arxiv: 2605.09401 · v1 · submitted 2026-05-10 · 🌊 nlin.PS · cond-mat.dis-nn· nlin.AO· nlin.CD· physics.comp-ph

Recognition: no theorem link

Classification of Chimera States via Fourier Analysis and Unsupervised Learning

Rommel Tchinda Djeudjo , Riccardo Muolo , Thierry Njougouo , Timoteo Carletti

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:51 UTC · model grok-4.3

classification 🌊 nlin.PS cond-mat.dis-nnnlin.AOnlin.CDphysics.comp-ph

keywords chimera statesFourier analysisunsupervised clusteringRayleigh oscillatorsdynamical patternsclassificationsynchronizationcoupled oscillators

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The pith

Fourier analysis combined with unsupervised clustering distinguishes chimera state types in coupled oscillator networks.

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

Chimera states are patterns where some oscillators in a network behave coherently while others do not, and prior detection techniques often lack the precision needed to separate their varieties reliably. The paper develops a procedure that first uses Fourier analysis to pull out each oscillator's amplitude, phase, and frequency, then feeds normalized measures of how these quantities change across the network into an unsupervised clustering algorithm. This combination locates the regions of parameter space that produce chimera behavior and partitions those regions according to distinct chimera subtypes. The procedure is demonstrated on a network of Rayleigh oscillators already known to support many different dynamical regimes. If the separation works, researchers gain a concrete way to catalog complex mixed synchronization patterns without relying on visual inspection or ad-hoc thresholds.

Core claim

The central claim is that Fourier-derived amplitude, phase, and frequency, together with normalized total variations of these quantities, supply features sufficient for unsupervised clustering to both detect chimera states and classify them into their different types when applied to a network of coupled Rayleigh oscillators.

What carries the argument

Fourier extraction of amplitude, phase, and frequency followed by unsupervised clustering on the normalized total variations of those features across the network.

If this is right

Parameter-space diagrams can be drawn that mark both the presence and the specific subtype of chimera states.
The method separates multiple varieties of chimera behavior in the same Rayleigh-oscillator network without manual tuning of detection thresholds.
Existing limitations of purely visual or threshold-based chimera classifiers are bypassed for this class of oscillator networks.
The same feature set can be reused on other networks that exhibit rich dynamical patterns to produce comparable classifications.

Where Pith is reading between the lines

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

The same pipeline could be tested on other common oscillator models such as Kuramoto or van der Pol networks to check whether the feature set remains informative.
If the clusters align with physically observable differences, the method might help design control schemes that stabilize one chimera subtype over another.
Automated classification could reveal previously unnoticed intermediate or hybrid chimera states in large networks.

Load-bearing premise

The chosen Fourier features and their normalized spatial variations are enough to reveal all meaningful differences between chimera types so that clustering yields accurate partitions.

What would settle it

Running the clustering on simulated Rayleigh-oscillator data where distinct chimera types are already known from other methods and finding that the algorithm either merges those types or splits a single type into multiple clusters.

Figures

Figures reproduced from arXiv: 2605.09401 by Riccardo Muolo, Rommel Tchinda Djeudjo, Thierry Njougouo, Timoteo Carletti.

**Figure 1.** Figure 1: Results of the clustering methods k-means for the x–x coupling. Panel (a) shows the silhouette score as a function of the number of clusters k, while panel (b) displays the Davies–Bouldin index. Panel (c) presents the corresponding three-dimensional cluster distributions in the feature space (V(⟨θ⟩), V(⟨a⟩), V(⟨ω⟩)). Panel (d) shows the associated cluster assignments in the (p, ε) parameter plane. The red … view at source ↗

**Figure 2.** Figure 2: Analysis of the normalized total variations for the two clusters shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Results of the refined clustering methods k-means for the x–x coupling. Panel (a) shows a three-dimensional representation of the two subclusters in the feature space, previously being identified with a single cluster 2. Panel (b) displays the corresponding subdivision in cluster into the (p, ε) parameter plane. Cluster 1 Cluster 2 Cluster 3 (a) 0 0.05 0.1 0.15 0.2 V ( h 3 i ) Cluster 1 Cluster 2 Cluster 3… view at source ↗

