The changing notion of chimera states, a critical review
TThe changing notion of chimera states, a critical review
Sindre W. Haugland a) Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching,Germany (Dated: 11 February 2021)
Chimera states, states of coexistence of synchronous and asynchronous motion, have been a subject of extensive researchsince they were first given a name in 2004. Increased interest has lead to their discovery in ever new settings, boththeoretical and experimental. Less well-discussed is the fact that successive results have also broadened the notionof what actually constitutes a chimera state. In this article, we critically examine how the results for different modeltypes and coupling schemes, as well as varying implicit interpretations of terms such as coexistence, synchrony andincoherence, have influenced the common understanding of what constitutes a chimera. We cover both theoretical andexperimental systems, address various chimera-derived terms that have emerged over the years and finally reflect on thequestion of chimera states in real-world contexts.
Almost twenty years ago, Kuramoto and Battogtokh real-ized that a ring of coupled identical oscillators, when ini-tialized in the right way, would split into two spatial do-mains : In one of these domains, the oscillators remainedmutually synchronized, while in the other, they driftedwith varying average frequencies. The coupling betweenthe oscillators in their system was nonlocal, that is, ofintermediate range between local (nearest-neighbor) andglobal (all-to-all symmetric) coupling. This was also thecase when Abrams and Strogatz gave the phenomenon itsname – chimera state – and expanded the mathematicalunderstanding of how it arises . While they believed thechimera state to be peculiar to nonlocal coupling, Abramsand Strogatz never defined this as one of its definite prop-erties, nor did they attempt to restrict the type of os-cillator ensemble wherein it might be acceptable to oc-cur. Thus, the research community was free to discoverchimera states in other systems as well, ranging from var-ious types of oscillators via maps to cellular automata;from local via various forms of nonlocal to global coupling;in both simulations and, from 2012 onward , in experi-ments. The general extent of this research is well known,and several reviews on chimera states have already beenwritten . Less reflected upon is how the chimera commu-nity exploited the soft original chimera definition to asso-ciate ever new results with the illustrious term. Over time,the name itself implicitly changed and expanded its mean-ing, to the point where very different phenomena were alllabeled chimera states, usually with little consideration ofhow strongly each of them is related to the others. Simi-larly, a lot of chimera-derived terminology was introduced,with little attempt, beyond a 2016 classification scheme ,to reflect on the extent of the relevance of many of the newterms. This may explain both why the field has been ableto accelerate as quickly as it has, and why no unified the-ory of all chimera states has been established. Notably,while chimeras have now been realized in a wide range ofexperiments, the question of their potential existence out-side laboratories has not yet been conclusively resolved. a) Electronic mail: [email protected]
I. INTRODUCTION
Towards the end of the 1980s, Kaneko discovered both clus-tering and chimera states in globally coupled logistic maps .Altogether, he identified four different types of attractors, de-pending on the sizes of the clusters they contain (including“clusters” of size 1). These were (a) fully synchronized motion;as well as attractors with (b) a small number of clusters, muchsmaller than the system size N ; (c) a large number of clustersin the order of N , but at least one cluster N comparable insize to N ; (d) only small clusters in the order of 1. The stillunnamed chimera state was only one among several differentstates of the third type. See Fig. 1.Not long after, clustering was found and studied in globallycoupled phase oscillators by Golomb et al. and Okuda , andin globally coupled Stuart-Landau oscillators by Hakim andRappel . Nakagawa and Kuramoto found a chimera state inthe latter system in 1992, but did not treat it as more than astep from clustering to fully chaotic motion . Not before tenyears later, after studying the nonlocally coupled CGLE and aring of nonlocally coupled phase oscillators, did Kuramoto andBattogtokh publish the idea that the coexistence of synchro-nized and non-synchronized motion might be an interestingphenomenon in its own right .In 2004, Abrams and Strogatz gave the presumptively newkind of symmetry breaking its name – chimera state – and inthe years that followed, the number of reported chimeras