Mechanism for Strong Chimeras
MMechanism for Strong Chimeras
Yuanzhao Zhang and Adilson E. Motter
1, 2 Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
Chimera states have attracted significant attention as symmetry-broken states exhibiting theunexpected coexistence of coherence and incoherence. Despite the valuable insights gained fromanalyzing specific systems, an understanding of the general physical mechanism underlying theemergence of chimeras is still lacking. Here, we show that many stable chimeras arise becausecoherence in part of the system is sustained by incoherence in the rest of the system. This mechanismmay be regarded as a deterministic analog of the phenomenon of noise-induced synchronization andis shown to underlie the emergence of strong chimeras. These are chimera states whose coherentdomain is formed by identically synchronized oscillators. Recognizing this mechanism offers a newmeaning to the interpretation that chimeras are a natural link between coherence and incoherence.
Chimera states are a remarkable phenomenon inwhich coherence and incoherence coexist in a system ofidentically-coupled identical oscillators [1, 2]. Initially re-garded as a state that requires specific nonlocal couplingstructure [3–5] and/or specially prepared initial condi-tions [6, 7], chimeras have since been shown to be ageneral phenomenon that can occur robustly as a sys-tem (upon parameter changes) transitions from coher-ence to incoherence [8–14]. Moreover, chimeras havebeen observed in a diverse set of physical systems, in-cluding those of optoelectronic [10, 15–17], electrochem-ical [12, 14, 18, 19], and mechanical [20] nature. Thereis even evidence pointing to chimera-like states in quan-tum systems [21] and in the brain [22]. Despite numer-ous efforts to elucidate the underlying principles [20, 23–29], currently no system-independent mechanistic expla-nation exists that can provide broad physical insight intothe emergence of chimeras.Our goal is to bridge this gap by proposing a gen-eral mechanism for chimeras that is not tied to specificnode dynamics, network structure, or coupling scheme.We consider the important class of permanently sta-ble chimera states whose coherent part is identically synchronized, which have been observed for both pe-riodic [12, 24, 30] and chaotic oscillators [16, 17, 31].We also focus on parameter regions where global co-herence is unstable , so the chimeras may be observedwithout the need of specially prepared initial conditions[8, 12, 17, 19, 24, 26, 29, 31, 32]. Here, chimera statesthat i) are permanently stable, ii) exhibit identically syn-chronized coherence, and iii) do not co-occur with stableglobal synchronization are referred to as strong chimeras.Such states represent a diverse range of chimeras andhave been observed in myriad systems [12, 17, 24, 26, 31].In this Letter, we characterize strong chimeras thatemerge between a globally synchronized state and a glob-ally incoherent state as a bifurcation parameter is var-ied (Fig. 1). In such chimera states, the coexistence ofa synchronous and an incoherent cluster challenges theintuition that inputs from the incoherent cluster wouldinevitably desynchronize the other cluster. Yet, our anal-
FIG. 1. Example scenario considered in this study. As abifurcation parameter is varied and the system transitionsfrom coherence to incoherence, an intermediate chimera re-gion emerges. ysis shows that incoherence in part of the system in factstabilizes the otherwise unstable coherence in the restof the system, thus preventing a direct transition fromcoherence to incoherence when the former becomes un-stable. This incoherence-stabilized coherence effect canbe seen as a deterministic analog of synchronization in-duced by common noise [33–36] and serves as a generalmechanism giving rise to strong chimeras.We first note that strong chimeras can be observed forboth periodic and chaotic node dynamics, and in net-works with either diffusive or non-diffusive coupling. Inall cases, the necessary and sufficient condition for thecoherent cluster to admit an identical synchronizationsolution is that each of its oscillators receives the sameinput from the incoherent cluster. This condition trans-lates generically to the coherent cluster belonging to anequitable partition [37] and not being intertwined [38]with the rest of the network, although intertwined clus-ters are still allowed elsewhere in the network. (For diffu-sive coupling, the partition can be further relaxed to be externally equitable [39].) The conditions above extendimmediately to strong chimeras consisting of multiple co-herent and incoherent clusters.The impact of the rest of the network on the coherentcluster of a strong chimera can be analyzed by consider-ing a network of N coupled oscillators described by x t +1 i = βf ( x ti ) + K N X j =1 M ij h ( x tj ) , i = 1 , . . . , N, (1) a r X i v : . [ n li n . AO ] J a n where x ti is the state of the i th oscillator at time t , func-tion f governs the dynamics of the uncoupled oscillators, β denotes the self-feedback strength of the oscillators, M = ( M ij ) is the coupling matrix representing the net-work structure, h is the coupling function, and K is theoverall coupling strength. We assume the oscillators tobe time-discrete and one-dimensional for simplicity, butthe analysis extends straightforwardly to continuous-timeand higher-dimensional systems. The matrix M can berather general, including both diffusive