**Figure 4.** Figure 4: Analysis of the normalized total variations for the three clusters obtained after the second clustering step. Panel (a) shows the case V(⟨θ⟩), panel (b) V(⟨a⟩), and panel (c) V(⟨ω⟩). The relative positions of the medians allow a direct identification of the dynamical regimes. Cluster 1 is characterized by low medians for all three observables and corresponds to the coherent state. Cluster 2 displays the hi… view at source ↗

**Figure 5.** Figure 5: Evolution of the normalized total variations as functions of the coupling strength ε for several representative values of the interaction range p. The blue, red, and green curves correspond to V(⟨θ⟩), V(⟨a⟩), and V(⟨ω⟩), respectively. The background color indicates the dynamical regime assigned by the clustering procedure: coherent state (blue), amplitude-mediated chimera (orange), and phase chimera (brown… view at source ↗

**Figure 8.** Figure 8: In the first row, the dynamical state can clearly be [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

**Figure 6.** Figure 6: Time series and Fourier features for the case p = 18. The top row (ε = 0.15) corresponds to an amplitude-mediated chimera, shown through the spatiotemporal diagram (a1), the amplitude profile (b1), the frequency profile (c1), and the phase profile (d1). The bottom row (ε = 0.65) corresponds to a coherent state, as evidenced by the corresponding snapshots in (a2)–(d2). This figure confirms the transition id… view at source ↗

**Figure 7.** Figure 7: Example of amplitude-mediated chimera, ϵ = 0.834, p = 5 and the use of a rotational matrix. Panel (a) shows the space–time plots, panels (b), (c) and (d), display respectively amplitude, frequency and phase, computed with the Fourier method. The center-of-mass is reported in panel (e). From the data reported in those panels we can conclude that the system exhibits an amplitude-mediated chimera. The remaini… view at source ↗

**Figure 8.** Figure 8: Example of amplitude-mediated chimera, ϵ = 0.8 (top row), phase-amplitude chimera ϵ = 2.0 (bottom row), p = 5 and the use of a rotational matrix. The first column presents the space–time plots, followed by the amplitude (second column), frequency (third column), phase (fourth column), and center-of-mass (last column) profiles. From the analysis of the top row panels one can conclude that the system exhibit… view at source ↗

read the original abstract

Chimera states are among the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Many methods have been proposed to detect chimera states and to distinguish their different types. However, such methods often suffer from important limitations that prevent sufficiently precise classification. In this work, we overcome the issue by considering a method based on Fourier analysis to determine key signal characteristics such as amplitude, phase, and frequency, jointly with an unsupervised clustering step acting on normalized total variations, measures of local spatial changes of the above-mentioned dynamical features. The proposed method allows us to identify regions in parameter space returning chimera states, but also to further distinguish between the different types. The method is applied to a network of Rayleigh oscillators, which has been shown to exhibit a rich variety of dynamical patterns.

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

Fourier amplitude/phase/frequency plus normalized total variation clustering is a new pipeline for chimera typing but rests on untested assumptions about what those features capture.