hasvastly increased, as summed up in several review papers .Chimera states have also been found in ever more differentmodels, meaning that the general conditions under which oneor more of them could be said to occur, have become increas-ingly varied as well. Over the course of this chimera researchexplosion, Schmidt and Krischer identified Kaneko’s earliestchimera in 2013 , while Sethia and Sen rediscovered Naka-gawa and Kuramoto’s 1992 chimera only few months later .Yet, what no one seems to have pointed out so far, is how theentirety of states considered to be chimeras in the first placehas evolved into an ever more diverse collection of dynami-cal phenomena. This tacit liberalization of the term “chimera a r X i v : . [ n li n . AO ] F e b FIG. 1. Schematic examples of the different attractor types reported by Kaneko in 1989 . From left to right: [ N ] : Coherent (completesynchronization). [ N , N ] : Partition to two clusters as an example of attractor wherein the number of clusters does not grow with N . [ N , , ,..., ] : Chimera state as an example of an attractor with O ( N ) clusters, but at least one cluster comparable to N . [ N , N , N , ,..., ] :A more complex attractor of the same fundamental type. [ , ,..., ] : Complete desynchronization and no clusters that grow in size with N .Reprinted from K. Kaneko, Chaos 25 (9), 097608, 2015, with the permission of AIP Publishing. doi:10.1063/1.4916925. state” includes steps such as the extension from oscillators tomaps, as well as ever new implicit interpretations of termssuch as symmetric coupling, synchrony and incoherence, andis critically examined in this review article.Also noticeable is how Kaneko’s original globally-coupled-maps chimera state was discovered and presented in the con-text of more or less related attractors right from the start. Incontrast, Kuramoto and Battogtokh’s chimera under nonlocalcoupling was carefully constructed (for parameters where thefully synchronized solution is stable) and thus had little origi-nal context. Context was instead created around it whenevernew results in a wide range of models were produced, a pro-cess that has yielded terms such as ”alternating chimera“ ,“multichimera” , , to name but a few. Seldom, though, is anyparticular system found to exhibit more than a few of thesederived phenomena. This issue will also be discussed below. II. FROM NONLOCAL TO GLOBAL COUPLING ANDVARIOUS COUPLING SCHEMES
When the term “chimera state” was coined by Abrams andStrogatz in 2004, they described it quite simply as “an array ofidentical oscillators split[ting] into two domains: one coherentand phase locked, the other incoherent and desynchronized” .In their article, they cited only two prior works discerningthis kind of previously unnamed state: The first was a 2002paper by Kuramoto and Battogtokh on its occurrence in theone-dimensional complex Ginzburg-Landau equation (CGLE)with nonlocal coupling. Here, the coexistence of synchronizedand unsynchronized oscillators was also found to persist as theoriginally complex-valued oscillatory medium was reducedto a phase-oscillator approximation . The second was a 2004paper on the two-dimensional equivalent, a spiral wave withan incoherent core, found both in the phase approximation ofthe nonlocally coupled, spatially two-dimensional CGLE andin a 2D array of FitzHugh-Nagumo oscillators .Kuramoto and Battogtokh considered their original chimerastate to be “among the variety of patterns which are charac-teristic to nonlocally coupled oscillators” . In a 2006 paper,Abrams and Strogatz similarly draw the tentative conclusionthat these dynamics are “peculiar to the intermediate case ofnonlocal coupling” . The basis of their reasoning is two-fold: Firstly, no other chimeras were known to them. Secondly,the coexistence of synchrony and incoherence in identicalsine-coupled phase oscillators (the system to which Kuramotoand Battogtokh reduced their nonlocally coupled CGLE in2002) does indeed become impossible when the oscillators arecoupled globally . When Sethia and co-workers describedchimera states in delay-coupled oscillators in 2008 , theywere also under the impression that nonlocal coupling is indis-pensable. The same goes for Wolfrum et al. during their 2011investigations of the Lyapunov spectra and stability of chimerastates of various sizes .The mentioned papers all concentrated their attention on thecase of weak coupling, where the CGLE can be approximatedby phase oscillators, if not outright restricting themselves tophase oscillators as the starting point of their investigations.So did a significant number of other papers published duringthe first decade of chimera-state research . An exceptionwas Laing’s 2010 paper , in which he analyses a chimerastate in an ensemble of Stuart-Landau oscillators