and non-diffusivecoupling schemes. Now, suppose that C is the coherentcluster and that it consists of n nodes numbered from 1 to n . For oscillators in this cluster, the dynamical equationtakes the form x t +1 i = βf ( x ti ) + K n X j =1 M ij h ( x tj ) + I ( t ) , i = 1 , . . . , n, (2)where I ( t ) = K P Nj = n +1 M ij h ( x tj ) is the input from therest of the network, which does not depend on i sincethe cluster must belong to a partition that is at leastexternally equitable. Thus, the function I ( t ) is commonacross all nodes in C and the identical synchronizationstate s t in this cluster is given by s t +1 = βf (cid:0) s t (cid:1) + Kµ h (cid:0) s t (cid:1) + I ( t ) , (3)where the row sum µ = P nj =1 M ij is a constant not de-pending on i for any 1 ≤ i ≤ n . The stability of thisstate is determined by the largest transverse Lyapunovexponent (LTLE) Λ specified by the variational equations η t +1 i = h βf ( s t ) + K b λ i h ( s t ) i η ti , i = 2 , . . . , n, (4)where b λ i is the i th eigenvalue of the n × n sub-couplingmatrix c M = ( M ij ) ≤ i,j ≤ n and η i is the correspondingperturbation mode. The mode associated with b λ = µ is excluded as it corresponds to perturbations parallel tothe synchronization manifold. Equation (4) implicitly as-sumes that c M is diagonalizable, but this assumption canbe lifted using the Jordan canonical form of this matrix[40, 41].We explicitly examine Eq. (1) for the two most widelystudied coupling schemes, namely diffusive coupling de-fined by the Laplacian matrix L and non-diffusive cou-pling defined by adjacency matrix A . For Laplacian cou-pling, M = − L and thus µ is the negative of the indegreeof nodes in C due to connections from the rest of the net-work. The eigenvalue b λ i is given by b λ i = − λ i + µ , where λ i is the eigenvalue of the Laplacian matrix of C in iso-lation (i.e., consisting of intracluster connections only).For adjacency-matrix coupling, M = A , the factor µ isthe indegree of nodes in C when the cluster is consideredin isolation, and the eigenvalues are b λ i = λ i , where λ i arethe eigenvalues of the adjacency matrix of C in isolation. We first consider Laplacian coupling and, for concrete-ness, focus on networks composed of two identical clusterswith all-to-all intercluster coupling. These networks areknown to exhibit chimera states, which have been exten-sively studied in the literature [6, 18, 20, 30]. Assumethat as a bifurcation parameter q is increased, the coher-ent state of a two-cluster network becomes unstable ata critical value q c . Because this point marks the end ofcoherence, at least one cluster must become incoherentwhen q is further increased (see Supplemental Material[43] for details). Since both clusters are identical, onemight expect that both will become unstable at q c andthat the system will thus transition directly from coher-ence (both clusters synchronized) into incoherence (bothclusters incoherent). That is, due to the symmetry be-tween the two clusters, both clusters are expected to losesynchrony at the same time. Nevertheless, chimera statesoften emerge right at the instability transition, breakingthe symmetry between the clusters, with only one clusterbecoming incoherent while the other remains perfectlysynchronized. So, what prevents the system from evolv-ing directly into global incoherence? The short answer isthat, beyond q c , incoherence in one cluster can stabilizesynchronization in the other cluster, delaying the onset ofglobal incoherence and instead giving rise to a chimera.To further investigate this question [44], we focus onthe node dynamics and coupling function given by f ( x ) = h ( x ) = sin ( x + π/ , (5)which model optoelectronic oscillators that have been re-alized in synchronization experiments [17, 45]. While theintracluster coupling structure can be arbitrary in gen-eral, for clarity we focus on a network consisting of twoclusters of n = 3 nodes. The clusters have internal cou-pling of strength K and are connected to each other byall-to-all coupling of strength cK [Fig. 2(a)], where c isincluded in matrix M in the representation of Eq. (1).There is nothing special about this choice of dynamicsand cluster structure, and we show in the SupplementalMaterial [43] that our conclusions hold for other oscilla-tors (with both discrete and continuous dynamics) andnetworks. There, we also demonstrate the robustness ofour results against oscillator heterogeneity.Figure 2(b) shows the corresponding state diagram inthe β − K parameter space for c = 0 .
2. The classificationof states in the diagram is based on the linear stabilityanalysis of the coherent and chimera states as determinedby the corresponding LTLE [17]. A generic bifurcationscenario is depicted in Fig. 2(c): as a linear combina-tion of the parameters β and K is increased [dashed linein Fig. 2(b)], the system transitions from global coher-ence to a chimera state, and then from the chimera stateto global incoherence. In this example, the chimera isdefined by incoherence in cluster C and coherence incluster C . Starting from random initial conditions, it isequally likely for the clusters to exhibit swapped states, FIG. 2. Strong chimeras in a network of diffusively coupled oscillators given by Eq. (5). (a) Network consisting of two identicalclusters, C and C . (b) Diagram in the β − K parameter space marking the regions for which the system exhibits coherence(cyan), chimeras (purple), and incoherence (red) for c = 0 .
