Between Synchrony and Turbulence: Intricate Hierarchies of Coexistence Patterns
AA hierarchy of coexistence patterns mediating between low- andhigh-dimensional dynamics in highly symmetric systems
Sindre W. Haugland and Katharina Krischer a) Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1,D-85748 Garching, Germany (Dated: 27 January 2021)
Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony andincoherence. The 2002 discovery of so-called chimera states, states of coexisting synchronized and unsyn-chronized oscillators, provided a possible link between the two and definitely showed that different parts ofthe same ensemble can sustain qualitatively different forms of motion. Here, we demonstrate that globallycoupled identical oscillators can express a range of coexistence patterns more comprehensive than chimeras.A hierarchy of such states evolves from the fully synchronized solution in a series of cluster-splittings. At thefar end of this hierarchy, the states further collide with their own mirror-images in phase space – renderingthe motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistencepatterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asyn-chronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, arefound to be just one step in this transition from low- to high-dimensional dynamics.One of the big problems in physics is how high-dimensional disorder in space and time may emerge froma spatially ordered, in the simplest case uniform, statewith low-dimensional dynamics . Exploring differentpaths from order to spatiotemporal disorder and theiruniversal character is central for a deeper understand-ing of complex emergent behaviour such as spatiotempo-ral chaos in reaction-diffusion systems or turbulence inhydrodynamic flows .Ensembles of coupled oscillators are one class of ap-parently simple dynamical systems that yet may adoptstates ranging from full synchrony to complete incoher-ence, and which has provided insights in virtually anydiscipline, ranging from the natural sciences to sociol-ogy . During the last two decades, a kind of hybridphenomenon, in which synchronized and incoherent os-cillators coexist in an ensemble of identical oscillators ,coined a chimera state , has received considerable atten-tion (see Reviews and the references therein), notleast since it can be considered a “natural link betweencoherence and incoherence” . In an earlier study em-ploying globally coupled logistic maps , four differentclasses of behaviour were found, including a large vari-ety of partially ordered states, some of which were laterclassified as chimeras . Yet, the bifurcation structurebetween the different classes was not resolved.In this article, we study the bifurcations from syn-chrony, via clustered and partially clustered states tofull incoherence in a system of globally coupled oscilla-tors with nonlinear coupling, with simulations and bi-furcation analysis for an increasing number of oscilla-tors. Here, chimera states are just one of a multitudeof coexistence patterns, all consisting of clusters, that is,internally synchronized groups of oscillators, of widely a) Electronic mail: [email protected]. different sizes and dynamics, and possibly including oneor several single oscillators. The path towards completeincoherence begins with a symmetry-breaking cascadeof cluster-splitting period-doubling bifurcations, whereinthe currently smallest cluster is repeatedly split into two,leading to hierarchical clustering. Due to the high sym-metry of the system, each symmetry-breaking producesmany equivalent mirror-image variants of each outcomestate, multiplying the number of attractors and leadingto an ever more crowded phase space . At some point,each variant collides with some of its mirror-images, cre-ating larger attractors with higher symmetry. Usually,this blows up some of the clusters, the resulting singleoscillators henceforth moving similarly on average. Asuccession of such symmetry-increasing bifurcations de-stroys first the smallest clusters, and then the larger ones,partially mirroring the former cluster-splitting cascadeand ultimately creating a completely incoherent state. Achimera state, consisting of one synchronized cluster andotherwise only single, incoherent oscillators is often thesecond to last state of the sequence.The model we employ is an ensemble of N Stuart-Landau oscillators W k ∈ C , k = 1 , . . . , N , with nonlinearglobal coupling : d W k d t = W k − (1 + ic ) | W k | W k − (1 + iν ) h W i + (1 + ic ) h| W | W i , (1)where c and ν are real parameters and h . . . i =1 /N P Nk =1 . . . denotes ensemble averages. The Stuart-Landau oscillator itself is a generic model