zero at gamma_c = arcsec[K1(1 - alpha)/2] derived in Eq. (23), although the anti-phase branch must terminate at r = 0 there. Because Fig. 4 evaluates the Jacobian of Eqs. (19)-(21) at the r1 = r2 values from Eq. (22), the reported instability interval of the anti-phase state, and hence the interpretation of the chaotic window, should be recomputed with the correct branch and the text amended accordingly.
  3. [Sec. IV and Fig. 2] The central claim that homogeneous phase lags induce chaotic behavior in coupled oscillators is demonstrated for the reduced system (19)-(21), not directly for the original oscillator ensemble (2). The paper does not establish that the Ott-Antonsen manifold defined by Eq. (8) is transversely attracting at the parameters used, and the only finite-N evidence is the visual agreement of three time series in Fig. 2. Please add a quantitative validation of chaos in the full ensemble, for example a finite-time Lyapunov exponent from direct simulation or a convergence check of the order-parameter statistics as N increases, and discuss the conditions under which the Ott-Antonsen reduction is expected to be attracting for these parameters.
minor comments (5)
  1. [Fig. 2 caption] The caption says 'Figs. 1(a)-(c)' but the panels belong to Fig. 2; please correct the cross-reference.
  2. [Eqs. (15)-(16)] The notation psi_{2 sigma'} is undefined; since z_2^sigma = (z_1^sigma)^2, please write 2 psi_{sigma'} or define psi_2^sigma explicitly in the text.
  3. [II.C, Eq. (18)] The statement that K^{sigma sigma' sigma''}_2 'does not affect the dynamics' is confusing because terms with two equal indices, such as K^{122}_2, do enter Eqs. (19)-(21); please clarify that only the fully heterogeneous triadic coupling with three distinct communities is absent for C = 2.
  4. [III.A.1, around Eq. (22)] The text uses both Psi and Phi for the phase difference; please use Phi consistently, including in the sentence that sets the phase difference to pi for the anti-phase state.
  5. [Fig. 8] The basin classification in Fig. 8 uses a threshold of 0.02 on the range of r1 and r2; please state explicitly that no periodic orbit with amplitude above this threshold exists at gamma = 1.416, since such an orbit would also be classified as chaotic by this criterion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reduced equations are derived from the original model and their predictions are checked against direct simulation, so the central claims are not equivalent to their inputs.

full rationale

The paper's derivation chain is self-contained. The low-dimensional system (19)-(21) is obtained from the original oscillator equations (2) via the Ott-Antonsen ansatz (8), which is a stated modeling approximation rather than a restatement of the target result. The parameters K1, K2, alpha, and Delta are fixed a priori and are not fitted to any observed dynamics; only the phase lag gamma is varied. The central chaotic claim is supported by Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension of the reduced system, and then tested against direct numerical simulation of the full model (2) with N=10,000 in Fig. 2, which is an external benchmark not used to fit any parameter. The analytic anti-phase expression (22) and critical gamma (23) are checked against numerical solutions of the same reduced equations; this is an internal consistency check, not a fitted-input prediction. The self-citations [22,31] supply the model form and the gamma=0 steady states, but the new results (oscillatory and chaotic synchronization, bistability) are derived and validated within the present paper. The paper itself notes limitations in scope (specific coupling strengths and alpha values, two-community complete hypergraphs only), but these are scope limitations, not circular steps. The only substantial concern is that the Ott-Antonsen manifold's transverse stability is not established for the chaotic regime and full-system validation is sparse; this is a correctness or robustness issue, not circularity. Therefore the paper does not reduce to its inputs by construction.

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

The central claim depends on the OA ansatz and on the specific choices of coupling structure and parameters. No entirely new entities are introduced; the analysis is carried out in the reduced order-parameter space. The main burden is on the validity of the Ott-Antonsen reduction for chaotic states, which is verified only numerically for a few cases.

