pith. sign in

arxiv: 2606.07002 · v1 · pith:N6W5VLWFnew · submitted 2026-06-05 · ❄️ cond-mat.stat-mech · nlin.CD

Phase lag enhances synchronization in coupled oscillators with inertia

Sudo Yi , Cook Hyun Kim , Heetae Kim , B. Kahng This is my paper

Pith reviewed 2026-06-27 20:45 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.CD
keywords phase lagsynchronizationKuramoto modelinertiaoscillator clustersentrainmentsecond-order model
0
0 comments X

The pith

A phase lag applied to a subset of inertial oscillators steers their primary cluster to merge with others and raises global synchronization.

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

The second-order Kuramoto model with inertia produces lower synchronization than the first-order version because multiple clusters form at different frequencies. The paper examines whether an external phase lag applied to a subset of oscillators already in steady state can improve this situation. It reports that the lag directs the primary cluster along a path that allows it to merge with higher-order clusters. A sympathetic reader would care because the finding points to a controllable way to raise entrainment levels in systems where inertia creates persistent frequency separation.

Core claim

In the second-order Kuramoto model with inertia, multiple synchronized clusters with different frequencies lower overall synchronization. Applying a symmetry-breaking phase lag to a subset of oscillators steers the primary cluster along a specific path, enabling it to merge with higher-order clusters and thereby enhancing global synchronization.

What carries the argument

Selective phase lag applied to a subset of oscillators, which steers the primary cluster's path to permit merging with higher-order clusters.

Load-bearing premise

The system has already reached a steady state containing multiple synchronized clusters with different frequencies, and an external phase lag can be applied selectively to a subset without disrupting the underlying dynamics in unforeseen ways.

What would settle it

Numerical simulations in which the phase lag is introduced and the primary cluster remains at a distinct frequency from the higher-order clusters, with no measurable rise in the global order parameter.

Figures

Figures reproduced from arXiv: 2606.07002 by B. Kahng, Cook Hyun Kim, Heetae Kim, Sudo Yi.

Figure 1
Figure 1. Figure 1: Plot of the order parameter 𝑅 for the 2nd KM without (blue) and with (orange) phase lags as a function of the coupling constant 𝐾 for (a) 𝑚 = 0, (b) 𝑚 = 1, and (c) 𝑚 = 5 for a single realization (left column). For 𝑚 = 0 and 𝑚 = 1, imposing a phase lag reduces 𝑅, whereas for 𝑚 = 5, 𝑅 shows a crossover behavior as 𝐾 varies. Close to the transition point 𝐾𝑐 , synchronization is enhanced, while for sufficientl… view at source ↗
Figure 2
Figure 2. Figure 2: (Left column) Plots of the mean angular velocities of the primary and higher-order synchronized clusters versus intrinsic frequencies for (a) 𝑚 = 0, (b) 𝑚 = 1, and (c) 𝑚 = 5 with 𝑁 = 5000, and 𝐾 = 5, starting from the phase-lag-free state (𝛼0 = 0) (blue) and after introducing the phase-lag with 𝛼0 = 0.3𝜋 in the steady states (orange). In all cases, the phase lag effectively shifts the boundary of the prima… view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of the instantaneous frequencies ̇𝜑𝑖 (𝑡) of selected oscillators with different intrinsic frequencies 𝜔𝑖 , (a) and (b) show the cases 𝑚 = 1 and 𝑚 = 5, respectively, with 𝑁 = 5000, 𝐾 = 5, 𝛼0 = 0.3𝜋, and 𝜂 = 0.5. For 𝑡 < 0, the system evolves without phase lag and is in a steady state. At 𝑡 = 0, the phase lag is introduced to an 𝜂 fraction of oscillators. The system relaxes to a new steady sta… view at source ↗
read the original abstract

The second-order Kuramoto model with inertia exhibits different dynamical behaviors than the first-order KM without inertia. A central difference is its lower synchronization due to the emergence of multiple synchronized clusters with different frequencies. We aim to investigate how such lowered synchronization can be improved by applying external perturbations to the system in a steady state, for example, a symmetry-breaking phase lag to a subset of oscillators. We find that this phase lag steers the primary cluster along a specific path and enables it to merge with higher-order clusters, thereby enhancing global synchronization. Our results reveal a mechanism by which controlled phase lag can improve entrainment in inertial oscillator systems, with possible implications for synchronization control in inertial oscillator networks.

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

2 major / 1 minor

Summary. The paper examines the second-order Kuramoto model with inertia, noting its reduced synchronization relative to the first-order case due to the formation of multiple clusters with distinct frequencies. It proposes that applying an external symmetry-breaking phase lag selectively to a subset of oscillators, once the system has reached a steady state with these clusters, steers the primary cluster along a path that enables merger with higher-order clusters and thereby improves global synchronization. The central result is presented as a control mechanism for entrainment in inertial oscillator systems.