read the letter

The main takeaway is that the authors extract amplitude, phase, and frequency per oscillator via Fourier analysis, compute normalized total variations of those quantities across the network, and feed the results into unsupervised clustering to locate chimera regions and label their types in a Rayleigh-oscillator network. This specific combination of steps is not described in the earlier literature they cite, so the pipeline itself counts as new. The choice of Rayleigh oscillators is sensible because that system already shows multiple chimera regimes, giving the method a nontrivial test case. The approach stays data-driven and avoids reducing the classification to a single fitted parameter, which is a plus. The description of the workflow is clear at the level of the abstract. The central weakness is that normalized total variation only tracks first-order spatial roughness. It does not encode higher-order correlations, phase-amplitude coupling, or intermittency that the chimera literature uses to separate amplitude chimeras from phase chimeras or multi-headed states. If two qualitatively different regimes produce similar variation profiles, the clustering step will merge them. The abstract supplies no error rates, no comparison against prior classifiers, and no example partitions, so the link between the features and accurate type distinction cannot be checked. If the full paper contains quantitative benchmarks or manual verification, that would change the picture; otherwise the claim that the method distinguishes types remains an assumption. This work is for people already working on chimera states in coupled oscillators who need an automated labeling tool for large networks. A reader focused on nonlinear dynamics or applications in physics and neuroscience could extract a usable method if the validation holds. It is solid enough on its own terms to deserve referee time rather than a desk reject, mainly to test whether the chosen features actually recover the known dynamical distinctions.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a method to classify chimera states in networks of coupled oscillators by extracting amplitude, phase, and frequency via Fourier analysis, then applying unsupervised clustering to the normalized total variations of these features across the network. The approach is used to identify parameter regions producing chimeras and to distinguish among chimera types in a network of Rayleigh oscillators.

Significance. If the feature set and clustering step prove sufficient, the method could offer an automated, quantitative alternative to visual or ad-hoc detection schemes for chimeras, with potential applicability to other oscillator networks exhibiting coexistence of coherent and incoherent dynamics. The choice of Rayleigh oscillators as a testbed is appropriate given their documented rich bifurcation structure.

major comments (2)

[Abstract] Abstract: the central claim that the method 'allows us to identify regions in parameter space returning chimera states, but also to further distinguish between the different types' is unsupported by any quantitative validation, accuracy metrics, confusion matrices, or direct comparison against existing chimera classifiers; without these the data-to-claim link cannot be evaluated.
[Method] Feature-construction and clustering steps: normalized total variation of the Fourier-derived scalars quantifies only first-order spatial roughness and therefore may fail to separate chimera types whose distinctions rely on higher-order spatial correlations, phase-amplitude coupling, or temporal intermittency (e.g., amplitude chimeras versus multi-headed phase chimeras).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and indicate the revisions we will make to strengthen the quantitative support and clarify the method's scope.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the method 'allows us to identify regions in parameter space returning chimera states, but also to further distinguish between the different types' is unsupported by any quantitative validation, accuracy metrics, confusion matrices, or direct comparison against existing chimera classifiers; without these the data-to-claim link cannot be evaluated.

Authors: We agree that explicit quantitative validation would strengthen the central claim. The manuscript demonstrates the classification through representative trajectories, parameter scans, and clustering visualizations that align with known chimera regimes in the Rayleigh network. In the revision we will add clustering validation metrics (silhouette scores and adjusted Rand index against manually labeled states from the bifurcation diagram), a confusion matrix for the identified chimera types, and a brief comparison to order-parameter-based detection schemes. These additions will make the data-to-claim link explicit. revision: yes
Referee: [Method] Feature-construction and clustering steps: normalized total variation of the Fourier-derived scalars quantifies only first-order spatial roughness and therefore may fail to separate chimera types whose distinctions rely on higher-order spatial correlations, phase-amplitude coupling, or temporal intermittency (e.g., amplitude chimeras versus multi-headed phase chimeras).

Authors: The normalized total variation is applied separately to the spatially resolved amplitude, phase, and frequency fields obtained from the Fourier analysis. For the Rayleigh oscillator network, the primary distinctions among chimera types appear as differences in the number and location of spatial transitions in these fields; the total-variation measure directly quantifies the cumulative magnitude of those transitions and therefore separates single-headed, multi-headed, and amplitude-chimera regimes in the studied parameter ranges. We acknowledge that the measure is first-order and may not capture higher-order correlations or strong intermittency in other systems. In the revision we will add a paragraph discussing this scope limitation and note that the feature set can be extended with second-order statistics if needed for broader applicability. revision: partial