2. (c) State transitions as parameter q is varied quasi-staticallyalong the dashed line in (b) [42], showing the abrupt change from coherence to a chimera at q c and then from the chimera toincoherence at q c . The top and middle panels show the states x ti in each cluster colored by the individual oscillators, and thebottom panel shows the time-averaged synchronization errors (obtained from separate simulations for fixed values of q ), wherethe error is defined by the standard deviation among the oscillator states. corresponding to a chimera in q c < q < q c that has in-coherence in C and coherence in C .We can now establish a theoretical foundation for themechanism underlying the onset of such chimeras by ex-amining Eqs. (3)-(4). Crucially, the input from the restof the network is irregular temporally but uniform spa-tially and does not affect the variational equations of C directly, since I ( t ) does not appear in Eq. (4). Yet, itindirectly impacts synchronization stability by changingthe synchronous state s t according to Eq. (3). It is en-tirely through the change it causes to s t that incoherencein the rest of the network stabilizes coherence in C , givingrise to a stable chimera.To establish this rigorously we note that the Lya-punov exponents of Eq. (4) can be written as Λ ( i ) =ln |− K b λ i − β | +Γ s , where Γ s = lim T →∞ ln (cid:12)(cid:12) Π Tt =1 f ( s t ) (cid:12)(cid:12) T for f ( x ) = h ( x ) as in the systems explicitly examinedhere. Since Γ s is generally finite, the associated masterstability function (MSF) [46] e Λ( α, β ) = ln | α − β | + Γ s defines a finite stability region, and synchronization in C is stable if and only if | − K b λ i − β | < e − Γ s , i = 2 , . . . , n. (6)This equation explains what happens at the interface be-tween global coherence and a chimera state. As the bifur-cation parameter q is varied and the desynchronizationtransition is approached from the left, the condition inEq. (6) is violated by at least one transversal mode andΛ = max i ≥ { Λ ( i ) } vanishes at q = q c . But past thispoint, desynchronization in the other clusters alters s t dramatically, and Γ s changes (abruptly) from Γ cos to Γ ins according to its dependence on Eq. (3). If Γ ins < Γ cos ,the stability region defined by Eq. (6) expands (i.e., Λbecomes negative again for q > q c ), and a chimera regionthen emerges due to stabilization caused by the incoher-ent input I ( t ). In the example of Fig. 2(b), in particular,Γ cos = − .
92 and Γ ins = − .
6, confirming our phenomeno- logical observation that incoherence in C stabilizes co-herence in C . Figure 3 shows the impact of this changeon e Λ [Fig. 3(a)] as well as the co-dependence of Λ and Γ s as the bifurcation parameter is varied [Fig. 3(b)].To further validate the hypothesis that the synchro-nization stability in the coherent cluster can be inducedby the incoherent driving, we model the effective input Kµ h (cid:0) s t (cid:1) + I ( t ) in Eq. (3) as a driving noise term ξ ( t ).The synchronization trajectory s t in the coherent clusteris then s t +1 = βf (cid:0) s t (cid:1) + ξ ( t ) , (7)with the corresponding variational equations given byEq. (4) for b λ i = − λ i − cn . Figure 4(a) shows the result ofour stability analysis for β = 1 . K = 1 .
2, and c = 0 . c ). We see that synchronization in the coherentcluster is unstable (Λ >
0) in the absence of external driv-ing ξ ( t ), but it can be stabilized (Λ <
0) by a Gaussianwhite noise over a range of mean ν and standard devia-tion σ . Moreover, the effective input from the incoherent FIG. 3. Impact of the incoherent cluster on the stability ofthe coherent one. (a) MSF for the coherent cluster before(red) and after (green) the other cluster transitions to inco-herence, showing a widening of the stable region. (b) LTLEof the coherent cluster (purple) as the parameter q = q ( β ) inFig. 2(b) is varied, showing a discontinuous transition at q c and a continuous one at q c due to the corresponding changesin Γ s (red). The system and other parameters are as in Fig. 2. FIG. 4. Incoherence-stabilized coherence for the influence ofthe incoherent cluster modeled as a noise driving. The systemis the same as in Fig. 2 for β = 1 . K = 1 .
2, and C as thecoherent cluster. (a) LTLE of the coherent cluster for c = 0 . ν and standard deviation σ . Also marked are thecorresponding ( ν , σ ) of the effective input when the othercluster is incoherent (blue cross) or coherent (red cross). (b)Typical time series of the effective input corresponding to theblue cross. cluster has ν = − .
30 and σ = 0 .
