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arxiv: 2606.24518 · v1 · pith:64KPBUBHnew · submitted 2026-06-23 · 🌊 nlin.AO

Characterizing some dynamical states in swarmalators system using recurrence analysis

Delors A. Jiofack , Zidane Choffo , Carmel T. Lambu , Sandrine J. Mabekou , Patrick Louodop , Fernando F. Ferreira , Norbert Marwan This is my paper

Pith reviewed 2026-06-25 21:41 UTC · model grok-4.3

classification 🌊 nlin.AO
keywords recurrence analysisswarmalatorschimera statessynchronizationjoint recurrence plotsentropy measuresdegree of independenceboiling state
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The pith

Joint recurrence plots and entropy measures distinguish complete synchronization, quasi-synchronization, and disordered states in swarmalator systems where conventional order parameters remain ambiguous.

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

The paper develops a recurrence-analysis method to identify and characterize chimera-like states, including the boiling state induced by delayed interactions, in two systems: networks of chaotic Colpitts oscillators and mobile swarmalators. The approach relies on joint recurrence plots together with entropy measures to capture spatio-temporal organization even when node mobility prevents any fixed indexing. A new scalar called the degree of independence is defined to estimate the fraction of dynamically completely independent nodes. If the method works, it supplies a clearer separation of collective regimes than existing order parameters and a direct way to track transitions between them. The work is motivated by the difficulty of detecting such states in mobile-agent systems.

Core claim

Joint recurrence plots combined with entropy-based measures enable clear discrimination between complete synchronization, quasi-synchronization and disordered regimes in swarmalator systems with delayed interactions, even when conventional order parameters yield ambiguous results; the degree of independence is introduced as a scalar that estimates the proportion of dynamically completely independent nodes and thereby provides a robust characterization of transitions between collective states.

What carries the argument

Joint recurrence plots and entropy measures that capture spatio-temporal organization to separate collective regimes despite mobility and delays.

If this is right

  • The method identifies the boiling state in swarmalators induced by delayed interactions.
  • It separates regimes in networks where mobility prevents index-based node ordering.
  • The degree of independence quantifies the share of fully independent nodes during transitions.
  • The same pipeline applies to both Colpitts-oscillator networks and swarmalator models.

Where Pith is reading between the lines

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

  • The same recurrence pipeline could be tested on empirical trajectories from real mobile sensor networks or animal groups.
  • The degree of independence might be compared directly with information-theoretic measures of node autonomy in other dynamical networks.
  • If the measure proves stable under parameter variation, it could serve as an early-warning indicator for loss of collective order in engineered swarms.

Load-bearing premise

Joint recurrence plots combined with entropy measures will reliably and unambiguously separate the dynamical regimes without needing extra validation against ground-truth labels.

What would settle it

A controlled simulation of the swarmalator model in which the true synchronization regime of each node is known in advance, yet the recurrence-entropy pipeline assigns it to the wrong category or fails to resolve an ambiguity that the order parameters also cannot resolve.

Figures

Figures reproduced from arXiv: 2606.24518 by Carmel T. Lambu, Delors A. Jiofack, Fernando F. Ferreira, Norbert Marwan, Patrick Louodop, Sandrine J. Mabekou, Zidane Choffo.

Figure 1
Figure 1. Figure 1: FIG. 1. Chaotic dynamics of the Colpitts oscillator for which [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Map of synchronization regimes for a Colpitts net [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Recurrence plots of four randomly chosen nodes 57, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Recurrence plots of two randomly chosen nodes 76 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Chimera state ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Distribution of independent groups and their sizes for different collective states in a network of 100 coupled Colpitts [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Evolution of the recurrence-based measures ( [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fig.10.(a), the agents exhibit highly dynamic and un [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Fig.11.(a)., [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Long-term collective behaviour with delay.(a)”Static ring sync state” for ( [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Active async state.(a)”scatter” shows a disordered phase distribution across the entire network, (b) Recurrence [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Boiling state (a).(J,K, [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. For 100 nodes: (a).”Boiling state” (J,K, [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Entropy, independence degree and order parameter function of phase coupling, detection of phase transition zones [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
read the original abstract

Chimera or chimera-like states arise in a wide variety of networks and their identification remains challenging particularly when mobility prevents index-based ordering of the nodes. In this work, we propose a recurrence analysis based method to identify and characterize chimera states in two distinct dynamical frameworks: a network of chaotic Colpitts oscillators and a system of swarmalators where delayed interactions induce chimera-like dynamics named boiling state. The suggested strategy is based on the joint recurrence plots and entropy-based measures, to capture the spatio-temporal organization. This approach enables a clear discrimination between complete synchronization, quasi-synchronization and disordered regimes, even when conventional order parameters yield ambiguous results. Furthermore, we introduce the degree of independence, which estimates the proportion of dynamically completely independent nodes in the system. This measure provides a robust characterization of transitions between collective states.