Circularity Check

0 steps flagged

No circularity: feature extraction and unsupervised clustering are applied to external simulation data without self-referential reduction

full rationale

The paper derives chimera classification by computing Fourier amplitude, phase and frequency per oscillator, then feeding normalized total variations of these quantities into standard unsupervised clustering. This pipeline operates on raw time-series output from the Rayleigh-oscillator network and does not define any quantity in terms of the final cluster labels, nor does it fit parameters to a subset and relabel the fit as a prediction. No equations or steps reduce the reported partitions to the input features by algebraic identity or by construction. The method therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the standard assumption that Fourier analysis reliably extracts amplitude, phase, and frequency from finite time series of coupled oscillators, and that normalized total variation is a meaningful local spatial descriptor for clustering. No new physical entities or ad-hoc constants are introduced in the abstract.

axioms (2)

domain assumption Fourier analysis accurately recovers amplitude, phase, and frequency from the oscillator signals
Invoked as the first step of the classification pipeline.
domain assumption Normalized total variations of the extracted features capture the spatial structure needed to separate chimera types
Central to the unsupervised clustering step.

pith-pipeline@v0.9.0 · 5470 in / 1424 out tokens · 50579 ms · 2026-05-12T01:51:35.878479+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 1 internal anchor

[1]

K. Kaneko. Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermit- tency in coupled logistic lattice: towards a prelude of a ”field theory of chaos”.Progress of Theoretical Physics, 72(3):480– 486, 1984

work page 1984
[2]

Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements.Physica D: Nonlinear Phenomena, 41(2):137–172, 1990

Kunihiko Kaneko. Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements.Physica D: Nonlinear Phenomena, 41(2):137–172, 1990

work page 1990
[3]

Dynamics of the glob- ally coupled complex ginzburg-landau equation.Physical Re- view A, 46(12):R7347, 1992

Vincent Hakim and Wouter-Jan Rappel. Dynamics of the glob- ally coupled complex ginzburg-landau equation.Physical Re- view A, 46(12):R7347, 1992

work page 1992
[4]

Collective chaos in a population of globally coupled oscillators.Progress of Theo- retical Physics, 89(2):313–323, 1993

Naoko Nakagawa and Yoshiki Kuramoto. Collective chaos in a population of globally coupled oscillators.Progress of Theo- retical Physics, 89(2):313–323, 1993

work page 1993
[5]

Collective chaos and noise in the globally coupled complex ginzburg-landau equation.Physica D: Nonlinear Phenomena, 103(1-4):273–293, 1997

Marie-Line Chabanol, Vincent Hakim, and Wouter-Jan Rap- pel. Collective chaos and noise in the globally coupled complex ginzburg-landau equation.Physica D: Nonlinear Phenomena, 103(1-4):273–293, 1997

work page 1997
[6]

Scaling behavior of turbulent oscillators with non-local interaction.Progress of Theoretical Physics, 94(3):321–330, 1995

Yoshiki Kuramoto. Scaling behavior of turbulent oscillators with non-local interaction.Progress of Theoretical Physics, 94(3):321–330, 1995

work page 1995
[7]

Origin of power-law spa- tial correlations in distributed oscillators and maps with nonlo- cal coupling.Physical review letters, 76(23):4352, 1996

Yoshiki Kuramoto and Hiroya Nakao. Origin of power-law spa- tial correlations in distributed oscillators and maps with nonlo- cal coupling.Physical review letters, 76(23):4352, 1996

work page 1996
[8]

Power-law spatial cor- relations and the onset of individual motions in self-oscillatory media with non-local coupling.Physica D: Nonlinear Phenom- ena, 103(1-4):294–313, 1997

Yoshiki Kuramoto and Hiroya Nakao. Power-law spatial cor- relations and the onset of individual motions in self-oscillatory media with non-local coupling.Physica D: Nonlinear Phenom- ena, 103(1-4):294–313, 1997

work page 1997
[9]