15, which is inside thestable region in Fig. 4(a). Figure 4(b) shows a typicaltime series of the effective input, which indeed closelyresemble a noisy signal. Our finding shares similaritieswith noise-induced synchronization, but it is worth not-ing that certain negative ν can stabilize coherence evenwhen σ = 0, which can potentially explain chimera deathstates [47]. On the other hand, the scenario in which bothclusters are synchronized corresponds to ν = σ = 0 fordiffusive coupling, which is equivalent to not having ex-ternal driving, and is thus unstable for the given param-eters. More generally, for diffusive coupling, coherence inone cluster can benefit from common driving only whenthe other cluster is not in the same state.We now turn to adjacency matrix coupling for f ( x ) = h ( x ) + ( π/ /β = [1 − cos( x )] / π/ /β, (8)which is a closely-related class of optoelectronic oscilla-tors for which this type of coupling has been implementedexperimentally [38]. Here, the dynamical variables areconstrained to the interval [0 , π ) by taking mod 2 π ateach iteration. Our primary goal with this model is to il-lustrate a coupling scheme for which the intercluster cou-pling term does not vanish in the coherent state. But wealso want to show that our results do not depend on thecoherent and incoherent clusters being equal. To facili-tate comparison with the literature, we adopt a networkand parameter setting for the system in Eq. (8) first con-sidered in Ref. [31]. The network consists of a ring of sixnodes coupled to their first and second nearest neighbors[Fig. 5(a)], and the parameters are set to β = 2 π/ − K .In Fig. 5(b), we perform a comparative analysis andplot the LTLE for coherence in cluster C (comprisingfour oscillators) when cluster C (comprising two oscilla-tors) is assumed to be coherent and incoherent, respec-tively. We see that incoherence in C significantly de-lays the instability transition in C from K = − .
65 to K = − .
80 and, as a consequence, gives rise toa much wider chimera region than previously expected[31]. A representative time series for the chimera stateat K = − .
72, whose coherent cluster is stabilized by theincoherent one, is shown in Fig. 5(a).The analysis presented above reveals a physical mech-anism underlying the emergence of strong chimeras. Theself-consistency of such states was partially elucidatedby the previous demonstration that desynchronization inone cluster does not necessarily lead to the concurrentdesynchronization in another cluster [31, 38]. Here, wehave been able to go one step further and demonstratethat incoherence in one cluster can in fact stabilize co-herence in the other cluster. This incoherence-stabilizedcoherence adds a new dimension to the proposition thatchimera states are the natural link between coherent andincoherent states [8, 48, 49].Our results have potential implications for seizuresthat arise from the excessive synchronization of large neu-ronal populations [50], which have been linked to chimerastates [51–54]. In particular, the discovery that the co-herent domain is often stabilized by its interaction withthe incoherent domain can help explain why corpus cal-losotomy [55], surgery that separates the brain into twodisconnected hemispheres, is an effective treatment forepilepsy and seizures. Incoherence-stabilized coherencealso provides insights on the counter-intuitive phenom-ena that desynchronization is often observed precedingseizures [52, 56, 57] and that high levels of synchronycan facilitate seizure termination [54, 58].As a promising direction for future research, we notethat chimera states not meeting the conditions for strongchimeras have been studied in the literature, includingthose with coherent domains that are not identically syn-chronized [5, 10, 14, 59] and those that co-occur with sta-ble global coherence [6, 7, 23, 29]. It remains to be shownhow the mechanism uncovered for strong chimeras mayprovide insight into those states. The key challenge isthat oscillators in the coherent domain are typically notdriven by a common input signal when they are not iden-tically synchronized, although they often remain approx-imately synchronized and receive similar input [5, 48].Given the prevalence of related phenomena such as noise-induced synchronization [60–63], we believe the coopera-tive relation between incoherence and coherence revealedby our analysis can be a general mechanism giving riseto a wide range of chimera states.The authors thank Z. G. Nicolaou, J. D. Hart, and R.Roy for insightful discussions. This work was supportedby ARO Grant No. W911NF-19-1-0383. [1] M. J. Panaggio and D. M. Abrams, Chimera states: Co-existence of coherence and incoherence in networks of
FIG. 5. Strong chimeras in a network of non-diffusively coupled oscillators given by Eq. (8). (a) Network of identically-coupled oscillators organized into an incoherent ( C ) and a coherent ( C ) cluster. Right: representative chimera trajectory for K = − .
