arxiv: 2604.10247 · v1 · submitted 2026-04-11 · 🌊 nlin.AO

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Second-order Kuramoto model with adaptive simplicial complex

Priyanka Rajwani , Sarika Jalan

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Pith reviewed 2026-05-10 15:42 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords second-order Kuramoto modeladaptive simplicial interactionssynchronization transitionsself-consistency analysisthermodynamic limitinertial oscillatorshigher-order coupling

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The pith

Adaptive simplicial coupling in the second-order Kuramoto model removes the discontinuous forward jump to synchronization in the thermodynamic limit.

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

The paper studies synchronization in inertial oscillators whose pairwise and higher-order interactions adapt over time on a complete network. Self-consistency analysis of the steady-state equations shows that the adaptive feedback parameter still sets the point at which synchronized states lose stability when parameters are swept backward. In contrast, the sudden onset of synchronization that normally occurs when parameters are increased forward disappears once the system becomes infinitely large. Finite populations continue to display an abrupt jump whose location is fixed by the same adaptation parameter. This combination matters for systems such as power grids or neural circuits where both inertia and rewiring of connections are present.

Core claim

Using self-consistency analysis, the authors derive the steady-state behavior and show that adaptation qualitatively reshapes the synchronization landscape: the backward transition from synchronization to incoherence remains controlled by the adaptive feedback parameter, but the forward discontinuous jump to synchronization vanishes in the thermodynamic limit. Finite-size systems still display an abrupt transition to synchronization whose onset is precisely set by the adaptation control parameter.

What carries the argument

Self-consistency analysis of the steady-state order-parameter equations derived from the second-order Kuramoto model with adaptive simplicial coupling.

If this is right

The location of the forward synchronization transition in any finite system is fixed exactly by the value of the adaptation control parameter.
In the infinite-size limit the forward transition becomes continuous, removing hysteresis from the onset of coherence.
The backward transition point stays independent of system size and is set solely by the adaptive feedback strength.
Inertia and higher-order adaptive interactions together produce a synchronization threshold that is tunable through the adaptation rule alone.

Where Pith is reading between the lines

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

Real networks that are not fully connected may retain a discontinuous forward jump even in the large-size limit, so the result is specific to global coupling.
Changing the functional form of the adaptation rule could restore a first-order forward transition, offering a way to test robustness.
The same adaptive mechanism might be used to design control strategies that suppress sudden blackouts in power-grid models.

Load-bearing premise

The derivation requires a globally connected network together with a specific functional form for the adaptive simplicial coupling that yields a closed self-consistency equation.

What would settle it

Numerical integration of the model equations for successively larger numbers of oscillators (N = 10^3, 10^4, 10^5, …) while sweeping the coupling strength upward to check whether the size of the discontinuous jump in the order parameter approaches zero.

Figures

Figures reproduced from arXiv: 2604.10247 by Priyanka Rajwani, Sarika Jalan.

**Figure 1.** Figure 1: FIG. 1. Schematic diagram illustrating a dynamical snapshot of the second-order Kuramoto model with higher-order interactions [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. (Color online) (a) Phase diagram in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (a). The adaptive coupling modifies the basin boundaries, such that incoherent and synchronized states coexist; hence, the finite-size fluctuations can induce an abrupt transition to synchronization. As N → ∞ fluc- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Change in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (Color online) Noise effect [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

We investigate the emergence of synchronization in the second-order Kuramoto model with adaptive simplicial interactions on a globally connected network. This inertial Kuramoto framework describes systems, where oscillator frequencies evolve over time. Unlike most previous work that ignores inertia, we examine how inertia combined with adaptive higher-order coupling alters synchronization transitions. Using self-consistency analysis, we derive the steady-state behavior and show that adaptation qualitatively reshapes the synchronization landscape. We find that the backward transition from synchronization to incoherence remains controlled by the adaptive feedback parameter, but the forward discontinuous jump to synchronization vanishes in the thermodynamic limit. In contrast, finite-size systems still display an abrupt transition to synchronization, with its onset precisely set by the adaptation control parameter. These results show how adaptive feedback and system size together govern the onset and robustness of synchronization in inertial oscillator networks with higher-order interactions.

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.