Significance. If the mechanism is rigorously demonstrated, the work identifies a concrete perturbation strategy that can enhance synchronization in systems where inertia naturally produces frequency-offset clusters. This could inform control protocols in physical networks of inertial oscillators, provided the perturbation leaves the dynamics of non-targeted clusters unaltered.

major comments (2)
  1. [Abstract and the section describing the external perturbation] The functional form of the applied phase lag is not specified (whether it enters as a constant offset inside the sine coupling, shifts the natural frequency of the targeted subset, or acts as an additive torque). Without this definition it is impossible to confirm that the frequency offsets and inertia dynamics of the remaining clusters remain unchanged after the perturbation is switched on, which is required for the merger claim.
  2. [Results and Methods] No equations, simulation parameters, quantitative synchronization measures (e.g., order-parameter time series with error bars), or explicit demonstration that non-targeted clusters retain their original frequencies appear in the provided text. These are load-bearing for the assertion that selective phase lag produces merger without side effects on other clusters.
minor comments (1)
  1. [Abstract] The abstract states the central finding but supplies no supporting equations or data; the full manuscript should include these in the main text rather than relying on the abstract alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [Abstract and the section describing the external perturbation] The functional form of the applied phase lag is not specified (whether it enters as a constant offset inside the sine coupling, shifts the natural frequency of the targeted subset, or acts as an additive torque). Without this definition it is impossible to confirm that the frequency offsets and inertia dynamics of the remaining clusters remain unchanged after the perturbation is switched on, which is required for the merger claim.

    Authors: We agree that the functional form of the symmetry-breaking phase lag must be defined explicitly to support the claim regarding unchanged dynamics in non-targeted clusters. The revised manuscript will include the precise mathematical implementation of the perturbation. revision: yes

  2. Referee: [Results and Methods] No equations, simulation parameters, quantitative synchronization measures (e.g., order-parameter time series with error bars), or explicit demonstration that non-targeted clusters retain their original frequencies appear in the provided text. These are load-bearing for the assertion that selective phase lag produces merger without side effects on other clusters.

    Authors: We acknowledge that the current manuscript text omits the governing equations, simulation parameters, quantitative measures, and explicit verification of unchanged frequencies for non-targeted clusters. These elements will be added to the revised manuscript, including order-parameter time series and supporting demonstrations. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper reports numerical observations on how an external phase lag applied to a subset of oscillators in the inertial Kuramoto model can promote cluster mergers and global synchronization. No load-bearing derivations, parameter fits renamed as predictions, self-citation chains, or ansatzes imported from prior author work appear in the abstract or described claims. The central result is presented as an empirical finding from the model's dynamics under perturbation and remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no information on free parameters, background axioms, or new postulated entities; the ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5644 in / 1024 out tokens · 15956 ms · 2026-06-27T20:45:10.930033+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

28 extracted references · 12 canonical work pages

  1. [1]

    2512.24765 , archivePrefix=

    Predicting the Oscillatory Regimes of Global Synchrony Induced by Secondary Clusters , author=. 2512.24765 , archivePrefix=

    arXiv

  2. [2]

    Explosive Synchronization Transitions in Scale-Free Networks , author =. Phys. Rev. Lett. , volume =. 2011 , month =. doi:10.1103/PhysRevLett.106.128701 , url =

    work page doi:10.1103/physrevlett.106.128701 2011

  3. [3]

    Explosive synchronization enhanced by time-delayed coupling , author =. Phys. Rev. E , volume =. 2012 , month =. doi:10.1103/PhysRevE.86.016102 , url =

    work page doi:10.1103/physreve.86.016102 2012

  4. [4]

    Physical Review E , author =

    Self-consistent method and steady states of second-order oscillators , volume =. Physical Review E , author =. 2018 , pages =. doi:10.1103/PhysRevE.98.042201 , number =

    work page doi:10.1103/physreve.98.042201 2018

  5. [5]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , author =

    Synchronized clusters in globally connected networks of second-order oscillators:. Chaos: An Interdisciplinary Journal of Nonlinear Science , author =. 2021 , pages =. doi:10.1063/5.0057125 , abstract =

    work page doi:10.1063/5.0057125 2021

  6. [6]

    Physical Review Letters , author =

    First. Physical Review Letters , author =. 1997 , pages =. doi:10.1103/PhysRevLett.78.2104 , number =

    work page doi:10.1103/physrevlett.78.2104 1997

  7. [7]

    Physica D: Nonlinear Phenomena , author =

    Self-synchronization of coupled oscillators with hysteretic responses , volume =. Physica D: Nonlinear Phenomena , author =. 1997 , pages =. doi:10.1016/S0167-2789(96)00193-5 , abstract =

    work page doi:10.1016/s0167-2789(96)00193-5 1997

  8. [8]