72. (b) LTLE of the coherent cluster as a function of K , where the other cluster is taken to be in a coherent state(red) or an incoherent one (green). In the shaded region, coherence in C is stabilized by incoherence in C .coupled oscillators, Nonlinearity , R67 (2015).[2] O. E. Omel’chenko, The mathematics behind chimerastates, Nonlinearity , R121 (2018).[3] Y. Kuramoto and D. Battogtokh, Coexistence of coher-ence and incoherence in nonlocally coupled phase oscil-lators, Nonlinear Phenom. Complex Syst. , 380 (2002).[4] Y. Kuramoto, Nonlinear Dynamics and Chaos: Wheredo we go from here? (CRC Press, 2002) pp. 209–227.[5] D. M. Abrams and S. H. Strogatz, Chimera states forcoupled oscillators, Phys. Rev. Lett. , 174102 (2004).[6] D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A.Wiley, Solvable model for chimera states of coupled os-cillators, Phys. Rev. Lett. , 084103 (2008).[7] E. A. Martens, Bistable chimera attractors on a triangu-lar network of oscillator populations, Phys. Rev. E ,016216 (2010).[8] O. E. Omel’chenko, Y. L. Maistrenko, and P. A. Tass,Chimera states: The natural link between coherence andincoherence, Phys. Rev. Lett. , 044105 (2008).[9] G. Bordyugov, A. Pikovsky, and M. Rosenblum, Self-emerging and turbulent chimeras in oscillator chains,Phys. Rev. E , 035205 (2010).[10] A. M. Hagerstrom, T. E. Murphy, R. Roy, P. H¨ovel,I. Omelchenko, and E. Sch¨oll, Experimental observationof chimeras in coupled-map lattices, Nat. Phys. , 658(2012).[11] G. C. Sethia, A. Sen, and G. L. Johnston, Amplitude-mediated chimera states, Phys. Rev. E , 042917 (2013).[12] L. Schmidt, K. Sch¨onleber, K. Krischer, and V. Garc´ıa-Morales, Coexistence of synchrony and incoherence in os-cillatory media under nonlinear global coupling, Chaos , 013102 (2014).[13] E. A. Martens, M. J. Panaggio, and D. M. Abrams,Basins of attraction for chimera states, New J. Phys. ,022002 (2016).[14] C. Bick, M. Sebek, and I. Z. Kiss, Robust weak chimerasin oscillator networks with delayed linear and quadraticinteractions, Phys. Rev. Lett. , 168301 (2017).[15] L. Larger, B. Penkovsky, and Y. Maistrenko, Virtualchimera states for delayed-feedback systems, Phys. Rev.Lett. , 054103 (2013).[16] J. D. Hart, K. Bansal, T. E. Murphy, and R. Roy, Ex-perimental observation of chimera and cluster states ina minimal globally coupled network, Chaos , 094801(2016).[17] Y. Zhang, Z. G. Nicolaou, J. D. Hart, R. Roy, and A. E.Motter, Critical switching in globally attractive chimeras, Phys. Rev. X , 011044 (2020).[18] M. R. Tinsley, S. Nkomo, and K. Showalter, Chimera andphase-cluster states in populations of coupled chemicaloscillators, Nat. Phys. , 662 (2012).[19] J. F. Totz, J. Rode, M. R. Tinsley, K. Showalter, andH. Engel, Spiral wave chimera states in large populationsof coupled chemical oscillators, Nat. Phys. , 282 (2018).[20] E. A. Martens, S. Thutupalli, A. Fourri`ere, and O. Hal-latschek, Chimera states in mechanical oscillator net-works, Proc. Natl. Acad. Sci. U.S.A. , 10563 (2013).[21] V. Bastidas, I. Omelchenko, A. Zakharova, E. Sch¨oll, andT. Brandes, Quantum signatures of chimera states, Phys.Rev. E , 062924 (2015).[22] K. Bansal, J. O. Garcia, S. H. Tompson, T. Verstynen,J. M. Vettel, and S. F. Muldoon, Cognitive chimera statesin human brain networks, Sci. Adv. , eaau8535 (2019).[23] G. C. Sethia and A. Sen, Chimera states: The existencecriteria revisited, Phys. Rev. Lett. , 144101 (2014).[24] A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Chimera-like states in an ensemble of globally coupled oscillators,Phys. Rev. Lett. , 144103 (2014).[25] N. Semenova, A. Zakharova, E. Sch¨oll, and V. An-ishchenko, Does hyperbolicity impede emergence ofchimera states in networks of nonlocally coupled chaoticoscillators?, Europhys. Lett. , 40002 (2015).[26] L. Schmidt and K. Krischer, Clustering as a prerequisitefor chimera states in globally coupled systems, Phys. Rev.Lett. , 034101 (2015).[27] N. Semenova, A. Zakharova, V. Anishchenko, andE. Sch¨oll, Coherence-resonance chimeras in a network ofexcitable elements, Phys. Rev. Lett. , 014102 (2016).