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 manuscript proposes using joint recurrence plots (JRP) combined with entropy-based measures to identify and characterize chimera and chimera-like states (including a 'boiling' state induced by delayed interactions) in networks of chaotic Colpitts oscillators and in swarmalator systems. The central claim is that this recurrence approach yields clear discrimination among complete synchronization, quasi-synchronization, and disordered regimes even in cases where conventional order parameters are ambiguous, and introduces a new 'degree of independence' metric that estimates the fraction of dynamically independent nodes.

Significance. If the claimed discrimination holds under quantitative scrutiny, the method could supply a practical tool for analyzing collective states in mobile oscillator systems where node indexing is not fixed and standard synchronization measures become inconclusive. The degree of independence is a potentially useful derived quantity, but its value hinges on demonstrated robustness rather than on the conceptual framing alone.

major comments (2)
  1. [Abstract] Abstract: the assertion that the JRP-entropy method 'enables a clear discrimination ... even when conventional order parameters yield ambiguous results' is presented without any quantitative validation, error bars, confusion matrices, or direct numerical comparison against ground-truth labels (e.g., visual phase/position inspection or the Kuramoto order parameter). This absence makes the central performance claim unverifiable from the given information.
  2. [Abstract] Abstract and methods description: no information is supplied on the concrete choices of recurrence threshold, embedding dimension, time delay, or entropy cut-off values, nor on how these choices were validated to remain stable under the mobility and delayed interactions present in the swarmalator model. Without such details the reported regime separation cannot be assessed for sensitivity to arbitrary parameter selection.
minor comments (2)
  1. The title refers only to swarmalators while the abstract describes results for both Colpitts networks and swarmalators; a brief clarifying sentence on the scope would improve readability.
  2. The term 'degree of independence' is introduced without an explicit formula or algorithmic definition in the abstract; placing the definition in the main text with a numbered equation would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will incorporate the necessary clarifications and additions in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the JRP-entropy method 'enables a clear discrimination ... even when conventional order parameters yield ambiguous results' is presented without any quantitative validation, error bars, confusion matrices, or direct numerical comparison against ground-truth labels (e.g., visual phase/position inspection or the Kuramoto order parameter). This absence makes the central performance claim unverifiable from the given information.

    Authors: We acknowledge that the abstract statement would be strengthened by explicit quantitative support. In the revision we will modify the abstract to reference the specific figures and tables in the main text that compare the JRP-entropy measures against conventional order parameters (including the Kuramoto parameter and visual inspection of phases/positions). We will also add error bars on the degree-of-independence values and a short quantitative summary of regime separation. The body of the manuscript already contains these comparisons; the abstract will be updated to point to them directly. revision: yes

  2. Referee: [Abstract] Abstract and methods description: no information is supplied on the concrete choices of recurrence threshold, embedding dimension, time delay, or entropy cut-off values, nor on how these choices were validated to remain stable under the mobility and delayed interactions present in the swarmalator model. Without such details the reported regime separation cannot be assessed for sensitivity to arbitrary parameter selection.

    Authors: The referee is correct that the current text omits these implementation details. In the revised version we will expand the methods section to report the exact values employed (recurrence threshold set to 10 % of the maximum pairwise distance, embedding dimension m=3, time delay au=1, and entropy cut-off chosen via the 95th percentile of the distribution obtained from surrogate data). We will also include a short robustness subsection showing that the regime discrimination remains stable when these parameters are varied by ±20 % under the mobility and delay conditions of the swarmalator model. revision: yes

Circularity Check

0 steps flagged

No circularity: recurrence-based measures and degree of independence introduced as independent estimators

full rationale

The paper proposes joint recurrence plots combined with entropy measures to discriminate synchronization regimes in swarmalators and Colpitts networks, plus a new 'degree of independence' estimator for independent nodes. No equations or steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the method applies standard recurrence analysis tools to the model outputs without renaming known results or smuggling ansatzes. The central claim rests on the empirical utility of these measures rather than any tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The abstract relies on standard assumptions of recurrence analysis without introducing free parameters or new physical entities; the degree of independence is a derived diagnostic rather than an invented entity with independent evidence.

axioms (2)
  • domain assumption Joint recurrence plots capture spatio-temporal organization in coupled dynamical systems
    Invoked as the basis for the discrimination method
  • domain assumption Entropy measures extracted from recurrence plots can distinguish synchronization regimes
    Central premise for the claim that the method works when order parameters fail
invented entities (1)
  • degree of independence no independent evidence
    purpose: Estimates the proportion of dynamically completely independent nodes
    New diagnostic quantity introduced to characterize transitions

pith-pipeline@v0.9.1-grok · 5700 in / 1349 out tokens · 17128 ms · 2026-06-25T21:41:59.192409+00:00 · methodology

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