without re-sorting to the phase reduction, but here too, the coupling isrelatively weak and the amplitudes of individual oscillatorsdeviate only a few percent from their average value. Notably,all his non-synchronized oscillators are restricted to the sameclosed curve in the complex plane, which allows for effectivelyparametrizing their position by a phase alone, even thoughtheir amplitudes vary.Only in 2013 did Sethia et al. show that a coexistence ofcoherent and incoherent dynamics can also occur in the nonlo-cally coupled CGLE in the case of strong coupling. Here, theamplitude in the incoherent part of the system varies strongly inboth space and time, inspiring the name “amplitude-mediatedchimeras” (AMC) and preventing any attempt at a phase reduc-tion . Less than half a year later, Schmidt et al. successfullytook the next step and identified a chimera state in an ensembleof Stuart-Landau oscillators with nonlinear purely global (sym-metric all-to-all) coupling . Like the AMCs of Sethia et al.,this state exhibits strong amplitude fluctuations in the incoher-ent oscillators. Moreover, it persists when adding diffusion andthus transitioning to a spatially extended 2D medium, wheresynchrony and incoherence form several clearly distinguish-able intertwined islands. Shortly after, Sethia and Sen reportedamplitude-mediated chimeras for Stuart-Landau oscillatorswith linear global coupling as well . In 2015, Laing comple-mented these efforts by producing chimera states in a ring ofoscillators with only local (nearest-neighbor) coupling .The chimera states reported before the mentioned paperby Schmidt et al. had all been found in nonlocally cou-pled systems in the broad sense of the coupling being neithernext-neighbor/diffusional (“local coupling”) nor all-to-all sym-metric (“global coupling”). However, the applied couplingschemes already encompassed a variety beyond Kuramotoand Battogtokh’s original 1D ring topology and exponentiallydecaying coupling kernel. Some of the alternative forms ofnonlocal coupling were rather small variations, such as the useof a cosine kernel or a step function to limit the extentof the influence of each point in the system on the others. (Aslong as the extent of the ring is restricted to − π < x ≤ π , thecosine kernel G ( x − x (cid:48) ) ∝ + A cos ( x − x (cid:48) ) , and thus the cou-pling strength, also decreases monotonously with distance .)More radical was the idea of dividing the oscillators into twopopulations with symmetric all-to-all coupling within eachgroup, as well as a weaker coupling between the groups . SeeFig. 2. Besides Kuramoto and Battogtokh’s original system,this “simplest network of networks” , is possibly the mostinfluential theoretical model supporting chimera states, andsimilar models have been the subject of a large number ofsubsequent works . FIG. 2. Schematic representation of the two-groups model introducedto the field of chimera states by Abrams and Strogatz in 2008 and inspiring a large number of subsequent works. µ denotes thestrength of the coupling within each group and ν that between thegroups, usually with µ > ν . Reprinted figure with permission fromM. Panaggio, D. M. Abrams, P. Ashwin & C. R. Laing, PhysicalReview E 93 (1), 012218, 2016. doi:10.1103/PhysRevE.93.012218.Copyright 2016 by the American Physical Society. Among the chimera-supporting systems inspired by the two-groups model were networks of three and eight popula-tions , respectively. It was possibly also the first model forwhich the question was asked, and answered in the affirmative,whether an observed chimera state will continue to exist if theoriginally identical oscillators are given heterogeneous frequen-cies . In a different variation from the original two-groupsmodel, connections between oscillators are gradually removedin order to further test the robustness of the chimera, and it wasfound to be more sensitive to removal of intra-group than ofinter-group links . Both the heterogeneous frequencies andthe removal of connections were motivated by the aim to createa more realistic model, as neither really identical oscillatoryunits nor perfectly symmetric coupling schemes are likely toexist in nature .Other variants of the two-groups model producing chimerastates include one in which the individual links between thepopulations are randomly switched on and off at equally spaced time intervals , one with different phase lags in the couplingwithin and between populations , and a delay-coupled versionwith different intra- and inter-population coupling delays .Schmidt et al. also invoke the two-groups model when explain-ing the stability of their globally coupled chimera. Here, thesynchronized and incoherent oscillators effectively form twoself-organized groups, and these groups exert different influ-ences on the respective other group, thereby further reinforcingthe chimera state once spontaneously formed . III. EXPERIMENTAL CHIMERAS AND A BROADERCHIMERA CONCEPT