Multiaffine chemical turbulence.Physical review letters, 81(16):3543, 1998

Yoshiki Kuramoto, Dorjsuren Battogtokh, and Hiroya Nakao. Multiaffine chemical turbulence.Physical review letters, 81(16):3543, 1998

work page 1998
[10]

Multi-scaled turbulence in large populations of oscillators in a diffusive medium.Physica A: Statistical Mechanics and its Ap- plications, 288(1-4):244–264, 2000

Yoshiki Kuramoto, Hiroya Nakao, and Dorjsuren Battogtokh. Multi-scaled turbulence in large populations of oscillators in a diffusive medium.Physica A: Statistical Mechanics and its Ap- plications, 288(1-4):244–264, 2000

work page 2000
[11]

Coexistence of coherence and incoherence in nonlocally coupled phase oscil- lators.arXiv preprint cond-mat/0210694, 2002

Yoshiki Kuramoto and Dorjsuren Battogtokh. Coexistence of coherence and incoherence in nonlocally coupled phase oscil- lators.arXiv preprint cond-mat/0210694, 2002

work page internal anchor Pith review arXiv 2002
[12]

Chimera states for coupled oscillators.Physical review letters, 93(17):174102, 2004

Daniel M Abrams and Steven H Strogatz. Chimera states for coupled oscillators.Physical review letters, 93(17):174102, 2004

work page 2004
[13]

Chimera states in networks of van der pol oscil- lators with hierarchical connectivities.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 26(9):094825, 2016

Stefan Ulonska, Iryna Omelchenko, Anna Zakharova, and Eck- ehard Sch¨oll. Chimera states in networks of van der pol oscil- lators with hierarchical connectivities.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 26(9):094825, 2016

work page 2016
[14]

Sergey A Bogomolov, Andrei V Slepnev, Galina I Strelkova, Eckehard Sch ¨oll, and Vadim S Anishchenko. Mechanisms of appearance of amplitude and phase chimera states in ensem- bles of nonlocally coupled chaotic systems.Communications in Nonlinear Science and Numerical Simulation, 43:25–36, 2017

work page 2017
[15]

Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling.Physical review E, 89(5):052914, 2014

R Gopal, VK Chandrasekar, A Venkatesan, and M Laksh- manan. Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling.Physical review E, 89(5):052914, 2014

work page 2014
[16]

Loss of coherence in dynamical networks: spatial chaos and chimera states.Physical review letters, 106(23):234102, 2011

Iryna Omelchenko, Yuri Maistrenko, Philipp H ¨ovel, and Eck- ehard Sch ¨oll. Loss of coherence in dynamical networks: spatial chaos and chimera states.Physical review letters, 106(23):234102, 2011

work page 2011
[17]

Correlation analysis of the coherence-incoherence transition in a ring of nonlocally cou- pled logistic maps.Chaos: an interdisciplinary journal of non- linear science, 26(9), 2016

Tatiana E Vadivasova, Galina I Strelkova, Sergey A Bogo- molov, and Vadim S Anishchenko. Correlation analysis of the coherence-incoherence transition in a ring of nonlocally cou- pled logistic maps.Chaos: an interdisciplinary journal of non- linear science, 26(9), 2016

work page 2016
[18]

Does hyperbolicity impede emergence of chimera states in net- works of nonlocally coupled chaotic oscillators?Europhysics Letters, 112(4):40002, 2015

N Semenova, A Zakharova, E Sch ¨oll, and V Anishchenko. Does hyperbolicity impede emergence of chimera states in net- works of nonlocally coupled chaotic oscillators?Europhysics Letters, 112(4):40002, 2015

work page 2015
[19]

Asllani, B.A

M. Asllani, B.A. Siebert, A. Arenas, and J.P. Gleeson. Symmetry-breaking mechanism for the formation of cluster chimera patterns.Chaos, 32:013107, 2022

work page 2022
[20]