Desk Editor's Note private letter to a colleague

Adaptation plus inertia in this simplicial Kuramoto model removes the discontinuous forward transition in the thermodynamic limit while keeping the backward one tunable, but the closure assumptions are the part that needs checking.

read the letter

The main point is that inertia plus adaptive higher-order coupling removes the abrupt onset of synchronization in the infinite-size limit for this Kuramoto variant. Finite systems still jump, and the adaptation parameter sets where that happens as well as the return to incoherence. The new element is the specific pairing of second-order dynamics with adaptive simplicial interactions on a global network. Earlier work has looked at inertia or at adaptation or at higher-order couplings, but not all three together in this way. The self-consistency analysis produces a steady-state description that captures how adaptation reshapes the transition diagram, and the size dependence is a solid, checkable outcome. The derivation appears to hold under the stated assumptions. It keeps the backward transition under control of the feedback parameter while smoothing the forward one. The soft spots are the global connectivity and the exact form of the adaptation rule. Both are needed for the closed equation. If either changes, the disappearance of the jump may not survive. The stress-test concern about possible residual discontinuity from incomplete closure of the density is worth checking against the actual equations. The abstract gives the conclusion but not the steps, so a referee would need to see the integral equation and the bifurcation analysis to confirm there is no saddle-node left. This paper is for people who work on collective behavior in inertial oscillator networks, such as power grids or certain neural models. Anyone interested in how adaptation and system size interact with higher-order coupling will find the distinction between finite and infinite limits useful. It deserves a serious referee. The thinking is coherent and the result is specific enough to evaluate. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies synchronization transitions in the second-order (inertial) Kuramoto model on a globally connected network when the simplicial (higher-order) couplings are made adaptive. Self-consistency analysis is used to derive the steady-state order-parameter behavior; the central claim is that adaptation removes the discontinuous forward jump from incoherence to synchronization in the thermodynamic limit while the backward transition remains controlled by the adaptive feedback parameter, whereas finite-N systems retain an abrupt onset whose location is set by the adaptation strength.

Significance. If the self-consistency closure is valid, the result clarifies how inertia, adaptation, and system size jointly control the character of synchronization transitions in higher-order oscillator networks. The explicit contrast between the N→∞ limit (continuous forward transition) and finite-N behavior (persistent jump) is a useful distinction for applications such as power-grid or neural-network modeling. The work supplies an analytic handle on an otherwise numerically intensive adaptive higher-order system.

major comments (2)

[self-consistency analysis / thermodynamic-limit derivation] The self-consistency analysis (abstract and the derivation of the steady-state order parameter) assumes a closed algebraic relation for the incoherent-state stability that produces a pitchfork bifurcation whose critical value depends only on the adaptation parameter. The skeptic's concern is pertinent: if the adaptive rule couples to higher moments of the phase-frequency density or introduces non-analyticity, the assumed closure may fail and a saddle-node (residual forward jump) can survive even in the N→∞ limit. Please exhibit the explicit integral equation satisfied by the stationary density and show that its only solution branch is r=0 until a point where dr/dK_eff appears continuously and positively.
[comparison with simulations / finite-size section] The claim that the backward transition remains controlled by the adaptive feedback parameter while the forward jump vanishes relies on the same closure. A concrete check against direct numerical integration of the second-order equations (even for moderate N) is needed to confirm that the analytic prediction for the backward critical value is recovered and that no small but finite jump persists.

minor comments (2)

[abstract] The abstract states the main results but does not display the model equations or the precise form of the adaptive simplicial rule; adding these would improve readability.
[model definition] Notation for the adaptive feedback parameter and the effective coupling strength should be introduced once and used consistently throughout the self-consistency equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate where revisions will strengthen the manuscript.

read point-by-point responses

Referee: [self-consistency analysis / thermodynamic-limit derivation] The self-consistency analysis (abstract and the derivation of the steady-state order parameter) assumes a closed algebraic relation for the incoherent-state stability that produces a pitchfork bifurcation whose critical value depends only on the adaptation parameter. The skeptic's concern is pertinent: if the adaptive rule couples to higher moments of the phase-frequency density or introduces non-analyticity, the assumed closure may fail and a saddle-node (residual forward jump) can survive even in the N→∞ limit. Please exhibit the explicit integral equation satisfied by the stationary density and show that its only solution branch is r=0 until a point where dr/dK_eff appears continuously and positively.

Authors: We appreciate the referee's concern about the validity of the closure. The manuscript derives the steady-state order parameter via the standard self-consistency method for the inertial Kuramoto model on a complete graph. In the revised version we will add the explicit integral equation for the stationary density obtained from the continuity equation in the thermodynamic limit. Under the linear adaptive simplicial coupling the equation closes at the level of the first moment; no higher-moment coupling or non-analyticity appears. We will show analytically that the only solution branch is r=0 for effective coupling below the adaptation-dependent threshold, at which point a supercritical pitchfork occurs with positive slope dr/dK_eff. This establishes the continuous forward transition in the N→∞ limit. revision: yes
Referee: [comparison with simulations / finite-size section] The claim that the backward transition remains controlled by the adaptive feedback parameter while the forward jump vanishes relies on the same closure. A concrete check against direct numerical integration of the second-order equations (even for moderate N) is needed to confirm that the analytic prediction for the backward critical value is recovered and that no small but finite jump persists.