for a sys-tem close to a Hopf bifurcation, that is, to the onsetof self-sustained oscillations . Networks of such oscilla-tors have previously been found to exhibit a wide rangeof dynamics, many of them occurring for linear globalcoupling . The nonlinear global coupling in equa-tion (1) stands out by featuring two qualitatively different a r X i v : . [ n li n . AO ] J a n chimera states, each of them deduced to somehow emergefrom a corresponding type of two-cluster solution . Be-cause the oscillators are identical and the coupling isglobal, the system is S N -equivariant: If W ( t ) ∈ C N isa solution, then so is γ W ( t ) ∀ γ ∈ S N , where S N is the symmetric group of all permutations of the N oscilla-tors . Further, the average h W i is confined to simpleharmonic motion with frequency ν , as shown by takingthe ensemble average of the whole equation: h d W k d t i = dd t h W i = − iν h W i ⇒ h W i = η e − iνt , (2)where η ∈ R is an additional parameter, implicitly setby choosing the initial condition. This constraint alsoimplies that for a Poincaré map defined by samplingthe system with frequency ν , the average of the N com-ponents of the map will always be constant. Thus thenonlinear constraint in the time-continuous equation (1)becomes a linear constraint in the time-discrete map.The fully synchronized solution W k = η e − iνt ∀ k al-ways exists and is stable for sufficiently large values of η . It loses stability in either an equivariant pitchfork bi-furcation, producing separate clusters that continue toorbit the origin with frequency ν at different fixed ampli-tudes, or an equivariant Hopf bifurcation to a T torus,producing separate modulated-amplitude clusters thathenceforth oscillate with two superposed frequencies ν and ω H .We will focus on the latter and the dynamicsarising from these.The Hopf bifurcation occurs at η H = 1 / √ for suit-able values of c and ν . For ν = 0 . , which we keep fixedthroughout, it does for c < − . . At η H , several dif-ferently balanced two-cluster solutions emerge from thesynchronized solution, some as stable and some as unsta-ble, depending on the value of c . The balanced N/ − N/ solution, with an equal number of oscillators in each clus-ter, is shown in Fig. 1 a. The dashed circle marks the en-forced path of the ensemble average h W i = η e − iνt , whichthe two clusters orbit on opposite sides as it circles theorigin.Because h W i is independent of the individual oscilla-tor dynamics, the value of any oscillator in the frame ofreference of the ensemble average is always given by thesimple transformation W k = η e − iνt (1 + w k ) ⇒ w k = W k η − e iνt − , (3)where w k is the value of W k in the co-rotating frame.There, the N/ − N/ solution from Fig. 1 a is simply pe-riodic with frequency ω H and looks as in Fig. 1 b. Anunbalanced modulated-amplitude cluster solution withcluster sizes N = 3 N/ and N = N/ appears as inFig. 1 c. The average of all oscillators in the co-rotatingframe of h W i is of course always zero. Notably, the globalcoupling ensures that all solutions for an ensemble size N are also solutions for N = nN, n ∈ N , with every clus-ter scaled up by a factor of n . For solutions that containonly clusters N i ≥ , the stability properties will also bethe same for different n . FIG. 1.
Two-cluster solutions and their bifurcations.a , Trajectory of an N/ − N/ modulated-amplitude clus-ter solution for c = − . and η = 0 . . b , The solutionin a when viewed in a frame co-rotating with the ensembleaverage h W i = η e − iνt . Here, the two clusters follow thesame trajectory. c , Unbalanced N/ − N/ two-cluster so-lution in the rotating frame at c = − . and η = 0 . . d , N/ − N/ − N/ three-cluster solution at c = − . ,emerging from the solution in a , b in a period-doubling bifur-cation. e , Bifurcations destabilizing the two-cluster solutionas a function of c and the relative size of the larger cluster N /N for η = 0 . . The blue line denotes a period-doubling(PD) that splits the smaller cluster and the green line a sub-critical pitchfork (PF) that blows it up. The vertical blackline marks the c -incremented simulation in Fig. 5. If we start at a point in parameter space where the N/ − N/ solution is stable, keep η < η H sufficientlyclose to the Hopf bifurcation and gradually increase c ,one of the two clusters will break up into two smallerclusters. A possible outcome is shown in Fig. 1 d. Thetrajectory of the two new clusters is no longer simplyperiodic, but period-2, with a small and a large loop.The N/ − N/ solution has thus become unstable ina symmetry-breaking period-doubling bifurcation, giv-ing rise to a stable N/ − N/ − N/ three-cluster so- FIG. 2.