[28] Z. G. Nicolaou, H. Riecke, and A. E. Motter, Chimerastates in continuous media: Existence and distinctness,Phys. Rev. Lett. , 244101 (2017).[29] T. Kotwal, X. Jiang, and D. M. Abrams, Connecting thekuramoto model and the chimera state, Phys. Rev. Lett. , 264101 (2017).[30] M. J. Panaggio, D. M. Abrams, P. Ashwin, and C. R.Laing, Chimera states in networks of phase oscillators:The case of two small populations, Phys. Rev. E ,012218 (2016).[31] Y. S. Cho, T. Nishikawa, and A. E. Motter, Sta-ble chimeras and independently synchronizable clusters,Phys. Rev. Lett. , 084101 (2017).[32] J. Sieber, O. E. Omel’chenko, and M. Wolfrum, Control-ling unstable chaos: stabilizing chimera states by feed-back, Phys. Rev. Lett. , 054102 (2014). [33] C. Zhou, J. Kurths, I. Z. Kiss, and J. L. Hudson, Noise-enhanced phase synchronization of chaotic oscillators,Phys. Rev. Lett. , 014101 (2002).[34] D. S. Goldobin and A. S. Pikovsky, Synchronization ofself-sustained oscillators by common white noise, PhysicaA , 126 (2005).[35] K. H. Nagai and H. Kori, Noise-induced synchronizationof a large population of globally coupled nonidentical os-cillators, Phys. Rev. E , 065202 (2010).[36] A. V. Pimenova, D. S. Goldobin, M. Rosenblum, andA. Pikovsky, Interplay of coupling and common noise atthe transition to synchrony in oscillator populations, Sci.Rep. , 38518 (2016).[37] I. Belykh and M. Hasler, Mesoscale and clusters of syn-chrony in networks of bursting neurons, Chaos , 016106(2011).[38] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E.Murphy, and R. Roy, Cluster synchronization and iso-lated desynchronization in complex networks with sym-metries, Nat. Commun. , 4079 (2014).[39] M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne,R. Lambiotte, and M. Barahona, Graph partitions andcluster synchronization in networks of oscillators, Chaos , 094821 (2016).[40] T. Nishikawa and A. E. Motter, Maximum performanceat minimum cost in network synchronization, Physica D , 77 (2006).[41] J. D. Hart, Y. Zhang, R. Roy, and A. E. Motter, Topo-logical control of synchronization patterns: Trading sym-metry for stability, Phys. Rev. Lett. , 058301 (2019).[42] In Fig. 2(c), q is varied according to β = 1 + 4 × − t , K = 0 . − t for 10 iterations.[43] See Supplemental Material for more examples of strongchimeras, their robustness against oscillator heterogene-ity, characterization of the desynchronization transition,and more details on the effective input.[44] Julia code to interactively explore the chimera dy-namics is available at https://github.com/y-z-zhang/chimera_mechanism .[45] J. D. Hart, D. C. Schmadel, T. E. Murphy, and R. Roy,Experiments with arbitrary networks in time-multiplexeddelay systems, Chaos , 121103 (2017).[46] L. M. Pecora and T. L. Carroll, Master stability functionsfor synchronized coupled systems, Phys. Rev. Lett. ,2109 (1998).[47] A. Zakharova, M. Kapeller, and E. Sch¨oll, Chimeradeath: Symmetry breaking in dynamical networks, Phys.Rev. Lett. , 154101 (2014).[48] I. Omelchenko, Y. Maistrenko, P. H¨ovel, and E. Sch¨oll,Loss of coherence in dynamical networks: Spatial chaosand chimera states, Phys. Rev. Lett. , 234102 (2011).[49] I. Omelchenko, B. Riemenschneider, P. H¨ovel, Y. Maistrenko, and E. Sch¨oll, Transition from spa-tial coherence to incoherence in coupled chaotic systems,Phys. Rev. E , 026212 (2012).[50] P. Jiruska, M. De Curtis, J. G. Jefferys, C. A. Schevon,S. J. Schiff, and K. Schindler, Synchronization and desyn-chronization in epilepsy: Controversies and hypotheses,J. Physiol. , 787 (2013).[51] A. Rothkegel and K. Lehnertz, Irregular macroscopic dy-namics due to chimera states in small-world networksof pulse-coupled oscillators, New J. Phys. , 055006(2014).[52] R. G. Andrzejak, C. Rummel, F. Mormann, andK. Schindler, All together now: Analogies betweenchimera state collapses and epileptic seizures, Sci. Rep. , 23000 (2016).[53] T. Chouzouris, I. Omelchenko, A. Zakharova, J. Hlinka,P. Jiruska, and E. Sch¨oll, Chimera states in brain net-works: Empirical neural vs. modular fractal connectivity,Chaos , 045112 (2018).[54] C. Lainscsek, N. Rungratsameetaweemana, S. S. Cash,and T. J. Sejnowski, Cortical chimera states predictepileptic seizures, Chaos , 121106 (2019).