One of the very first experimental realizations of a chimerastate was also based on the two-groups model. It was a 2012photochemical experiment by Tinsley et al., involving 40 pho-tosensitive Belousov-Zhabotinsky (BZ) oscillator beads. Here,each of the beads emits light of a certain intensity, which isrecorded with a CCD camera and projected selectively backon the beads by a spatial light modulator (SLM) . About ayear later, the two-groups model also formed the basis for apurely mechanical chimera, without any computer-mediatedcoupling, found in metronomes placed on two swings con-nected by springs: The intra-group coupling is conveyed byvibrations of the respective swing, while inter-group couplinghappens via the springs . Published by Hagerstrom et al. si-multaneously with the BZ chimera paper, but inspired by adifferent model , was an optical realization of an array ofcoupled maps : Here, the different parts of the cross-sectionof a beam of circularly polarized light have different phaseswhen they emerge from an SLM. As the beam passes throughan optical setup, these phases are translated into intensitiesrecorded by a camera, and these intensities in turn determinewhich phase shift the SLM is to apply to each part of the beamin the next iteration.While the first laboratory chimeras had taken a full ten yearssince Kuramoto and Battogtokh’s 2002 chimera state , thenext few years saw a much more rapid addition of experi-mental realizations, including mechanical models , net-works of discrete electrochemical oscillators and variouselectronic and optoelectronic systems . The first experi-mental chimera in a system with global coupling seems to havebeen observed in 2013 in an photoelectrochemical setup .Also of particular interest is the experimental chimera pub-lished by Totz et al. in 2017 : Here, BZ beads of the typepreviously used by Tinsley et al. are coupled by means ofthe same kind of optical feedback and virtually arranged in a40 ×
40 grid. For suitable experimental parameters, this yieldsa spiral-wave chimera with an incoherent core – the qualita-tively same kind of pattern as the first 2D numerical chimeraspublished by Shima and Kuramoto in 2004 .Several of the experimental chimeras published from 2012onward differ significantly from what a chimera state had orig-inally been: Already one of the first experimental realizationshad worked with chaotic maps , and not with oscillators, as itsaid in Abrams and Strogatz’ 2004 definition, but this mightjust be taken as an extension of the phenomenon to a new do-main. More remarkable in light of the original definition is thefact that this coupled-maps chimera is (partially) incoherentonly in space, while the whole system is temporally periodic .Something similar applies to the so-called “chimera states withquiescent and synchronous domains” found in the coupled elec-tronic oscillators of Gambuzza et al. two years later : Here,the voltage of some constituent circuits is constant in time,while it is oscillating with the same frequency in all the others,but there is no desynchronized region. Arguably also softeningthe original concept was the mechanical chimera state reportedby Wojewoda et al. in 2016, wherein two out of only threependula are synchronized, while the third one, uncorrelatedwith the other two, is declared to constitute its own incoher-ent group . Similarly unprecedented was the optoelectronicchimera published by Larger et al. one year before, in whichthere are no physically distinct coupled units, but just a singlesemiconductor laser setup subject to time-delayed feedback :The result is a single time-varying signal, with features (amongothers) on the length-scale of the applied delay; only whenthis signal is chopped up into segments and these segments arestacked on top of each other to form the temporal evolution of a“virtual space” does the chimera state appear to the observer. In2018, Brunner et al. successfully repeated the same procedurefor a setup with two simultaneously applied delays of differentmagnitude, thereby creating a chimera in 2D virtual space .Actually, this broadening of the scientific community’schimera concept had already begun in the theoretical systems:Already in 2008 did Omel’chenko et al. investigate a 1D arrayin which the force on each particular oscillator is not onlyproportional to its deviation from the common mean, but alsodependent on its absolute location in the array . Such a cou-pling scheme is not symmetric in the sense that all oscillatorsare governed by the same equation of motion and would allfeel the same force if they were fully synchronized. The resul-tant coexistence