Muolo, J.D

R. Muolo, J.D. O’Brien, T. Carletti, and M. Asllani. Persistence of chimera states and the challenge for synchronization in real- world networks.The European Physical Journal B, 97(1):6, 2024

work page 2024
[21]

Prabhu, Axion production in pulsar magneto- sphere gaps, Physical Review D104, 10.1103/phys- revd.104.055038 (2021)

Ga ¨el R Simo, Thierry Njougouo, RP Aristides, Patrick Louodop, Robert Tchitnga, and Hilda A Cerdeira. Chimera states in a neuronal network under the action of an electric field.Physical Review E, 103(6):062304 , DOI: 10.1103/Phys- RevE.103.062304, 2021

work page doi:10.1103/phys- 2021
[22]

Bick and E.A

C. Bick and E.A. Martens. Controlling chimeras.New Journal of Physics, 17(3):033030, 2015

work page 2015
[23]

Controlling chimera states: The influence of excitable units.Physical Review E, 93(2):022217, 2016

Thomas Isele, Johanne Hizanidis, Astero Provata, and Philipp H¨ovel. Controlling chimera states: The influence of excitable units.Physical Review E, 93(2):022217, 2016

work page 2016
[24]

Gambuzza and M

L.V . Gambuzza and M. Frasca. Pinning control of chimera states.Physical Review E, 94(2):022306, 2016

work page 2016
[25]

Controlling chimera states via minimal coupling modification.Chaos: an interdisciplinary journal of nonlinear science, 29(5), 2019

Giulia Ruzzene, Iryna Omelchenko, Eckehard Sch¨oll, Anna Za- kharova, and Ralph G Andrzejak. Controlling chimera states via minimal coupling modification.Chaos: an interdisciplinary journal of nonlinear science, 29(5), 2019. 11

work page 2019
[26]

Pinning control of chimera states in systems with higher-order interactions.Nonlinear Dynamics, pages 1– 23, 2025

Riccardo Muolo, Lucia Valentina Gambuzza, Hiroya Nakao, and Mattia Frasca. Pinning control of chimera states in systems with higher-order interactions.Nonlinear Dynamics, pages 1– 23, 2025

work page 2025
[27]

Chimera and phase-cluster states in populations of coupled chemical oscillators.Nature Physics, 8(9):662–665, 2012

Mark R Tinsley, Simbarashe Nkomo, and Kenneth Showal- ter. Chimera and phase-cluster states in populations of coupled chemical oscillators.Nature Physics, 8(9):662–665, 2012

work page 2012
[28]

Experimen- tal observation of chimeras in coupled-map lattices.Nature Physics, 8(9):658–661, 2012

Aaron M Hagerstrom, Thomas E Murphy, Rajarshi Roy, Philipp H¨ovel, Iryna Omelchenko, and Eckehard Sch ¨oll. Experimen- tal observation of chimeras in coupled-map lattices.Nature Physics, 8(9):658–661, 2012

work page 2012
[29]

Chimera states in mechanical oscillator networks.Proceedings of the National Academy of Sciences, 110(26):10563–10567, 2013

Erik Andreas Martens, Shashi Thutupalli, Antoine Fourriere, and Oskar Hallatschek. Chimera states in mechanical oscillator networks.Proceedings of the National Academy of Sciences, 110(26):10563–10567, 2013

work page 2013
[30]

Spatially orga- nized dynamical states in chemical oscillator networks: Syn- chronization, dynamical differentiation, and chimera patterns

Mahesh Wickramasinghe and Istv ´an Z Kiss. Spatially orga- nized dynamical states in chemical oscillator networks: Syn- chronization, dynamical differentiation, and chimera patterns. PloS one, 8(11):e80586, 2013

work page 2013
[31]

Spatially orga- nized partial synchronization through the chimera mechanism in a network of electrochemical reactions.Physical Chemistry Chemical Physics, 16(34):18360–18369, 2014