[55] A. A. Asadi-Pooya, A. Sharan, M. Nei, and M. R. Sper-ling, Corpus callosotomy, Epilepsy Behav. , 271 (2008).[56] T. I. Netoff and S. J. Schiff, Decreased neuronal synchro-nization during experimental seizures, J. Neurosci. ,7297 (2002).[57] F. Mormann, T. Kreuz, R. G. Andrzejak, P. David,K. Lehnertz, and C. E. Elger, Epileptic seizures are pre-ceded by a decrease in synchronization, Epilepsy Res. ,173 (2003).[58] K. Schindler, H. Leung, C. E. Elger, and K. Lehnertz,Assessing seizure dynamics by analysing the correlationstructure of multichannel intracranial eeg, Brain , 65(2007).[59] S. A. Bogomolov, A. V. Slepnev, G. I. Strelkova,E. Sch¨oll, and V. S. Anishchenko, Mechanisms of appear-ance of amplitude and phase chimera states in ensemblesof nonlocally coupled chaotic systems, Comm. NonlinearSci. Numer. Simulat. , 25 (2017).[60] C. Zhou and J. Kurths, Noise-induced phase synchro-nization and synchronization transitions in chaotic oscil-lators, Phys. Rev. Lett. , 230602 (2002).[61] J.-N. Teramae and D. Tanaka, Robustness of the noise-induced phase synchronization in a general class of limitcycle oscillators, Phys. Rev. Lett. , 204103 (2004).[62] H. Nakao, K. Arai, and Y. Kawamura, Noise-inducedsynchronization and clustering in ensembles of uncou-pled limit-cycle oscillators, Phys. Rev. Lett. , 184101(2007).[63] E. Ullner, J. Buceta, A. D´ıez-Noguera, and J. Garc´ıa-Ojalvo, Noise-induced coherence in multicellular circa-dian clocks, Biophys. J. , 3573 (2009). Supplemental Material
Mechanism for Strong ChimerasYuanzhao Zhang and Adilson E. Motter
CONTENTS
S1. Strong chimeras in coupled logistic maps 1S2. Strong chimeras in continuous-time dynamical systems 2S3. Robustness against oscillator heterogeneity 3S4. Nature of the Desynchronization Transition 4S5. Characteristics of the effective input and its impact on stability 5
S1. STRONG CHIMERAS IN COUPLED LOGISTIC MAPS
Here, we show that the same mechanism described in the main text gives rise to strong chimeras in networksand oscillator models other than the ones investigated in Figs. 2 and 5. In Fig. S1, we consider coupled logistic mapsand show their transition from coherence to incoherence through chimera states. The system can be described by thedynamical equation x t +1 i = n rx ti (1 − x ti ) − K N X j =1 L ij x tj (1 − x tj ) o mod 1 , i = 1 , . . . , N, (S1)where L = { L ij } is the Laplacian matrix representing a two-cluster network with nearest-neighbor intraclustercoupling and all-to-all intercluster coupling [Fig. S1(a)]. FIG. S1. Analog of Fig. 2 in the main text for a two-cluster network of logistic maps. The value of c is again set to 0 . q quasi-statically along the dashed line in (b)( r = 2 . × − t , K = 1 . − t for 10 iterations). The bottom panel in (c) show the time-averaged synchronizationerrors in the two clusters for fixed values of q . Each data point is averaged over 10 iterations, and the final state from theprevious data point is used as the initial condition for the next data point (with increased q ). The orange points are shiftedslightly to not completely overlap with the blue points before q c . a r X i v : . [ n li n . AO ] J a n S2. STRONG CHIMERAS IN CONTINUOUS-TIME DYNAMICAL SYSTEMS
We further demonstrate that incoherence-stabilized coherence is not limited to discrete maps and it can also giverise to strong chimeras in continuous-time dynamical systems. For this purpose, we consider the multilayer networkdepicted in Fig. S2(a), where the intracluster and intercluster coupling are mediated by different types of interactions.Each layer consists of six identical Lorenz oscillators interacting through the coupling function H = (0 , , z ) | . Inaddition, the two layers are all-to-all coupled through the coupling function H = (0 , , x ) | . The oscillators in thefirst layer are thus described by the equations˙ x (1) i = α ( y (1) i − x (1) i ) , ˙ y (1) i = x (1) i ( ρ − z (1) i ) − y (1) i , ˙ z (1) i = x (1) i y (1) i − βz (1) i + K ( z (1) i +1 + z (1) i − − z (1) i ) + cK X j =1 ( x (2) j − x (1) i ) , (S2)where the superscripts identify the layers and we set α = 10, β = 2, and c = 0 .