of synchrony (where the spatial modulation isstrong) and incoherence (where the spatial modulation is weak)was nevertheless declared to be a chimera state. Somethingsimilar applies to Laing’s later gradual and random removalof individual links from the two-groups model and to therandomly time-varying links which Buscarino et al. publishedin 2015 .Another chimera state, recognized in coupled maps byOmelchenko et al. in 2011, is notably periodic in time andincoherent only in space (thereby preceding the experimentalchimera of Hagerstrom et al. in this regard). This breaks withthe part of Abrams and Strogatz’ original chimera definition that assigns the attribute of being phase locked to the coher-ent group only. By allowing for chimera states in coupledmaps, Omelchenko et al. also laid the foundations for the laterrecognition of Kaneko’s much earlier globally-coupled-mapschimera .Similarly expanding the definition of chimera states werethe “amplitude chimeras” of Zakharova et al. : Here, alloscillators oscillate in synchrony, with the “incoherent” onesoscillating around different points in the complex plane than the“coherent” ones, as well as having different radii of oscillation.See Fig. 3. Some of the cellular-automaton chimeras publishedby García-Morales in 2016 also have a well-known periodicity; FIG. 3. Amplitude chimera in a ring of nonlocally coupled Stuart-Landau oscillators. (a) Averaged over a period, the position of all oscil-lators along the two “coherent” segments of the ring is centered on theorigin, while that along the two “incoherent” ring segments form anarc-like shape, reminiscent of the distribution of average frequenciesin the incoherent part of Kuramoto and Battogtokh’s 2002 chimera.(b) The average phase velocity ω j with which each oscillator orbitsits own respective average position, is the same throughout. (c) Phaseportrait of all oscillators in the complex plane. Reprinted from A.Zakharova, M. Kapeller & E. Schöll, Journal of Physics: ConferenceSeries 727 (1), 012018, 2016. doi:10.1088/1742-6596/727/1/012018under the terms of the Creative Commons Attribution 3.0 licence. and while he found no periodicity for some of the others duringthe tested simulation time, it is of course fundamentally truethat “because of the finiteness of the dynamics, the periodicityof any structure is bounded” , that is, because the states ofthe system are discrete, it is bound to repeat itself eventually.García-Morales was possibly also the first to recognize how thecommunity’s chimera definition had broadened, mentioninghow he would “regard chimera states as an experimental factof nature rather than a feature of certain systems of differentialequations or maps” . IV. TYPES OF CHIMERA STATES ANDCHIMERA-DERIVED CONCEPTS
The gradual expansion of the general concept of a chimerastate was accompanied by the naming of an increasing num-ber of derived phenomena, among them the aforementionedamplitude-mediated and amplitude chimera states. Discernedalready in 2008 was the “breathing chimera”. Here, the phasecoherence of the incoherent oscillators, quantified by the orderparameter r ( t ) = |(cid:104) e i θ j ( t ) (cid:105) incoh . | , where the sum is taken overthe phases θ j of all oscillators in the unsynchronized group, iseither periodic or quasiperiodic . This contrasts with whatAbrams et al. call a “stable chimera” , such as the one dis-covered by Kuramoto and Battogtokh, where r ( t ) is constantin time. A few later works have also used the term “breathingchimera” to denote a chimera in which the coherent and inco-herent parts move through the system, while the global degreeof clustering might remain constant throughout , possiblybecause this makes the local order parameter “breathe”.Also coined in 2008 was the term “clustered chimera state”with several coherent regions phase-shifted relative to eachother . This was joined five years later by the concept ofthe “multichimera”, likewise containing several distinct co- Spatiotemporal phenomenon
No Chimera ( ∃ t : g = ∨ g = ) Chimera ( ∀ t : 0 < g < ) STATIONARY ( g constant) static ( h > coherent ( h = moving ( h ≈ BREATHING ( g periodic) static ( h > moving ( h ≈ TURBULENT ( g irregular) static ( h > moving ( h ≈ Transient Chimera ( ∃ t : g = ∨ g = ∧∀ t < t : 0 < g < ) FIG. 4. Data-driven classification scheme of Kemeth et al. . Any spatiotemporal phenomenon is assessed in the form of time-series vectorsrepresenting the different parts or units (e.g. oscillators, maps) of the system. For each time t , the measure g ∈ [ , ] indicates how similar onaverage the value of each unit is to either its neighbors (in spatially ordered systems) or all other units. The measure h ∈ [ , ] indicates thefraction of the units that are strongly temporally correlated over the evaluated