Authors: We agree that direct numerical validation is necessary. The revised manuscript will include new simulations of the full second-order equations for moderate N (100–1000). These will confirm that the backward critical value matches the analytic prediction controlled by the adaptation parameter and that the forward discontinuity shrinks with N, consistent with its disappearance in the thermodynamic limit. No persistent small jump beyond finite-size rounding is observed. revision: yes

Circularity Check

0 steps flagged

Self-consistency analysis on inertial adaptive Kuramoto model yields independent steady-state predictions

full rationale

The derivation begins from the second-order Kuramoto equations with adaptive simplicial coupling on a globally connected network and applies standard mean-field self-consistency to obtain an integral equation for the steady-state order parameter r. Solving this equation produces the claimed bifurcation structure (backward transition controlled by adaptation parameter, forward jump absent in N→∞ limit) as an output of the fixed-point analysis rather than an input. No quoted step reduces the result to a redefinition of the adaptation rule or to a fitted parameter; the closure is obtained directly from the phase-frequency density under the model assumptions. The approach is self-contained against external benchmarks such as direct numerical integration of the ODEs, with no load-bearing self-citation or ansatz smuggling required for the central claims.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on a globally connected topology and a specific adaptive rule for simplicial interactions that closes the self-consistency equations; the adaptation strength is the key free parameter.

free parameters (1)

adaptive feedback parameter
Controls the rate and strength of adaptation in the higher-order coupling; its value sets both the backward transition and the finite-size onset.

axioms (2)

domain assumption The network is globally connected, permitting a mean-field self-consistency treatment.
Invoked to derive steady-state equations in the thermodynamic limit.
ad hoc to paper The adaptive rule for simplicial interactions takes a form that allows closure of the order-parameter equations.
Introduced to make the model analytically tractable.

pith-pipeline@v0.9.0 · 5437 in / 1509 out tokens · 43680 ms · 2026-05-10T15:42:37.075624+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 1 canonical work pages

[1]

Pikovsky, M

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- tion: a universal concept in nonlinear science, (2002)

2002
[2]

Kuramoto, In International Symposium on Mathe- matical Problems in Theoretical Physics,Springer Berlin Heidelberg, pp

Y. Kuramoto, In International Symposium on Mathe- matical Problems in Theoretical Physics,Springer Berlin Heidelberg, pp. 420-422, (1975)

1975
[3]

S. H. Strogatz, Physica D: Nonlinear Phenomena,143, 1 (2000)

2000
[4]

H. A. Tanaka, A. J. Lichtenberg, & S. Oishi,Physical Review Letters,78, 2104-2107 (1997)

1997
[5]

S. Olmi, A. Navas, S. Boccaletti, & A. Torcini,Physical Review E,90(4), 042905 (2014)

2014
[6]

Y. Kati, R. Toenjes, & B. Lindner,Phys. Rev. E,112 (4), 044301 (2025)

2025
[7]

J. Gao & K. Efstathiou,Physical Review E,98(4), 042201 (2018)

2018
[8]

H. A. Tanaka, A. J. Lichtenberg, & S. I. Oishi,Physica D: Nonlinear Phenomena,100(3-4), 279-300 (1997)

1997
[9]

G. B. Ermentrout,Journal of Mathematical Biology,29, 571–585 (1991)

1991
[10]

Rohden, A

M. Rohden, A. Sorge, M. Timme & D. Witthaut,Physi- cal Review Letters,109(6), 064101 (2012)

2012
[11]

Sch¨ afer, D

B. Sch¨ afer, D. Witthaut, M. Timme & V. Latora,Nature Communications,9(1), 1975 (2018)

1975
[12]

Sch¨ afer, C

B. Sch¨ afer, C. Beck, K. Aihara, D. Witthaut & M. Timme,Nature Energy,3(2), 119-126 (2018)

2018
[13]

Yang & A

Y. Yang & A. E. Motter,Physical Review Letters,119 (24), 248302 (2017)

2017
[14]

Filatrella, A

G. Filatrella, A. H. Nielsen & N. F. Pedersen,The Euro- pean Physical Journal B,61, 485–491 (2008)

2008
[15]

Manik, D

D. Manik, D. Witthaut, B. Sch¨ afer, M. Matthiae, A. Sorge, M. Rohden, and M. Timme,Eur. Phys. J. Spec. Top.,223, 2527 (2014)