Cluster solutions and bifurcations for N = 16 . Sizes of filled circles mirror sizes of clusters. a , − − solution inthe co-rotating frame of h W i for c = − . and η = 0 . . b , − − − solution for c = − . and η = 0 . . c , Bifurcationdiagram of − -derived solutions with PD=period-doubling, PF=pitchfork. Dashed lines mark bifurcations of the − − solution ( a ), solid lines those of the − − (Fig. 1 d) and the solutions emerging from it. Labeled crosses mark parametervalues of depicted complex-plane portraits. The black line marks the c -incremented simulation in Fig. 3 a,b. d , − − − solution for c = − . and η = 0 . . e , f , − − − − − solution for c = − . and η = 0 . , including time series of realpart of each cluster and single oscillator. lution. This bifurcation also destabilizes less balancedtwo-cluster solutions, for which the smaller of the twoclusters is split. Its position in parameter space dependson the relative sizes of the clusters, as shown in Fig. 1 e.For very unbalanced solutions, the smallest cluster is ex-plosively destroyed in a subcritical pitchfork bifurcation. HIERARCHICAL CLUSTERING THROUGH PERVASIVESTEPWISE SYMMETRY BREAKING
The first period-doubling bifurcation curve is followedby a mesh of additional cluster-splitting bifurcationscurves, creating a hierarchy of successively less symmetricmulti-cluster solutions with various periodicities. Eachbifurcation involves the breakup of either one clusteror two similarly behaving clusters and produces several qualitatively different solutions, differing by how the os-cillators of the splitting cluster(s) distribute. (For exam-ple, the − solution for N = 8 can split into either − − , − − , − − − , − − − or − − − .)However, all these solutions will usually not be co-stable.Fig. 2 c shows the stability boundaries of several solu-tions for N = 16 . The − solution is stable in the upperleft. When increasing c for η > . , it gives rise tostable − − and − − solutions (Figs. 1 d and 2 a,respectively) at the leftmost blue period-doubling line.The − − solution is stable within the two dashed lines.Below the dashed green line, it in turn produces a stable − − − and unstable − − − and − − − solutions.At the dashed blue line, the − − solution undergoesanother period-doubling cluster split to an − − − period-4 solution.The solid lines all affect the − − solution and its FIG. 3.
Cluster-splitting cascades for different N . a , Maxima of Re( w k ) in an N = 16 c -incremented simulationat η = 0 . . Labels on the figure mark the clusters reaching the different maxima as the solution changes from − − via − − − to − − (4 × . Labels on the abscissa mark occurring bifurcations. The additional smallest yellow maximumappearing at c ≈ − . is caused by the continuous deformation of the oscillator trajectories and not by a bifurcation. At c ≈ − . , the torus bifurcation to three-frequency dynamics manifests itself in a distinct broadening of the formerly discretemaxima. b , Schematic portrayal of the period-doubling cluster splittings in the simulation in a . c , d , Like b for c -incrementedsimulations at η = 0 . with N = 32 and N = 256 , respectively, based on quantitative results in Supplementary Figures 1-3. descendants. At the solid green line from c ≈ − . to c ≈ − . in the lower left, it produces stable − − − and − − − (Fig. 2 b) solutions, as well as unstable − − − and − − − solutions. Like the dashedgreen line, this is an equivariant pitchfork bifurcation,splitting clusters, but not altering the overall periodicityof the ensemble. Below these lines, the above-mentionedfour-cluster solutions also emerge directly from the − solution at the leftmost period-doubling line.At the solid blue line directly to the right of the dashedblue one, the − − solution undergoes a period-doublingbifurcation analogous to that of the − − solution,producing a stable − − − (Fig. 2 d) and an unstable − − − period-4 solution. The former becomes unstable atthe bottom diagonal green pitchfork line at c ≈ − . ,where it produces a stable − − − − solution. Thissolution also emerges directly from the − − − solutionin a period-doubling bifurcation not shown.At the rightmost blue period-doubling line (see inset),the − − − solution produces an unstable − − − − and a stable − − − − − period-8 solution (Fig. 2e-f). At the red line in Fig. 2 c, the − − − − − solution undergoes a torus bifurcation, whereby a thirdfrequency is added to the dynamics, while all clusters