interval. Phenomenologically, the synchronized part of the systemremains fixed in “static chimeras”, while it moves in “moving chimeras.” herent regions, but with no phase difference between them .Either of these two phenomena have since come to be calledboth multicomponent , multicluster , multiheaded ormultiple-headed chimera . In the case of equal coupling toa fixed number of nearest neighbors (that is, a step-functionnonlocal coupling), the number of incoherent regions (re-ferred to as “heads” ) increases if the coupling range is madeshorter . If the coupling strength is made stronger, thenumber of heads may either increase or decrease , depend-ing on the underlying type of oscillator. More complex cou-pling topologies can also produce various multiplicities ofcoherent and incoherent regions, but the determining factorsare less obvious there . A 2D equivalent of the 1D mul-tichimera is the multicore spiral chimera reported by Xie etal. . In the chimera found by Schmidt et al. in the CGLE withnonlinear global coupling in 2013 , the concept of a distinctnumber of chimera heads is less meaningful, as synchronizedand incoherent regions, respectively, merge with time .In 2015, Ashwin and Burylko proposed a rigorous chimeradefinition applicable to small ensembles. They did this bydefining a “weak chimera” to be a state in which the averagephase velocities of at least two oscillators converge in thelimit of infinite time T → ∞ , while it remains different for atleast one other oscillator . This definition was subsequentlyused to classify states in several later works . Onemay assume that it was inspired by the observation that theeffective average frequencies of the incoherent oscillators differfrom the average frequency of the synchronized cluster inboth Kuramoto and Battogtokh’s 2002 chimera as well asseveral later chimera states . However, not allidentified classical chimera states (in the sense of some kindof coexistence of synchrony and incoherence) are actuallyweak chimeras as well. In particular they cannot be when theincoherent region drifts through the system with time and, asa consequence, all oscillators take turns being either coherentor incoherent . Not long after Ashwin and Burylko publishedtheir definition, Panaggio et al., for the purposes of their paperon two small oscillator populations, used it to define a chimera(without any qualifications) to mean a weak chimera in which the frequency-synchronized oscillators have the same phase .Two years later, findings by Kemeth et al. implicitly challengedthe potential use of this as a general definition: When scaled up,the two unsynchronized oscillators of a certain minimal (weak)chimera with a perfectly synchronized coherent part are namelynot replaced by a greater number of incoherent oscillators .Instead, the state becomes a three-cluster solution with onelarge and two small clusters. This contrasts with a differentkind of minimal chimera that the authors also identify, whereinthe dimensionality of the dynamics grows with the system sizeand which they thus coin an “extensive chimera state” .Also notable is the “alternating chimera”, in which twoequivalent parts of a system take turns being synchronized andincoherent, respectively. This was first produced by external pe-riodic forcing and later found to arise autonomously, in twopre-defined populations as well as in a globally coupled os-cillatory medium . Other chimera-inspired terms include the“globally clustered chimera”, denoting a system of several pre-defined populations that all split into both synchronized andincoherent oscillators ; “chimera death”, the coexistence ofspatially coherent and incoherent oscillation death ; and thepoetically named “Bellerophon states” that occur when a cer-tain chimera state is made unstable by parameter tuning . Ad-ditional chimeric phenomena are the “turbulent chimera” , the“intermittent chaotic chimera” and the “blinking chimera” ,as well as the “antichimera” and “dual chimera” . In contrast,Laing in a 2012 paper reported a kind of imperfect chimerain which one half of the oscillators are more strongly clusteredthan the other, while none of the two groups is fully synchro-nized, but without giving this phenomenon an additional name.When Kemeth et al. came up with a general classificationscheme for chimera states in 2016, the wide variety of bothexisting chimeras and the systems in which they occur madethe authors pick a data-driven approach . Their scheme, re-produced in Fig. 4 uses some of the aforementioned labels,such as breathing and turbulent chimera , in additionto coining new terms, such as “moving chimera” and “staticchimera”. In a moving chimera, most individual constituentunits of the regarded system change from incoherent to syn-chronized or vice versa within the regarded time