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arxiv: 2605.15628 · v1 · pith:YYHODSXInew · submitted 2026-05-15 · 🌊 nlin.AO

Multi-cluster chimeras in phase oscillators with repulsive nonlocal coupling

Ayushi Saxena , Sangeeta Rani Ujjwal , Ram Ramaswamy This is my paper

Pith reviewed 2026-05-19 18:21 UTC · model grok-4.3

classification 🌊 nlin.AO
keywords phase oscillatorschimera statesnonlocal couplingrepulsive couplingmulti-cluster chimerasOtt-Antonsen reductionstability analysis
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The pith

Phase oscillators with repulsive nonlocal coupling form multi-cluster chimeras where synchronized clusters differ by 2π/n.

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

This paper shows that nonlocal repulsive coupling in phase oscillators on a ring produces multi-cluster chimera states. In these states several synchronized clusters coexist with desynchronized oscillators, and consecutive clusters are either antiphase or in splay, differing in phase by exactly 2π/n. The relation is derived from the symmetry of the repulsive interaction and stands in contrast to the phase arrangements seen with attractive nonlocal coupling. The states are located through numerical simulation, analytical solution of the cluster equations, and linear stability analysis, with the Ott-Antonsen reduction supplying an independent check on the numerics.

Core claim

For phase oscillators on a ring with nonlocal piecewise-linear repulsive coupling that includes a fixed phase lag, the multi-cluster chimera states are characterized by n synchronized clusters whose consecutive members differ in phase by 2π/n. This antiphase or splay relation follows directly from the repulsive sign and the nonlocal spatial structure; it is confirmed both numerically and by solving the reduced equations for the cluster phases and amplitudes. Stability analysis then maps the regions of coupling strength and phase lag in which the states persist, and the Ott-Antonsen ansatz reproduces the same phase relation and stability boundaries.

What carries the argument

Multi-cluster chimera state in which n consecutive synchronized clusters maintain a fixed phase offset of 2π/n under repulsive nonlocal coupling.

If this is right

  • Stability boundaries in the space of coupling strength and phase lag identify where the 2π/n states can be observed.
  • The phase-offset rule is tied to the repulsive sign of the coupling and does not appear in the corresponding attractive case.
  • Ott-Antonsen reduction reproduces both the phase relation and the stability regions found in direct simulation.
  • The chimera states consist of n synchronized clusters coexisting with a desynchronized background.

Where Pith is reading between the lines

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

  • The same phase-locking pattern may appear in other oscillator networks once repulsion is made spatially nonlocal.
  • Changing the range or functional shape of the repulsion could produce different cluster numbers or phase offsets.
  • The mechanism supplies a route to engineer partial synchronization by tuning only the sign and range of the coupling.

Load-bearing premise

The coupling function is assumed to be piecewise linear, nonlocal, repulsive, and to include a fixed phase lag.

What would settle it

A numerical integration or experiment in which the phase differences between consecutive synchronized clusters deviate from 2π/n while the coupling remains piecewise linear, nonlocal, and repulsive would falsify the central phase-relation claim.

Figures

Figures reproduced from arXiv: 2605.15628 by Ayushi Saxena, Ram Ramaswamy, Sangeeta Rani Ujjwal.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic diagram showing the nonlocal piecewise [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Figure showing asymptotic phase profile [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Figure showing various forms of nonlocal piecewise [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Synchronized frequency Ω is shown with the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: it can be seen that αth varies across coupling con￾figurations. The αth also changes with the system size N. For G(x) containing attractive coupling (locally or laterally) without repulsive coupling, αth saturates to a particular value as N increases ( [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Figure showing the switching of two cluster in-phase [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Space-time plots showing the emergence of [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Figure showing the existence of various dynamical [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The fraction of oscillators forming the [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Log-log plot of [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The first column shows the coupling function [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The first column shows the coupling function [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: The first row [(a)-(f)] shows the coupling kernel [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
read the original abstract

Local repulsive coupling tend to a desynchronize ensembles of globally coupled oscillators, but when the repulsive coupling is nonlocal, multi-cluster chimeras can result. In this case, several groups of synchronized oscillators (the so-called clusters) are formed, and these coexist with a set of desynchronized oscillators. For phase oscillators on a ring with nonlocal piecewise linear repulsive coupling that also involves a phase lag, we find that in the multi-cluster chimera state the synchronized clusters are either antiphase or in splay with respect to each other, namely the n consecutive synchronized clusters differ in phase by 2{\pi}/n. This is in contrast to multi-cluster chimeras that are formed with nonlocal attractive coupling. The synchronized solutions are studied numerically as well as analytically and by analysing their stability, we identify the parameter regions where these can be observed. Our numerical results are validated by dimensional reduction using the Ott-Antonsen analysis.