stayintact. The resultant three-frequency motion is resistantto the addition of small random numbers over a nonzero c interval, which is notable in light of a prior proofthat quasiperiodic dynamics with three or more frequen-cies are in general structurally unstable . Such stablequasi-periodic motion on T has also been observed inStuart-Landau oscillators with linear global coupling and could be caused be due to the rotational invarianceof the differential equations.If we initialize the − − solution at c = − . and η = 0 . and slowly increase c along the horizontal blackline in Fig. 2 c, the maxima of Re( w k ) for k = 1 , . . . , develop as in Fig. 3 a: Initially, there are one maximumof the oscillators in the cluster of eight (blue) and twoshared maxima of the two clusters of four (red). Whenone of these clusters splits up into two clusters of two at PD , their maxima appear as four distinct yellow lines.In the next period-doubling cluster split, these lines splitup into eight.From the fully synchronized solution to the − − (4 × solution, four discrete steps of symmetry breaking havetaken place. The three last of these steps are shownschematically in Fig. 3 b. Similar stepwise symmetry-breaking is observed both for larger N and when thesmallest cluster does not break up into equal-sized parts(Fig. 3 c). The larger N is, the more steps occur, atever closer parameter values, and for N = 256 , as manyas seven steps can be observed (see Fig. 3 d). The two-cluster solutions thus give rise to a cluster-splitting cas-cade, producing a multitude of coexisting multi-clusterstates and, most notably, hierarchical clustering. SYMMETRY-INCREASING BIFURCATION ANDPRECISION-DEPENDENT CLUSTERING
At the end of a cascade of cluster-splitting period-doubling bifurcations, a torus bifurcation usually occurs.The resultant T motion is usually stable for a nonzeroparameter interval, before being superseded by less regu-lar dynamics in a symmetry-increasing bifurcation : Be-cause equation (1) is S N -equivariant, any solution re-mains a solution when any of the oscillators are inter-changed, and each solution (except the fully synchronizedone) exists in the form of several distinct symmetric vari-ants in phase space. (For example, if we interchangedan oscillator from the blue cluster in Fig. 1 a with onefrom the red, the outcome would be such a different, butequivalent variant.)If two or more variants grow to touch each other, theymerge to become a single instance of a new solution, ofwhich there are fewer distinct mirror-image variants intotal. The attractor on which the new solution lives iscorrespondingly more symmetric than the attractors ofthe colliding variants. One symmetry-increasing bifurca-tion can in general be followed by another, further in-creasing the attractor symmetry. In the N = 16 case inFig. 3 a, the first symmetry-increasing bifurcation onlydisrupts the former cyclic order of the four single os-cillators, inherited from the solution in Fig. 2 e-f (i.e.,that the purple oscillator trails the yellow one, whichtrails the pink, and so on). In other cases, some of theintact clusters of a colliding variant contain oscillatorsthat are in a different cluster in some of the other col-liding variants. Then, the symmetry-increasing bifurca-tion destroys these clusters. (Because the colliding vari-ants have identical properties, they must namely all betreated equally by the collision, and if two oscillators thatare clustered in only some of these variants were to re-main permanently together, this would not be the case.)Hence, as the attractor symmetry is increased, the num-ber of single oscillators, in general, grows, in a sense alsodecreasing the overall order of the ensemble.Past the N = 32 cluster-splitting cascade in Fig. 3 c, the long-term cluster-size distribution at some point be-comes − − (7 × . A time series of the resulting solutionis shown in Fig. 4 a: Here, the seven single oscillators inyellow move similarly to the clusters of four, two and onein the former − − − − solution, being close todeep minima when the red cluster of nine is at a shallowminimum and vice versa. They also repeatedly congre-gate into temporary agglomerations of four, three andtwo oscillators, respectively, as illustrated in Fig. 4 b-d.Here, the cross-correlation between all oscillator trajec-tories is calculated every time steps. Two oscillatorsare said to be in the same cluster if their cross-correlation FIG. 4.