free parameters (4)
  • K1 = 10
    Dyadic intra-community coupling strength, chosen as a representative value in all simulations (Sec. III). The existence of the reported dynamics is demonstrated only at this value.
  • K2 = 10
    Triadic intra-community coupling strength, chosen in all simulations (Sec. III).
  • Delta = 1
    Width of the Lorentzian frequency distribution, set to 1 throughout; this sets the time scale and affects bifurcation locations.
  • alpha = -0.5 and 0.5
    Community structure parameter; two representative values are explored, one negative and one positive, demonstrating the effect in both regimes.
assumptions (4)
  • domain assumption Ott-Antonsen ansatz: the Fourier coefficients of the oscillator density take the form f_n^sigma(omega,t) = (b^sigma(omega,t))^n for all n.
    Invoked in Sec. II.B, Eq. (8), to close the moment hierarchy. For chaotic attractors the validity of this ansatz is not proven; it is verified a posteriori by direct simulation for selected parameters.
  • domain assumption Natural frequencies are drawn from identical Lorentzian distributions with the same mean and width in both communities.
    Assumed in Sec. II.C to isolate the effect of phase lags; the frequency distributions are set identical with zero mean-frequency difference.
  • domain assumption The coupling structure scales with alpha as in Eqs. (17)-(18), with intra-community coupling stronger than inter-community coupling.
    Taken from Ref. [31] and applied to the triadic coupling; this defines the community structure but is not derived within the paper.
  • domain assumption The triadic coupling term in Eq. (2) is the relevant higher-order interaction, descending from a phase reduction of the complex Ginzburg-Landau equation.
    Stated in Sec. II.A with references [19,33]; the paper does not derive this from a microscopic model.

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Pith. "Pith review of Macroscopic dynamics of oscillator ensembles with communities, higher-order interactions, and phase lags." pith.science (2026). https://pith.science/paper/C5BSNJ6I

@misc{pith2026250707956,
  author       = {Pith},
  title        = {Pith review of: Macroscopic dynamics of oscillator ensembles with communities, higher-order interactions, and phase lags},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C5BSNJ6I}},
  note         = {Machine review of arXiv:2507.07956}
}
read the original abstract

We study the effects of phase-frustrated, higher-order interactions in a system of coupled phase oscillators with two communities. We use dimensionality reduction techniques to derive a low-dimensional system of ODEs to describe the macroscopic behavior of the system. By analyzing this system we show that, in addition to the fixed point solutions present in a system of oscillators with higher order interactions and community structure only, the system also exhibits oscillatory or chaotic synchronization behavior for some phase lag values. Moreover, some phase lag values give rise to multistability of solutions, where both fixed point solutions and oscillatory or chaotic behavior of the order parameters can be observed, depending on initial conditions.

Figures

Figures reproduced from arXiv: 2507.07956 by the authors.

Figure 1
Figure 1. , where the red and blue filled circles show the steady states of the two communities, i.e., the limiting values of r1 (red) and r2e iΨ (blue). When the order pa￾FIG. 1. Six possible steady states for negative α. Schematic illustration of the first six possible stable states in the two community case for negative α: (i) incoherent state, (ii) skew￾phase synchronized state, (iii) anti-phase synchronized state, (iv) a… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

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Works this paper leans on

48 extracted references · 47 canonical work pages

  1. [28]

    M. J. Panaggio and D. Abrams, Chimera states: coexis- tence of coherence and incoherence in networks of coupled oscillators, Nonlinearity 28, R67 (2015)

    work page 2015

  2. [1]

    For γ > 0, the communities exhibit seven different possible dynamics: (i) The incoherent state with r1 =r2 = 0

    States observed and bifurcation diagram First, we note that for γ = 0 the two communities reach the steady states found in [31]: an incoherent state with r1 = r2 = 0 and a skew-phase synchronized state with r1 = r2 > 0 and 0 < Φ < π/ 2. For γ > 0, the communities exhibit seven different possible dynamics: (i) The incoherent state with r1 =r2 = 0. (ii) The ...

    work page

  3. [2]

    Chaotic Dynamics Now we focus our attention on the chaotic dynamics [case (vi)] observed for γ ∈ [1. 4068, 1. 4163]. In Fig. 5, we show phase attractors for γ = 1. 406, 1. 40685, and 1. 41, generated by plotting r1 cos (Φ) on the horizontal axis andr2 sin (Φ) on the vertical axis. Consistent with Figs. 2 and 3, the attractor for γ = 1. 406 looks periodic,...

    work page

  4. [3]

    This is not apparent in Fig

    Bistability There are various regions of parameter space in which different solutions coexist. This is not apparent in Fig. 3, which was produced by adiabatically increasing γ, but can be seen by comparing this bifurcation diagram with another produced by decreasing γ, or alternatively by choosing initial conditions at random for a given set of parameters....