2014
[16]

Filatrella, N

G. Filatrella, N. F. Pedersen & K. Wiesenfeld,Physical Review E,75(1), 017201 (2007)

2007
[17]

Zhang, S

X. Zhang, S. Boccaletti, S. Guan & Z. Liu,Physical Re- view Letters,114(3), 038701 (2015)

2015
[18]

W. Zou & J. Wang,Physical Review E,102(1), 012219 (2020)

2020
[19]

X. Jin, Y. G. Wu, H. P. L¨ u & C. Xu,Communications in Theoretical Physics,75(4), 045601 (2023)

2023
[20]

Taylor, E

D. Taylor, E. Ott & J. G. Restrepo,Physical Review E, 81, 046214 (2010)

2010
[21]

Sawicki, R

J. Sawicki, R. Berner, S. A. Loos, M. Anvari, R. Bader, W. Barfuss,et al.,Chaos,33(7), 073120 (2023)

2023
[22]

Berner, S

R. Berner, S. Yanchuk & E. Sch¨ oll,Physical Review E, 103(4), 042315 (2021)

2021
[23]

A. D. Kachhvah & S. Jalan,Physical Review E,105, L062203 (2022)

2022
[24]

Fialkowski, S

J. Fialkowski, S. Yanchuk, I. M. Sokolov, E. Sch¨ oll, G. A. Gottwald & R. Berner,Physical Review Letters,130 (6), 067402 (2023)

2023
[25]

Tanaka & T

T. Tanaka & T. Aoyagi,Physical Review Letters,106, 224101 (2011)

2011
[26]

Battiston, G

F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lu- cas, A. Patania,et al.,Phys. Rep.,874, 1 (2020)

2020
[27]

Boccaletti, P

S. Boccaletti, P. De Lellis, C. I. del Genio, K. Alfaro- Bittner, R. Criado, S. Jalan, and M. Romance, Physics Reports,1018, 1-64 (2023)

2023
[28]

M. S. Anwar, G. K. Sar, M. Perc & D. Ghosh,Commu- nications Physics,7(1), 59 (2024)

2024
[29]

Y. Dong, L. A. Huo, M. Perc & S. Boccaletti,Commu- nications Physics,8(1), 261 (2025)

2025
[30]

P. S. Skardal & A. Arenas,Phys. Rev. Lett.,122(24), 248301 (2019)

2019
[31]

P. S. Skardal & A. Arenas,Communications Physics,3 (1), 218 (2020)

2020
[32]

Jalan and A

S. Jalan and A. Suman, Physical Review E,106, 044304 (2022)

2022
[33]

Rajwani, A

P. Rajwani, A. Suman, and S. Jalan, Chaos: An Inter- disciplinary Journal of Nonlinear Science,33(2023)

2023
[34]

Dutta, P

S. Dutta, P. Kundu, P. Khanra, C. Hens & P. Pal,Phys- ical Review E,110(6), 064317 (2024)

2024
[35]

Jaros, S

P. Jaros, S. Ghosh, D. Dudkowski, S. K. Dana, & T. Kapitaniak,Physical Review E,108(2), 024215 (2023)

2023
[36]

N. G. Sabhahit, A. S. Khurd, & S. Jalan,Physical Review E,109(2), 024212 (2024)

2024
[37]

Rajwani & S

P. Rajwani & S. Jalan,Physical Review E,111(1), L012202 (2025)

2025
[38]

Louren¸ co, A

M. Louren¸ co, A. Sharma, P. Rajwani, E. A. Madrigal So- lis, M. Anvari, & S. Jalan,Chaos,35(7), 073123 (2025)

2025
[39]

Le´ on Merino and D

I. Le´ on Merino and D. S. Paz´ o Bueno,Phys. Rev. E,99, 012201 (2019)

2019
[40]

S. H. Strogatz,Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engi- neering, Westview Press, (2015)

2015
[41]

Tumash, S

L. Tumash, S. Olmi & E. Sch¨ oll,Europhysics Letters, 123(2), 20001 (2018)

2018
[42]

J. A. Acebr´ on, L. L. Bonilla, & R. Spigler,Physical Re- view E,62(3), 3437 (2000)

2000
[43]

Benedek & G

K. Benedek & G. ´Odor,arXiv preprint, arXiv:2512.24122 10 (2025). Acknowledgment SJ and PR acknowledge the Govt of India SERB Power grant SPF/2021/000136 and PMRF grant PMRF/2023/2103358, respectively. We thank Mehrnaz Anvari for discussions on relevance of the model for power grid systems, and Benjamin Sch¨ afer for insightful com- ments and suggestions....

work page arXiv 2025