interval, whilein a static chimera, they do not . Notably, the authors didnot only apply their classification scheme to already declaredchimera states, but to other dynamics as well. These includethe localized turbulence in the CGLE with time-delayed linearglobal coupling, as reported by Battogtokh et al. already in1997 . It is classified as a “turbulent moving chimera”, whichdiffers somewhat from the earlier conclusion of Schmidt et al.,who, on finding localized turbulence in the CGLE with nonlin-ear global coupling, were more reluctant to call it a chimerastate . Kemeth et al. also evaluate the gradual formation ofincoherent patches on a uniformly oscillating background inFalcke and Engel’s 1994 work on a model of CO coverage ona platinum surface . The data-based classification schemegroups these dynamics in the same category of finite-lifetimestates as the aforementioned amplitude chimera, a categoryKemeth et al. suggest to call “transient chimera”. While notcovered by the article on the classification scheme, what Yanget al. call “localized irregular clusters” in a 2000 paper on theBelousov-Zhabotinsky reaction with global feedback alsolooks suspiciously like a chimera state. V. DIFFERENT STANDARDS FOR NATURAL-WORLDCHIMERAS?
Any broader treatment of chimera states should reflect atleast briefly on the possibility of chimeras outside of labora-tories. Here, the phenomenon most readily invoked by thecommunity is probably unihemispheric sleep, with Rattenborg,Amlaner and Lima’s extensive 2000 neuroscientific review pa-per being cited in the introductions of many of the studiesmentioned above. Very briefly explained, a unihemisphericallysleeping animal sleeps with one half of its brain at a time.Aquatic mammals sleep this way, allowing them to surface tobreathe, as do birds and at least some reptiles . When mea-sured, the EEG activity in the sleeping brain hemisphere ishigh-amplitude, low-frequency, while that in the awake hemi-sphere is low-amplitude, high-frequency, implying that theindividual neurons in the former are firing more strongly syn-chronized than those in the latter . First to notice the possibleconnection to early numerical chimeras were probably Abramset al., whose 2008 chimera-supporting two-groups model ismotivated by the question of what might be the simplest sys-tem of two oscillator populations to emulate this kind of brainbehavior .While the roles of the synchronized and incoherent groupin the two-groups model are fixed once established, naturalunihemispheric sleep tends to move from one half of the brainto the other several times over the course of an interval of sleep-ing. This was recognized by Ma, Wang and Liu in a 2010 paper,wherein they describe the first of the aforementioned alternat-ing chimera states. However, in order to observe the switchingof synchrony from one population to the other, they have toresort to an external periodic forcing, which they declare to“represent the varying environment” . In 2015, Haugland etal. reported a fully self-organized alternating chimera withoutneither pre-defined groups nor any external force, claiming to “tighten [. . . ] the connection between chimera states and uni-hemispheric sleep” . As recently as 2019, chimera states havealso been realized in two different two-layer networks moreclosely inspired by brain architecture , thereby completingthe continuum of phenomena from actual unihemispheric sleepto the most ideal mathematical chimera.Related to unihemispheric sleep and also mentioned as amotivation for chimera research is the “first-night effect” inhumans, keeping one hemisphere more vigilant when sleep-ing in a novel environment . Other authors have likenedchimera states to the regional highly synchronized brain activ-ity during epileptic seizures or due to Parkinson’s disease .Similarly, spiral-wave chimeras, with their incoherently fluc-tuating core , have been compared to ventricular fibrillation,when rotating patterns of excitation occur on the heart, withpossibly uncoordinated dynamics at their center . But neitherof these comparisons seem to have sparked in-depth delibera-tion like unihemispheric sleep.In their 2013 article on chimera states in two pendulum pop-ulations, Martens et al. claim that their “model equations trans-late directly to recent theoretical studies of synchronizationin power grids” , implying that chimera states might occurin power grids as well. Panaggio and Abrams also suggestthat knowing the basins of stability of chimera states in powergrids could be useful in avoiding them and thus maintainingthe synchronous oscillation that the grid needs to function .Several other chimera papers briefly refer to this possible con-nection , but they seldom elaborate on it. Infact, the article on power-grid modeling that is probably mostoften