    work page

  5. [4]

    B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J. G. Re- strepo, M. J. Holland, and A. M. Rey, Synchronization of interacting quantum dipoles, New J. Phys 17, 083063 (2015). 10

    work page 2015

  6. [5]

    Petri, P

    G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, Homological scaffolds of brain functional networks, J. R. Soc. Interface 11, 20140873 (2014)

    work page 2014

  7. [6]

    M. G. Kitzbichler, M. L. Smith, S. R. Christensen, and E. Bullmore, Broadband criticality of human brain net- work synchronization, PLoS Comput. Biol. 5, e1000314 (2009)

    work page 2009

  8. [7]

    Y. Penn, M. Segal, and E. Moses, Network synchroniza- tion in hippocampal neurons, Proc. Natl. Acad. Sci. U. S. A. 113, 6 (2016)

    work page 2016

Show all 48 references
  1. [8]

    C. M. Gray, Synchronous oscillations in neuronal sys- tems: mechanisms and functions, J. Comput. Neurosci. 1, 11 (1994)

    1994

  2. [9]

    S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Crowd synchrony on the millennium bridge, Nature 438, 43 (2005)

    2005

  3. [10]

    Fujino, B

    Y. Fujino, B. M. Pacheco, S.-I. Nakamura, and P. War- nitchai, Synchronization of human walking observed dur- ing lateral vibration of a congested pedestrian bridge, Earthq. Eng. Struct. Dyn. 22, 741 (1993)

    1993

  4. [11]

    Rohden, M

    M. Rohden, M. T. A. Sorge, and D. Witthaut, Self- organized synchronization in decentralized power grids, Phys. Rev. Lett. 109, 5 (2012)

    2012

  5. [12]

    Sarfati, J

    R. Sarfati, J. C. Hayes, ´E. Sarfati, and O. Peleg, Spatio- temporal reconstruction of emergent flash synchroniza- tion in firefly swarms via stereoscopic 360-degree cam- eras, J. R. Soc. Interface 17, 20200179 (2020)

    2020

  6. [13]

    Buck and E

    J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211, 562 (1966)

    1966

  7. [14]

    D¨ orfler, M

    F. D¨ orfler, M. Chertkov, and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. U.S.A. 110, 6 (2012)

    2012

  8. [15]

    Z. Lu, K. Klein-Carde˜ na, S. Lee, T. Antonsen, N. Gir- van, and E. Ott, Resynchronization of circadian oscilla- tors and the east-west asymmetry of jet-lag, Chaos 26, 094811 (2016)

    2016

  9. [16]

    Wiesenfeld, P

    K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchroniz a- tion transitions in a disordered josephson series array, Phys. Rev. Lett. 76, 404 (1996)

    1996

  10. [17]

    Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International sympo- sium on mathematical problems in theoretical physics (Springer, 1975) pp

    Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International sympo- sium on mathematical problems in theoretical physics (Springer, 1975) pp. 420–422

    1975

  11. [18]

    Sakaguchi and Y

    H. Sakaguchi and Y. Kuramoto, A soluble active rotater model showing phase transitions via mutual entertain- ment, Prog. Theor. Phys. 76, 576 (September 1986)

    1986

  12. [19]

    A. E. Sizemore, C. Giusti, A. Kahn, J. M. Vettel, and et al , Cliques and cavities in the human connectome, J. Comput. Neurosci. 44, 115 (2018)

    2018

  13. [20]

    Gibbs, S

    T. Gibbs, S. A. Levin, and J. M. Levine, Coexistence in diverse communities with higher-order interactions, Proc . Natl. Acad. Sci. U.S.A. 119, 015012 (2022)

    2022

  14. [21]

    Ashwin and A

    P. Ashwin and A. Rodrigues, Hopf normal form with sn symmetry and reduction to systems of nonlinearly cou- pled phase oscillators, Physica D 325, 12 (2016)

    2016

  15. [22]

    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 (2019)

    2019

  16. [23]

    P. S. Skardal and A. Arenas, Abrupt desynchronization and extensive multistability in globally coupled oscillat or simplexes, Phys. Rev. Lett. 122, 248301 (2019)

    2019

  17. [24]

    P. S. Skardal and A. Arenas, Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching, Commun. Phys. 3, 6 (2020)

    2020

  18. [25]