Partially clustered − − (7 × solutionpast the cascade in Fig. 3 c. a , Time series of Re( w k ) for c = − . and η = 0 . . The single oscillators (yellow)reach a deep minimum approximately when the cluster of nine(red) reaches a shallow minimum and the cluster of a max-imum. b-d , Clusters sizes detected when two oscillators aresaid to be clustered if their cross-correlation over an intervalof time units is > − ε for ε = 10 − , − and − , re-spectively. The large red and blue dots mark two exemplaryloose oscillator conglomerations at different sensitivities ε . is greater than − ε for ε = 10 − (b), ε = 10 − (c) or ε = 10 − (d). Sometimes, a temporary cluster of threebecomes a cluster of four for larger values of ε , such asthe blue cluster at t = 7 · and the red cluster at t = 1 . · . This probably marks less close approachesto the − − − − remains, when three out of four con-gregating oscillators are more correlated than the fourth.If c is sufficiently increased, the ensemble will oftenjump to an entirely different solution, and beyond thestate in Fig. 4, it jumps to the − − -derived branch.However, the end result can also be the destruction of allpermanent clusters and the motion of only single oscilla-tors on a fully symmetric chaotic attractor. See Supple-mentary Figures 4-6.Dynamics like those in Fig. 4 have previously been ob-served in globally coupled logistic maps when phase spacebecomes so full of mirror-image attractors that they in-evitably intrude upon each other . The outcome is aform of chaotic itinerancy , wherein the system mean-ders between the attractor ruins of previous attractors,each of them relatively low-dimensional, but connectedby higher-dimensional transitional motion .Also found in globally coupled maps is precision-dependent clustering , wherein maps that are unclusteredwhen distinguished with high precision appear to repeat-edly merge into the ever thicker branches of a clusteringtree when the precision is decreased . In our ensemble,this occurs as a consequence of the symmetry-breakingperiod-doubling cascade. For example, past the N = 256 cascade in Fig. 3 (at c ≈ − . ), we encounter a − − − (31 × itinerant solution that for small ε ≤ − is found to have an additional cluster of usually , sometimes or oscillators, while the remainingoscillators repeatedly form ephemeral smaller clusters ofstrongly fluctuating sizes. For ε = 10 − , a cluster of size is also sometimes detected, and for ε = 10 − the sizesare always − − − − , − − − − or − − − − . For ε = 10 − , they are − − throughout, and for ε = 10 − , − − . The same pat-tern to some extent already applies in the quasiperiodicdomain of Fig. 3 b-d, where clusters are most stronglycorrelated with those other clusters from which they mostrecently split. EMERGENCE OF A CHIMERA STATE
In our context, a chimera state is a N − (( N − N ) × solution. The modulated-amplitude chimeras previouslyfound in equation (1) have significantly more synchro-nized ( N ) than unsynchronized ( N − N ) oscillators .This suggests they have not evolved from balanced two-cluster solutions. Yet, our above results can be usedto explain how they are created. If we e.g. initializean N = 20 ensemble as a − solution (Fig. 5 a) for c = − . and η = 0 . , Fig. 1 e tells us it will undergoa cluster-splitting period-doubling bifurcation if c is in-creased. The resulting − − period-2 solution is shown in Fig. 5 b. Further up in c , the cluster of two is splitinto single oscillators (Fig. 5 c). Then, a torus bifurcationsmears the previously closed trajectories into continuousbands (Fig. 5 d).Finally, the current − − − variant collides withnine others in a symmetry-increasing bifurcation. Thisdestroys the cluster of three, resulting in a − (5 × chimera state (Fig. 5 e). Note how the three oscillatorsthat are temporarily close to each other in Fig. 5 e (red,yellow, grey, in the lower left) are not all the same threethat were clustered in Figs. 5 b-d (red, purple, grey).The ensemble is currently close to the ruin of a differ-ent − − − solution variant, and the chimera state isthus also an example of chaotic itinerancy. For N = 200 ,the transition from a − to a − (50 × solution pro-ceeds along a much more involved, but essentially similarpath. See Supplementary Figure 7. CONCLUDING REMARKS