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

1 major / 2 minor

Summary. The manuscript studies multi-cluster chimera states in a ring of phase oscillators with nonlocal piecewise-linear repulsive coupling that includes a fixed phase lag. Through numerical simulations, linear stability analysis of the synchronized cluster states, and Ott-Antonsen reduction, the authors report that the n synchronized clusters differ in phase by exactly 2π/n (antiphase for n=2, splay for larger n). This phase rule is contrasted with the behavior under attractive nonlocal coupling, and stable parameter regions in coupling strength and phase lag are identified.

Significance. If the central observations hold, the work supplies a concrete phase-organization rule for multi-cluster chimeras under repulsive nonlocal interactions, a regime less studied than its attractive counterpart. The combination of direct simulation, stability analysis, and dimensional reduction provides useful cross-validation, and the explicit mapping of stable regions is a practical contribution for future modeling or experiments.

major comments (1)
  1. [§3] §3 (model and effective interaction): the phase-difference rule and the stability of the 2π/n configurations are obtained by integrating the specific piecewise-linear kernel over the supports of the clusters. Because the sign and magnitude of these integrals are kernel-dependent, the manuscript should state explicitly that the reported rule and stability boundaries are tied to this functional form and nonlocality range; without such a statement the central claim risks being read as more general than the derivation supports.
minor comments (2)
  1. [Figures] Figure 4 and 5 captions: parameter values (coupling strength, phase lag, nonlocality range) used for each panel should be listed explicitly rather than referred to the text.
  2. [Model section] Notation: the phase lag is introduced as a fixed parameter but its symbol is not consistently defined in the first appearance in the model equation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We agree that the results are specific to the chosen coupling kernel and have revised the text accordingly to make this explicit.

read point-by-point responses

  1. Referee: [§3] §3 (model and effective interaction): the phase-difference rule and the stability of the 2π/n configurations are obtained by integrating the specific piecewise-linear kernel over the supports of the clusters. Because the sign and magnitude of these integrals are kernel-dependent, the manuscript should state explicitly that the reported rule and stability boundaries are tied to this functional form and nonlocality range; without such a statement the central claim risks being read as more general than the derivation supports.

    Authors: We agree with the referee. The phase-difference rule (clusters differing by exactly 2π/n) and the associated stability boundaries are obtained by explicit integration of the piecewise-linear kernel over the cluster supports, so both the sign and magnitude of the resulting effective coupling are kernel-dependent. In the revised manuscript we have added a clarifying paragraph in §3 (and a brief remark in the abstract and conclusions) stating that the reported 2π/n organization and the identified stable regions are tied to the present functional form and nonlocality range. We also note that other kernels would in general produce different integrals and therefore different phase rules, which would need to be analyzed separately. revision: yes

Circularity Check

0 steps flagged

No circularity: phase-difference rule obtained from independent numerical and reduced analysis of given model

full rationale

The paper obtains the 2π/n inter-cluster phase offset by direct numerical integration of the phase-oscillator equations on the ring, followed by explicit linear stability analysis of the resulting cluster states and cross-validation via the standard Ott-Antonsen reduction. These operations solve the dynamical system for the chosen piecewise-linear repulsive kernel plus fixed lag; the offset is an emergent property of the solved trajectories, not a re-statement of the kernel integral or a fitted parameter renamed as a prediction. No self-citation is invoked to justify uniqueness or to close the derivation, and the Ott-Antonsen step is an external, parameter-free technique whose assumptions do not presuppose the reported phase rule. The derivation chain is therefore self-contained against the model equations themselves.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Ott-Antonsen reduction for phase oscillators and on the assumption that the coupling kernel is piecewise linear and nonlocal; no new free parameters or invented entities are introduced beyond the usual coupling strength and phase-lag parameters.

free parameters (1)
  • coupling strength and phase lag
    Standard tunable parameters of the model; their specific values define the regions where the chimera states exist.
axioms (1)
  • domain assumption Ott-Antonsen reduction applies to the nonlocal repulsive kernel
    Invoked to validate numerical results; standard for phase-oscillator networks but requires the kernel to satisfy certain analyticity conditions.

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Reference graph

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