    Chaos, Solitons & Fractals , author =

    Cluster-mediated synchronization dynamics in globally coupled oscillators with inertia , volume =. Chaos, Solitons & Fractals , author =. 2025 , pages =. doi:10.1016/j.chaos.2025.116281 , abstract =

    work page doi:10.1016/j.chaos.2025.116281 2025

  9. [9]

    Nonlinear Phenomena in Complex Systems , volume =

    Yoshiki Kuramoto and Dorjsuren Battogtokh , title =. Nonlinear Phenomena in Complex Systems , volume =. 2002 , url =

    2002

  10. [10]

    Abrams and Steven H

    Daniel M. Abrams and Steven H. Strogatz , title =. Physical Review Letters , volume =. 2004 , month = oct, doi =

    2004

  11. [11]

    Abrams and Rennie Mirollo and Steven H

    Daniel M. Abrams and Rennie Mirollo and Steven H. Strogatz and Daniel A. Wiley , title =. Physical Review Letters , volume =. 2008 , month = aug, doi =

    2008

  12. [12]

    Chaos , year =

    Extended mean-field approach for chimera states in random complex networks , author =. Chaos , year =. doi:10.1063/5.0079471 , pmid =

    work page doi:10.1063/5.0079471

  13. [13]

    Nature Communications , volume =

    Network-induced multistability through lossy coupling and exotic solitary states , author =. Nature Communications , volume =. 2020 , publisher =. doi:10.1038/s41467-020-14417-7 , url =

    work page doi:10.1038/s41467-020-14417-7 2020

  14. [14]

    Nature Communications , year =

    Sajadi, Amirhossein and Kenyon, Rick Wallace and Hodge, Bri-Mathias , title =. Nature Communications , year =

  15. [15]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Low dimensional behavior of large systems of globally coupled oscillators , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. 2008 , publisher =

    2008

  16. [16]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Long time evolution of phase oscillator systems , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. 2009 , publisher =

    2009

  17. [17]

    Pinto and Alberto Saa , keywords =

    Rafael S. Pinto and Alberto Saa , keywords =. Synchrony-optimized networks of Kuramoto oscillators with inertia , journal =. 2016 , issn =. doi:https://doi.org/10.1016/j.physa.2016.07.009 , url =

    work page doi:10.1016/j.physa.2016.07.009 2016

  18. [18]

    Higher-order interactions in Kuramoto oscillators with inertia , author =. Phys. Rev. E , volume =. 2023 , month =. doi:10.1103/PhysRevE.108.024215 , url =

    work page doi:10.1103/physreve.108.024215 2023

  19. [19]

    Synchronization in the presence of time delays and inertia: Stability criteria , author =. Phys. Rev. E , volume =. 2022 , month =. doi:10.1103/PhysRevE.105.014210 , url =

    work page doi:10.1103/physreve.105.014210 2022

  20. [20]

    , title =

    Yi, Sudo and Kim, Cook Hyun and Kim, Heetae and Kahng, B. , title =. 2026 , note =

    2026

  21. [21]

    Physical Review E , year =

    Olmi, Simona and Navas, Aitor and Boccaletti, Stefano and Torcini, Alessandro , title =. Physical Review E , year =

  22. [22]

    Nature Physics , volume =

    Symmetry breaking in twisted double bilayer graphene , author =. Nature Physics , volume =. 2021 , publisher =

    2021

  23. [23]

    Independent Fast Phase Shifting Transformer: A Flexible and High-Precision Power Flow Controller , year=

    Yuan, Jiaxin and Zhang, Weizhe and Mei, Jiajun and Gan, Dongliang and Zhou, Hang and Zheng, Yuankun and Liu, Jiawei , journal=. Independent Fast Phase Shifting Transformer: A Flexible and High-Precision Power Flow Controller , year=

  24. [24]

    Chaos, Solitons & Fractals , volume =

    Hybrid synchronization with continuously varying exponent in modernized power grid , author =. Chaos, Solitons & Fractals , volume =. 2024 , doi =

    2024

  25. [25]

    Progress of Theoretical Physics , volume =

    Sakaguchi, Hidetsugu and Kuramoto, Yoshiki , title =. Progress of Theoretical Physics , volume =. 1986 , month =

    1986

  26. [26]

    On the stability of the center for time periodic solutions , author=. Trans. Moscow Math. Soc , volume=

  27. [27]

    2013 , publisher=

    Nonlinear oscillations, dynamical systems, and bifurcations of vector fields , author=. 2013 , publisher=

    2013

  28. [28]

    Tyloo and L

    M. Tyloo and L. Pagnier and P. Jacquod , title =. Science Advances , volume =. 2019 , doi =

    2019