Above, we have shown how a system of identical unitssuccessively splits up into ever more complex coexistencepatterns. The high symmetry of the equations is prob-ably central to the bifurcation scenario observed, as S N not only has many subgroups, but most of these sub-groups have many subgroups as well, and so on. Thisintricate subgroup structure is mirrored in the hierarchyof successively less symmetric quasiperiodic solutions.Moreover, because S N is much larger than those of itssubgroups corresponding to the more intricate solutions,there are many mirror-image variants of the latter, caus-ing the symmetry-increasing bifurcations and the itiner-ant coexistence patterns that these produce.In globally coupled maps, multi-cluster states,precision-dependent clustering, chimeras and chaoticitinerancy have all been previously reported, but with-out an overall explanation of how they are bifurcation-theoretically created and connected to each other .This suggests that the bifurcation scenario uncoveredhere is universal to S N -equivariant ensembles and pos-sibly to other highly symmetric systems.To trigger the observed symmetry-increasing bifurca-tion, it seems as if a certain dimensionality of the dynam-ics is needed. In our system, this is provided by the torusbifurcations, in the globally coupled maps by a period-doubling cascade to chaos. For Stuart-Landau oscillatorswith linear global coupling, a symmetry-increasing bifur-cation has also been found to succeed a period-doublingcascade in an N = 4 system . To test whether hierar-chical clustering and symmetry-increasing events mediatebetween synchrony and incoherence there as well, is anexciting task for future studies. FIG. 5.
Steps from − two-cluster solution to chimera state. N = 20 , ν = 0 . and η = 0 . , proceeding along theblack line in Fig. 1 e. a , − solution for c = − . . b , − − period-2 solution for c = − . . c , − − − period-4solution for c = − . . The two largest loops of the two single oscillators are almost equal. d , − − − three-frequencysolution for c = − . . e , − (5 × chimera at c = − . , after collision of mirror-image attractors. REFERENCES Argyris, J. H., Faust, G., Haase, M. & Friedrich, R.
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METHODS
The differential equations (1) were solved numericallyusing the Python programming language (version 2.7and later 3.8) and the implicit Adams method of the scipy.integrate.ode class of the SciPy library (ver-sion 1.6) with a time step of ∆ t = 0 . . The data wereheld and processed in the form of NumPy (version 1.19)arrays with complex-valued floating-point elements andvisualized using the Matplotlib library and graphics en-vironment (version 3.3) . The numerical results wereevaluated using custom-built functions drawing on theresources of these standard Python libraries, written byS.W.H. Simulations were carried out in the non-rotatingframe of equation (1), and results in the co-rotating frameof h W i were visualized applying equation (3) to the data after simulations had been carried out. When not other-wise stated, initial conditions of numerical solutions wererandom numbers on the real line, fulfilling the global con-straint that h W i = η e − iνt . This choice was inspired byearlier work .Figs. 1 e and 2 c were created using the dynamical-systems continuation software Auto07p to continuesolutions in parameter space. As Auto can only continuefixed-point and limit-cycle solutions, equation (1) had tobe formulated in the co-rotating frame of the ensembleaverage in order to carry out these continuations, yielding d a k d t = a k − νb k − η [ A k − c B k ]+ 1 N η X j [ A j − c B j ] , d b k d t = b k + νa k − η [ B k + c A k ]+ 1 N η X j [ B j + c A j ] , (4)where w k = a k + ib k with a k , b k ∈ R , k = 1 , . . . , N , and A k = 3 a k + 3 a k + a k + a k b k + b k ,B k = b k + 2 a k b k + a k b k + b k . (5)To obtain Fig. 2 c, the relevant N = 16 quasiperiodic so-lutions where first generated using Python simulations.The output data were transferred to the rotating frame,and a time series corresponding to one full period in thatframe was used as input for a c or η one-parameter con-tinuation of each periodic solution, in order to detect the location of the depicted bifurcations. Then, the detectedbifurcations were two-parameter continued in c and η toobtain the depicted bifurcation lines.To obtain Fig. 1 e, equation (4) was reduced to a two-cluster model by setting a k = a c1 and b k = b c1 for all k = 1 , . . . , N , where w c1 = a c1 + ib c1 is the value of thefirst cluster. All other oscillators k = N +1 , . . . , N arein the other cluster w c2 = a c2 + ib c2 . This yields thefollowing equation for the first cluster d a c1 d t = a c1 − νb c1 + η (1 − N N ) [ A c2 − A c1 − c ( B c2 − B c1 )] , d b c1 d t = b c1 + νa c1 + η (1 − N N ) [ c ( A c2 − A c1 ) + B c2 − B c1 ] , (6)with A c1 and B c1 analogous to equation (5), while w c2 = N N − N w c1 , because of the constraint that P k w k =0 . Thus, the reduced two-cluster model is only two-dimensional. The relative size of the first cluster, N /N ,becomes an effective fourth parameter, in addition to c , ν and η .Whereas (6) describes the motion of two clusters ofsizes N and N = N − N , respectively, it says noth-ing about intra-cluster stability and cannot model thebreakup of either cluster. To be able to evaluate theinternal stability of the clusters, we followed Ku et al. and added two effectively infinitesimal extra oscillators tothe model, which only feel the presence of the two macro-scopic clusters, but themselves neither affect the move-ment of each other, nor that of the macroscopic clusters.Their motion is given by d p , d t = p , − νq , − η [ P , − c Q , ]+ η (cid:20) N N ( A c1 − c B c1 )+ N − N N ( A c2 − c B c2 ) (cid:21) , d p , d t = q , + νp , − η [ Q , + c P , ]+ η (cid:20) N N ( B c1 + c A c1 )+ N − N N ( B c2 + c A c2 ) (cid:21) , (7)where P , and Q , denote composite expressions for thefirst and second infinitesimal oscillator, of the same formas A c1 , c2 and B c1 , c2 : P , := 3 p , + 3( p , ) + ( p , ) + p , ( q , ) + ( q , ) ,Q , := q , + 2 p , q , + ( p , ) q , + ( q , ) . (8)In the initial state of the continuation, one of these twoinfinitesimal oscillators is set to follow the same periodicorbit as either of the two clusters. If any bifurcations aredetected to make either infinitesimal oscillator leave themacroscopic cluster it started at, this means that clusterhas become unstable.Fig. 3 a was created by initializing the N = 16 en-semble in the − − configuration at c = − . andincrementing c by ∆ c = 10 − every ∆ T = 4 · timesteps until c = − . for ν = 0 . and η = 0 . . At thebeginning of each c step, random numbers ≤ − wereadded to the real and imaginary part of each oscillator toprovoke the breakup of potential unstable clusters. Max-ima of Re( w k ) were plotted for the last time stepsof simulation at each c steps.The schematic in Fig. 3 b was drawn based on au-tomatically detected cluster sizes at each c step inthe aforementioned c -incremented simulation. Thesecluster sizes were determined by calculating the pair-wise cross-correlations of the trajectories of all oscilla-tors over the last time steps at each c step, re-spectively. If the cross-correlation differed from 1 byless than (cid:15) = 10 − , the two oscillators were deemedto belong to the same cluster. To calculate the cross-correlations and obtain the clusters, we used SciPy’sbuilt-in scipy.cluster.hierarchy.linkage function.The schematic in Fig. 3 c was determined based onan analogous c -incremented simulations for N = 32 , ν = 0 . and η = 0 . , initialized in the − configura-tion at c = − . . Here, ∆ c = 2 · − and ∆ T = 10 .The simulation was performed until c = − . , pro-ducing the result in Supplementary Figure 1 a. Clusterswere calculated as in the N = 16 case based on the last time steps of simulation at each c step, producingSupplementary Figure 2 b.The schematic in Fig. 3 d was determined based ontwo analogous c -incremented simulations for N = 256 , ν = 0 . and η = 0 . . In the first of these, the en-semble was initialized in the − − configura-tion at c = − . , from where c was incrementedby ∆ c = 10 − every ∆ T = 2 · time steps until c = − . , producing the result in SupplementaryFigure 2 a. In a second c -incremented simulation for N = 256 , the ensemble was initialized at c = − . in the − − − − configuration found there inthe prior N = 256 simulation with ∆ c = 10 − . Fromthere on, c was incremented by ∆ c = 2 · − every ∆ T = 2 · time steps until c = − . , producingthe result in Supplementary Figure 3 a-b. For either sim-ulation, clusters were calculated based on the last time steps of simulation at each c step, producing Sup-plementary Figures 2 b and 3 b-c, respectively.Fig. 4 b-d were created by simulating the − − (7 × solution in Fig. 4 a for T = 10 time steps. Every time steps, the pairwise cross-correlation between alloscillators was calculated over an interval of timesteps, and if the cross-correlation of two oscillators wasfound to be greater than − ε for ε = 10 − (Fig. 4 b), ε = 10 − (Fig. 4 c) or ε = 10 − (Fig. 4 d), respectively,they were counted as being in the same cluster.The solutions in Fig. 5 were obtained by initializing the N = 20 ensemble in a − solution at c = − . , ν =0 . and η = 0 . , and incrementing c by ∆ c = 2 · − every ∆ T = 5000 time steps until c = − . . Supplemen-tary Figures 4 to 7 were created based on data obtainedanalogously to that in Figs. 3 and 5 with parameters asgiven in their respective captions. DATA AVAILABILITY
Data are available from the authors upon request.
