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arxiv: 2506.12511 · v5 · submitted 2025-06-14 · 🌊 nlin.PS · math-ph· math.MP· nlin.AO· nlin.CD· physics.comp-ph

Chimera states on m-directed hypergraphs

Rommel Tchinda Djeudjo , Timoteo Carletti , Hiroya Nakao , Riccardo Muolo This is my paper

Pith reviewed 2026-05-19 09:24 UTC · model grok-4.3

classification 🌊 nlin.PS math-phmath.MPnlin.AOnlin.CDphysics.comp-ph
keywords chimera statesm-directed hypergraphshigher-order interactionsnon-reciprocal couplingssynchronizationoscillatorsphase reduction
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0 comments X

The pith

Directionality in non-reciprocal higher-order interactions enables chimera states in oscillator systems that do not appear without it.

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

This paper examines how chimera states, where some oscillators synchronize while others do not, arise in systems with non-reciprocal many-body couplings modeled by m-directed hypergraphs. The authors show that the directionality introduced by these structures can produce types of chimeras not seen in undirected versions, and that these states persist across a wider set of parameters than when using only pairwise network couplings. A sympathetic reader would care because many real systems involve directed group interactions, so this helps explain when and why partial synchronization occurs in complex settings. The work compares higher-order and pairwise cases directly and uses phase reduction to confirm the findings.

Core claim

Chimera states can emerge on m-directed hypergraphs due to the directionality of the interactions, which had not been observed in the absence of such directionality. These states appear over a broader parameter range with higher-order interactions than in the corresponding network case. The nature of phase chimeras is validated through phase reduction theory.

What carries the argument

m-directed hypergraphs that encode non-reciprocal higher-order couplings among identical oscillators.

If this is right

  • Some chimera states require directionality to appear even in higher-order interaction structures.
  • Non-reciprocal higher-order couplings support chimera states across more parameter values than pairwise directed couplings.
  • Phase reduction theory can be applied to confirm the phase relationships in these chimeras.
  • The impact of directionality is stronger or different in higher-order settings compared to networks.

Where Pith is reading between the lines

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

  • Models of real systems with group interactions, such as in biology or social dynamics, might need to account for directionality to predict synchronization patterns accurately.
  • Studying the continuum limit of these hypergraphs could yield analytical descriptions of the chimera states.
  • Similar effects might appear in other oscillator models or with different coupling functions, suggesting broader applicability.

Load-bearing premise

The specific numerical results on finite-sized hypergraphs with chosen initial conditions and coupling functions will hold generally for larger systems and other similar setups.

What would settle it

A simulation or analysis showing that removing directionality from the hypergraph eliminates the observed chimera states or that the parameter range does not broaden with higher-order terms.

Figures

Figures reproduced from arXiv: 2506.12511 by Hiroya Nakao, Riccardo Muolo, Rommel Tchinda Djeudjo, Timoteo Carletti.

Figure 1
Figure 1. Figure 1: A 2-hyperring made of hyperedges and 10 nodes. Dcliqueprojectiondrawio (29)svg [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: We schematically represent a family of 1 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: We schematically represent a family of clique-projected networks, obtained from the 1-directed 2-hyperring presented in Fig. 2. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Analysis of the dynamics on a 1-directed 2-hyperring of 204 nodes. The first row shows the spatiotemporal diagrams, the second [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Analysis of the dynamics on a clique-projected network of 204 nodes. The first row shows the spatiotemporal diagrams, the second [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Analysis of the dynamics on a 1-directed 2-hyperring of 204 nodes. The first row shows the spatiotemporal diagrams, the second [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Analysis of the dynamics on a clique-projected network of 204 nodes. The first row shows the spatiotemporal diagrams, the second [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Analysis of the dynamics of the phase reduced model on a 1-directed 2-hyperring of 204 nodes. The first row shows the spatiotem [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Analysis of the dynamics of the phase reduced model on a clique-projected network of 204 nodes. The first row shows the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
read the original abstract

Chimera states are synchronization patterns in which coherent and incoherent regions coexist in systems of identical oscillators. This elusive phenomenon has attracted significant interest and has been widely analyzed, revealing several types of dynamical states. Most studies involve reciprocal pairwise couplings, where each oscillator exerts and receives the same interaction from neighboring ones, thus being modeled via symmetric networks. However, real-world systems often exhibit non-reciprocal, non-pairwise (many-body) interactions. Previous studies have shown that chimera states are more elusive in the presence of non-reciprocal pairwise interactions, while they are easier to observe when the interactions are reciprocal and higher-order (many-body). In this work, we investigate the emergence of chimera states on non-reciprocal higher-order structures, called m-directed hypergraphs, which we compare with their corresponding networks, and we observe that some types of chimera states can emerge due to directionality, which had not been previously observed in its absence. We also compare the effect of non-reciprocal interactions between higher-order and pairwise couplings, and we find numerically that chimera states appear over a broader parameter range when considering higher-order interactions than in the corresponding network case, demonstrating the impact of directionality and the effect of higher-order interactions. Finally, the nature of phase chimeras has been further validated through phase reduction theory.