Mahesh Wickramasinghe and Istv ´an Z Kiss. Spatially orga- nized partial synchronization through the chimera mechanism in a network of electrochemical reactions.Physical Chemistry Chemical Physics, 16(34):18360–18369, 2014

work page 2014
[32]

Experi- mental investigation of chimera states with quiescent and syn- chronous domains in coupled electronic oscillators.Physical Review E, 90(3):032905, 2014

Lucia Valentina Gambuzza, Arturo Buscarino, Sergio Chessari, Luigi Fortuna, Riccardo Meucci, and Mattia Frasca. Experi- mental investigation of chimera states with quiescent and syn- chronous domains in coupled electronic oscillators.Physical Review E, 90(3):032905, 2014

work page 2014
[33]

Transient scaling and resur- gence of chimera states in networks of boolean phase oscilla- tors.Physical Review E, 90(3):030902, 2014

David P Rosin, Damien Rontani, Nicholas D Haynes, Ecke- hard Sch¨oll, and Daniel J Gauthier. Transient scaling and resur- gence of chimera states in networks of boolean phase oscilla- tors.Physical Review E, 90(3):030902, 2014

work page 2014
[34]

Coherence and in- coherence in an optical comb.Physical review letters, 112(22):224101, 2014

Evgeny A Viktorov, Tatiana Habruseva, Stephen P Hegarty, Guillaume Huyet, and Bryan Kelleher. Coherence and in- coherence in an optical comb.Physical review letters, 112(22):224101, 2014

work page 2014
[35]

Chimera states in populations of nonlocally coupled chemical oscillators.Physical review letters, 110(24):244102, 2013

Simbarashe Nkomo, Mark R Tinsley, and Kenneth Showalter. Chimera states in populations of nonlocally coupled chemical oscillators.Physical review letters, 110(24):244102, 2013

work page 2013
[36]

Amplitude-mediated chimera states.Physical Review E, 88(4):042917, 2013

Gautam C Sethia, Abhijit Sen, and George L Johnston. Amplitude-mediated chimera states.Physical Review E, 88(4):042917, 2013

work page 2013
[37]

Amplitude chimera and chimera death induced by external agents in two-layer net- works.Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence, 30(4):043104, 2020

Umesh Kumar Verma and G Ambika. Amplitude chimera and chimera death induced by external agents in two-layer net- works.Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence, 30(4):043104, 2020

work page 2020
[38]

Chimera death: Symmetry breaking in dynamical networks

Anna Zakharova, Marie Kapeller, and Eckehard Sch ¨oll. Chimera death: Symmetry breaking in dynamical networks. Physical review letters, 112(15):154101, 2014

work page 2014
[39]

Stability of amplitude chimeras in oscillator networks.Europhysics Letters, 117(2):20001, 2017

Liudmila Tumash, Anna Zakharova, Judith Lehnert, Wolfram Just, and Eckehard Sch ¨oll. Stability of amplitude chimeras in oscillator networks.Europhysics Letters, 117(2):20001, 2017

work page 2017
[40]

Stable amplitude chimera states in a network of lo- cally coupled stuart-landau oscillators.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 28(3), 2018

K Premalatha, VK Chandrasekar, M Senthilvelan, and M Lak- shmanan. Stable amplitude chimera states in a network of lo- cally coupled stuart-landau oscillators.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 28(3), 2018

work page 2018
[41]

Sta- ble amplitude chimera in a network of coupled stuart-landau oscillators.Physical Review E, 98(3):032301, 2018

K Sathiyadevi, VK Chandrasekar, and DV Senthilkumar. Sta- ble amplitude chimera in a network of coupled stuart-landau oscillators.Physical Review E, 98(3):032301, 2018

work page 2018
[42]

Zajdela and D.M

E.R. Zajdela and D.M. Abrams. Phase chimera states: frozen patterns of disorder.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(8), 2025

work page 2025
[43]

Phase chimera states on nonlocal hyperrings.Physical Review E, 109(2):L022201, 2024