2, leaving the parameters ρ and K tobe varied. The oscillators in the second layer are described by similar equations. FIG. S2. Chimera states in continuous-time dynamical systems. (a) Multilayer network of Lorenz oscillators with differentintralayer and interlayer interactions, given by H and H , respectively. (b) Diagram in the ρ - K plane marking the regions forwhich the system exhibits coherence (cyan), chimeras (purple), and incoherence (red) according to our linear stability analysis.(c) State transitions as parameter q is varied quasi-statically along the dashed line in (b) ( ρ = 30+5 × − t , K = 2 for 10 timeunits). Incoherence-stabilized coherence gives rise to strong chimeras as the system transitions from coherence to incoherence.(d) Time-averaged synchronization errors in the two clusters for fixed values of q . Each data point is averaged over 10 timeunits, and the final state from the previous data point is used as the initial condition for the next data point for increasing q .The orange points are shifted slightly to not completely overlap with the blue points before q c . S3. ROBUSTNESS AGAINST OSCILLATOR HETEROGENEITY
Our results are robust to the presence of oscillator heterogeneity, which we demonstrate in Fig. S3 by introducingrandom mismatches with a standard deviation of 10 − (drawn from a Gaussian distribution) to the parameter β forthe same transitions studied in Fig. 2. FIG. S3. Analog of Fig. 2(c), with oscillator heterogeneity incorporated into the simulation to mimic a realistic aspect ofexperimental conditions.
S4. NATURE OF THE DESYNCHRONIZATION TRANSITION
Consider the two-cluster networks analyzed in the main text, where the clusters are identical and connected toeach other through all-to-all coupling of strength cK . The clusters, C and C , can be allowed to have arbitrary sizeand network structure. In such systems, as a bifurcation parameter is varied, global synchronization can lose stabilitythrough either intercluster desynchronization ( C and C go to different states) or intracluster desynchronization ( C and/or C desynchronize within themselves). In both cases, stability is determined by the master stability function e Λ( α, β ).For diffusively coupled oscillators, b λ = − cn for the intercluster case and b λ i = − λ i − cn for the intracluster case,where we used that µ = − cn and that λ i are the eigenvalues of the Laplacian matrix of the cluster for i = 2 , ..., n . Inthis analysis, Kµh ( s t )+ I ( t ) = 0 and thus Γ s = Γ , where Γ is the Lyapunov exponent of an isolated oscillator definedby f . Thus, the stability of the global synchronization state requires α = Kcn (intercluster) and α i = Kcn + Kλ i for i = 2 , . . . , n (intracluster) all to be in the negative region of e Λ( α i , β ).The stability conditions then take the form( − β + Kcn ) < e − for intercluster , (S3)( − β + Kcn + K Re λ i ) + ( K Im λ i ) < e − for intracluster , (S4)where Re λ i and Im λ i are the real and imaginary parts of the eigenvalue, respectively. Equation (S4) represents thecondition in Eq. (6), whereas Eq. (S3) corresponds to the perturbation mode parallel to the cluster synchronizationmanifold omitted in Eq. (6). Since c and Re λ i are nonnegative for the systems we consider, it follows that as K isincreased beyond the stability region the condition in Eq. (S4) is the first to be violated, and thus stability of globalsynchronization is first lost through intracluster desynchronization. Analogous description applies to the bifurcationparameter q in Fig. 2(b).The condition in Eq. (S3) is violated first when the stability boundary is reached by decreasing K , meaning thatfor small K stability can be lost through intercluster desynchronization if β > e − . Here we focus on intraclusterdesynchronization transitions as induced by any bifurcation parameter, given our focus on chimeras, but our resultsalso apply to transitions involving intercluster desynchronization as well as intercluster desynchronization followed byintracluster desynchronization.The only difference in the latter case is that, if intercluster desynchonization happens first and is followed byintracluster desynchronization (e.g., small K for chaotic oscillators) giving rise to a chimera, then the stability analysishas to be repeated twice since now s t changes due to nonvanishing Kµh ( s t )+ I ( t ) between the two desynchronizations.The analysis of the transition to the chimera is then to be performed in two steps by updating s t and thus Γ s . S5. CHARACTERISTICS OF THE EFFECTIVE INPUT AND ITS IMPACT ON STABILITY
For a more systematic understanding of the interaction between the two clusters, we vary the interclustercoupling strength by varying c while the other parameters are kept fixed. We first calculate the mean ν and standarddeviation σ of the effective input [Fig. S4(a)]. As c is increased from zero, ν steadily decreases while σ increases.Figure S4(b) shows the resulting LTLE for synchronization in the coherent cluster when this cluster is subject toGaussian white noise input or effective input from the incoherent cluster. For each value of c , the mean and standarddeviation of the noise input are set respectively to the values of ν = ν ( c ) and σ = σ ( c ) from Fig. S4(a). Both formsof input lead to the stabilization of synchronization for intermediate intercluster coupling strength, which is to becontrasted with the unstable state obtained if the other cluster is set to the same coherent state [shown in Fig. S4(b)as a reference]. This suggests that the stabilization effect of a common driving signal is relatively insensitive to theform of the signal, as long as the mean and standard deviation of the signal are suitable. FIG. S4. Dependence of effective input and synchronization stability on intercluster coupling strength. (a) Mean ν and standarddeviation σ of the effective input from the incoherent cluster as the coupling parameter c is varied. (b) LTLE of the coherentcluster as a function of c for noise input (purple), effective input based on direct simulation of the incoherent cluster (green),and no input (red). The system is the same as in Figs. 2–4 for β = 1 . K = 1 ..