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 / 2 minor

Summary. The paper investigates chimera states in systems of identical oscillators coupled via non-reciprocal higher-order interactions on m-directed hypergraphs. It claims that directionality on these structures enables certain chimera types not previously observed in undirected cases, that higher-order couplings widen the existence region relative to the corresponding pairwise networks, and that phase-reduction theory confirms the phase character of the observed states. These conclusions rest on numerical simulations of finite oscillator populations together with a phase-reduction approximation.

Significance. If the central claims hold, the work would be significant for extending chimera research beyond reciprocal pairwise networks to non-reciprocal many-body structures, highlighting how directionality and higher-order interactions can facilitate chimera formation. The use of phase reduction to validate the phase nature of the states is a clear strength that strengthens the interpretation of the numerics.

major comments (2)
  1. [§4 (Numerical results)] The central claim that directionality induces new chimera types and that higher-order interactions broaden the parameter range (abstract and §4) rests entirely on finite-N simulations with fixed coupling functions and initial conditions. No scaling analysis, mean-field reduction, or continuum limit is presented to show that the directionality-specific features and the widened existence region survive as N increases, which is load-bearing because chimera coexistence is known to be sensitive to system size.
  2. [§5 (Phase reduction)] The phase-reduction validation (mentioned in the abstract and §5) is used to confirm the phase character of the states, but the assumptions of the reduction (e.g., weak coupling, slow phase dynamics) and its range of applicability to the chosen m-directed hypergraph couplings are not stated explicitly; this leaves open whether the reduction supports the existence claims or only the phase interpretation.
minor comments (2)
  1. [§2] Notation for the m-directed hypergraph adjacency tensors and the coupling functions should be introduced with explicit definitions and compared side-by-side with the corresponding network case to improve readability.
  2. [Figures 2–5] Figure captions and legends should include the precise values of N, the coupling strength ranges, and the initial-condition protocols used for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments, which help clarify the presentation of our results on chimera states in m-directed hypergraphs. We address the two major comments point by point below, indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [§4 (Numerical results)] The central claim that directionality induces new chimera types and that higher-order interactions broaden the parameter range (abstract and §4) rests entirely on finite-N simulations with fixed coupling functions and initial conditions. No scaling analysis, mean-field reduction, or continuum limit is presented to show that the directionality-specific features and the widened existence region survive as N increases, which is load-bearing because chimera coexistence is known to be sensitive to system size.

    Authors: We agree that finite-size effects warrant explicit attention, as chimera coexistence can indeed depend on N. In the revised manuscript we will add simulations at larger system sizes (N up to several thousand) and include a short discussion in §4 showing that the directionality-induced chimera types and the expanded existence intervals remain qualitatively unchanged. A full mean-field or continuum limit lies outside the scope of the present work, which centers on concrete hypergraph constructions, but the phase-reduction analysis already supplies an N-independent analytical anchor for the phase character of the states. We will make these points explicit without overstating the numerical evidence. revision: yes

  2. Referee: [§5 (Phase reduction)] The phase-reduction validation (mentioned in the abstract and §5) is used to confirm the phase character of the states, but the assumptions of the reduction (e.g., weak coupling, slow phase dynamics) and its range of applicability to the chosen m-directed hypergraph couplings are not stated explicitly; this leaves open whether the reduction supports the existence claims or only the phase interpretation.

    Authors: We thank the referee for highlighting the need for greater clarity. In the revision we will explicitly state the standard assumptions of the phase reduction (weak coupling, small phase deviations from a common frequency, and slow phase dynamics) at the beginning of §5 and discuss their applicability to the m-directed hypergraph coupling functions. The reduction is used solely to confirm that the numerically observed states are phase chimeras; existence and the broadening of parameter ranges are established by the direct simulations. We will revise the text to separate these roles clearly and avoid any implication that the reduction alone proves the existence claims. revision: yes

Circularity Check

0 steps flagged

Numerical observations and phase reduction remain independent of fitted inputs or self-referential definitions

full rationale

The paper's central results derive from direct numerical integration of oscillator dynamics on finite m-directed hypergraphs versus their pairwise network counterparts, using fixed coupling functions and initial conditions to identify chimera coexistence regions. Phase reduction is invoked only as a post-hoc validation of the phase character of the observed states, constituting an independent approximation rather than a redefinition or fit of the same data. No equations or claims reduce a reported prediction to a parameter fitted from the target quantity, nor does any load-bearing step rest on a self-citation chain whose content is itself unverified within the manuscript. The comparison of parameter ranges and the attribution to directionality are therefore self-contained against the external benchmark of the performed simulations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim depends on the oscillator model, the precise definition of m-directed hypergraphs, and the choice of coupling functions; none of these are supplied with independent external validation in the abstract.

invented entities (1)
  • m-directed hypergraphs no independent evidence
    purpose: Model non-reciprocal higher-order interactions
    Introduced as the novel interaction structure; no independent evidence outside the paper is provided in the abstract.

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