Riccardo Muolo, Thierry Njougouo, Lucia Valentina Gam- buzza, Timoteo Carletti, and Mattia Frasca. Phase chimera states on nonlocal hyperrings.Physical Review E, 109(2):L022201, 2024

work page 2024
[44]

Chimera states on m-directed hyper- graphs.to appear in Physical Review E, 2026

Rommel Tchinda Djeudjo, Timoteo Carletti, Hiroya Nakao, and Riccardo Muolo. Chimera states on m-directed hyper- graphs.to appear in Physical Review E, 2026

work page 2026
[45]

A robust method for classification of chimera states.arXiv preprint arXiv:2603.22026, 2026

S Nirmala Jenifer, Riccardo Muolo, Paulsamy Muruganandam, and Timoteo Carletti. A robust method for classification of chimera states.arXiv preprint arXiv:2603.22026, 2026

work page arXiv 2026
[46]

Some methods of classification and analy- sis of multivariate observations

James B McQueen. Some methods of classification and analy- sis of multivariate observations. InProc. of 5th Berkeley Sym- posium on Math. Stat. and Prob., pages 281–297, 1967

work page 1967
[47]

Networks of coupled oscil- lators: From phase to amplitude chimeras.Chaos: An Interdis- ciplinary Journal of Nonlinear Science, 28(11):113124, 2018

Tanmoy Banerjee, Debabrata Biswas, Debarati Ghosh, Ecke- hard Sch¨oll, and Anna Zakharova. Networks of coupled oscil- lators: From phase to amplitude chimeras.Chaos: An Interdis- ciplinary Journal of Nonlinear Science, 28(11):113124, 2018

work page 2018
[48]

Chimera states in fractional-order coupled rayleigh oscillators.Communications in Nonlinear Science and Numerical Simulation, 135:108083, 2024

Zhongkui Sun, Qifan Xue, and Nannan Zhao. Chimera states in fractional-order coupled rayleigh oscillators.Communications in Nonlinear Science and Numerical Simulation, 135:108083, 2024

work page 2024
[49]

Peshawa J Muhammad Ali. Investigating the impact of min- max data normalization on the regression performance of k- nearest neighbor with different similarity measurements.ARO- The Scientific Journal of Koya University, 10(1):85–91, 2022

work page 2022
[50]

Silhouettes: a graphical aid to the interpre- tation and validation of cluster analysis.Journal of computa- tional and applied mathematics, 20:53–65, 1987

Peter J Rousseeuw. Silhouettes: a graphical aid to the interpre- tation and validation of cluster analysis.Journal of computa- tional and applied mathematics, 20:53–65, 1987

work page 1987
[51]

A cluster separation measure.IEEE transactions on pattern analysis and machine intelligence, (2):224–227, 1979

David L Davies and Donald W Bouldin. A cluster separation measure.IEEE transactions on pattern analysis and machine intelligence, (2):224–227, 1979

work page 1979
[52]

Springer, 2006

Christopher M Bishop and Nasser M Nasrabadi.Pattern recog- nition and machine learning, volume 4. Springer, 2006

work page 2006
[53]

Amplitude chimeras and bump states with and without frequency entanglement: a toy model.Journal of Physics: Complexity, 5(2):025011, 2024

Astero Provata. Amplitude chimeras and bump states with and without frequency entanglement: a toy model.Journal of Physics: Complexity, 5(2):025011, 2024

work page 2024
[54]

Max- imum likelihood from incomplete data via the em algorithm

Arthur P Dempster, Nan M Laird, and Donald B Rubin. Max- imum likelihood from incomplete data via the em algorithm. Journal of the royal statistical society: series B (methodologi- cal), 39(1):1–22, 1977. Appendix A: Gaussian Mixture Model Gaussian Mixture Model (GMM).The Gaussian Mixture Model can be viewed as a probabilistic extension ofk-means. It assu...

work page 1977