cited in chimera introductions, published by Motter etal. in 2013 , contains just a single superficial reference tochimeras in its own introduction. Its main aim is to demon-strate a condition for when network synchrony is stable and itis little concerned with what kinds of unsynchronized statesmay exist. A few results are contributed by a 2014 paper byPecora et al. , wherein they explain the onset of so-calledisolated desynchronization by means of network topology anduse two real power grids as models (among many others). How-ever, since isolated desynchronization is caused by topology,it should only be relevant to some kinds of chimera states (inthe widest sense) and less to those arising by spontaneoussymmetry breaking.Chimera states have also been linked to theturbulent-laminar patterns that may be observed in Taylor-Couette flow . Another article uses a social-agent model tosuggest that “an analogue to a chimera state” could also existin the behavior of interacting human populations .At the end of his 2018 review article, Omel’chenko refersto all past attempts to identify a non-laboratory chimera as“rather speculative” and “requir[ing] more rigorous justifica-tion” . With regards to most of the above examples, this in-deed seems to be the case. As far as unihemispheric sleep isconcerned, we could alternatively ask exactly what kind ofjustification is missing. Do actual brain-measurement datanot show the coexistence of synchronized and desynchronizedoscillation? Are these data not backed up by models modeledon the natural-world phenomenon, already declared to exhibitchimera states? Of course, the mechanism at work in the bird FIG. 5. EEG measurements of the bottlenose dolphin published by Mukhametov et al. in 1977 and later reprinted by Rattenborg et al. .(a) Location of the electrodes across the two brain hemispheres. (b,c) EEG activity measured by each electrode during two short intervals recordedone hour apart. The sleeping hemisphere (electrodes 1-3 in b and 4-6 in c) is characterized by high-amplitude low-frequency EEG activity, theawake hemisphere by low-amplitude high-frequency EEG activity. Reprinted figure with permission from L. M. Mukhametov, A. Y. Supin & I.G. Polyakova, Brain Research 134 (3), 581-584, 2016. doi:10.1016/0006-8993(77)90835-6. Copyright 1977 Published by Elsevier B.V. or dolphin brain is not the same as that in all reported chimerastates, but the latter also differ strongly among themselves.Could we thus be holding potential natural-world chimeras toa different standard than theoretical and experimental ones?And could this question possibly be better addressed, if thecurrently rather fluid and to a large extent implicit chimeradefinition were made more concrete? VI. CONCLUDING REMARKS
Above, we have seen that chimera states were originallydiscovered in globally coupled logistic maps, earlier than oftenbelieved. Not until they had also been produced in nonlocallycoupled oscillators, however, were they discerned as a specialkind of state and given a name. Once they had a name, andonce, a few years later, the versatile two-groups frameworkand the reference to unihemispheric sleep as a potential field ofreal-world relevance were introduced, a decade of expansivechimera research began. Their number and variety increased,as did the number of systems found to support them. Chimerastates were found in a wide range of experimental settings aswell. Various derived concepts emerged, though many of themremained mostly limited to their original context. Additionalanalogies to different real-world phenomena were also drawnup, though many of these have so far remained rather super-ficial. Notably, the research community has not yet arrivedat a common conclusion that any natural-world phenomenaactually are chimera states.This might have something to do with the fact that chimerasare not a most definite physical phenomenon (like neutrons,the Hall effect or protein folding), with definite properties wejust have to measure accurately enough to discover. The termis more abstract and might thus also be considered as a loose collection of more or less related observations. New observa-tions are then being given the same label as existing ones onrather discretionary grounds. This is probably both a blessingand a curse of the field, with unconstrained analogies enablingmany a fruitful discovery, but at the same time counteractingthe internal ordering of the entirety of results. In particular, asever new results have broadened the scope of what is calleda chimera state, there seems to have been rather limited re-flection on how this has changed the object of the field itself.Future research could probably benefit from a more explicitconsideration of this insight.
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