    Adhikari, J

    S. Adhikari, J. G. Restrepo, and P. S. Skardal, Synchro- nization of phase oscillators on complex hypergraphs, Chaos 33, 033116 (2023)

    2023

  19. [26]

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

    2020

  20. [27]

    O. E. Omel’chenko and M. Wolfrum, Nonuniversal tran- sitions to synchrony in the sakaguchi-kuramoto model, Phys. Rev. Lett. 109, 164101 (2012)

    2012

  21. [29]

    D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators, Phys. Rev. Lett. 93, 174102 (2004)

    2004

  22. [30]

    M. J. Panaggio, D. M. Abrams, P. Ashwin, and C. R. Laing, Chimera states in networks of phase oscillators: The case of two small populations, Phys. Rev. E 93, 012218 (2016)

    2016

  23. [31]

    C. Bick, M. J. Panaggio, and E. A. Martens, Chaos in kuramoto oscillator networks, Chaos 28, 071102 (2018)

    2018

  24. [32]

    Dutta, A

    S. Dutta, A. Mondal, P. Kundu, P. Khanra, P. Pal, and C. Hens, Impact of phase lag on synchronization in frus- trated kuramoto model with higher-order interactions, Phys. Rev. E 108, 034208 (2023)

    2023

  25. [33]

    A. B. Das, S. Dutta, and P. Pal, Effect of phase- lag on synchronization in adaptive multilayer networks with higher-order interactions (2025), arXiv:2507.01640 [nlin.AO]

    2025 arXiv

  26. [34]

    P. S. Skardal, S. Adhikari, and J. G. Restrepo, Multi- stability in coupled oscillator systems with higher-order interactions and community structure, Chaos 33, 023140 (2023)

    2023

  27. [35]

    Ott and T

    E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 6 (2008)

    2008

  28. [36]

    C. Bick, P. Ashwin, and A. Rodriquez, Chaos in generi- cally coupled phase oscillator networks with non-pairwise interaction, Chaos 26, 8 (2016)

    2016

  29. [37]

    D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable model for chimera states of coupled os- cillators, Phys. Rev. Lett. 101, 084103 (2008)

    2008

  30. [38]

    P. S. Skardal, E. Ott, and J. G. Restrepo, Cluster synchrony in systems of coupled phase oscillators with higher-order coupling, Phys. Rev. E 84, 036208 (2011)

    2011

  31. [39]

    Geist, U

    K. Geist, U. Parlitz, and W. Lauterborn, Comparison of different methods for computing lyapunov exponents, Prog. Theor. Phys. 83, 875 (1990)

    1990

  32. [40]

    Benettin, L

    G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelc yn, Lyapunov characteristic exponents for smooth dynami- cal systems and for hamiltonian systems; a method for computing all of them. part 2: Numerical application, Meccanica 15, 21 (1980)

    1980

  33. [41]

    Frederickson, J

    P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, The liapunov dimension of strange attractors, J. Diff. Eq. 49, 185 (1983)

    1983

  34. [42]

    R. C. Hilborn, Lyapunov Exponents: A Tool to Explore Complex Dynamics, Phys. Today 70, 62 (2017)

    2017

  35. [43]

    Nishikawa and A

    T. Nishikawa and A. E. Motter, Comparative analysis of existing models for power-grid synchronization, New J. Phys. 17, 015012 (2015). 11

    2015

  36. [44]

    Ott, Chaos in Dynamical Systems , 2nd ed

    E. Ott, Chaos in Dynamical Systems , 2nd ed. (Cam- bridge University Press, 2002)

    2002

  37. [45]

    S. H. Strogatz, Nonlinear dynamics and chaos : with ap- plications to physics, biology, chemistry, and engineerin g (CRC Press, 2018)

    2018

  38. [46]

    Eichhorn, S

    R. Eichhorn, S. J. Linz, and P. H¨ anggi, Transforma- tion invariance of lyapunov exponents, Chaos, Solitons & Fractals 12, 1377 (2001)

    2001

  39. [47]

    Pikovsky and A

    A. Pikovsky and A. Politi, Lyapunov Exponents: A Tool to Explore Complex Dynamics (Cambridge University Press, 2016)

    2016

  40. [48]

    Wiesenfeld, P

    K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in josephson arrays: Connection with the ku- ramoto model, Phys. Rev. E 57, 1563 (1998)

    1998

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.