ACKNOWLEDGMENTS
The authors thank Felix P. Kemeth and Maximil-ian Patzauer for fruitful discussions. Financial supportfrom the Studienstiftung des deutschen Volkes and theDeutsche Forschungsgemeinschaft, project KR1189/18“Chimera States and Beyond”, is gratefully acknowl-edged.
AUTHOR CONTRIBUTIONS
S.W.H. carried out the simulations and analysed thedata. Both authors discussed the results and wrote thepaper. K.K. supervised the project.
COMPETING INTERESTS
The authors declare no competing interests. upplementary Information: A hierarchy of coexistence patternsmediating between low- and high-dimensional dynamics in highlysymmetric systems
Sindre W. Haugland and Katharina Krischer a) Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1,D-85748 Garching, Germany (Dated: 27 January 2021) c -INCREMENTED SIMULATIONS BEHIND THESCHEMATICS IN FIG. 3 B-D Like the schematic in Fig. 3 b is based on the quan-titative simulation in Fig. 3 a, the schematic in Fig. 3 cis based on the quantitative simulation in Supplemen-tary Fig. 1, and the schematic in Fig. 3 d is based on thequantitative simulations in Supplementary Figs. 2 and 3.In Supplementary Fig. 1, the N = 32 ensemble is ini-tialized in a − configuration at c = − . , ν = 0 . and η = 0 . . This corresponds to the very left of thefigure, from where c was incremented by ∆ c = 2 · − every ∆ T = 10 time steps. The upper part of the figuredepicts how the maxima of the real part of each oscilla-tor evolve in the co-rotating frame of the ensemble aver-age h W i = η e − iνt . The lower part depicts the sizes ofclusters automatically detected at each c step. Furtherdetails are given in the figure caption.In Supplementary Fig. 2, the N = 256 ensemble isinitialized in a − − configuration at c = − . , ν = 0 . and η = 0 . . This corresponds to the very left ofthe figure, from where c was incremented by ∆ c = 10 − every ∆ T = 2 · time steps. Again, the upper part ofthe figure depicts how the maxima of the real part of eachoscillator evolve in the co-rotating frame of the ensembleaverage h W i = η e − iνt . The lower part depicts the sizes ofclusters automatically detected at each c step. Furtherdetails are given in the figure caption.Supplementary Fig. 3 depicts a continuation of the c -incremented simulation in Supplementary Fig. 2 with thefar smaller c step ∆ c = 2 · − . In the very left partof this figure, at c = − . , the ensemble was ini-tialized in the − − − − state found there inthe prior c -incremented simulation depicted in Supple-mentary Fig. 2. Further details are given in the figurecaption. SYMMETRY-BREAKING BIFURCATIONS TO FULLINCOHERENCE
For ν = 0 . and η = 0 . , c -incremented simulationslike those in Fig. 3 a and Supplementary Fig. 1 do not cre-ate stable states of fully incoherent single oscillators at a) Electronic mail: [email protected]. the end of the depicted bifurcation cascades. Instead, theensemble will at some point jump to some other cluster-solution that is co-stable with the last step of the simu-lated cascade. If for example the N = 16 c -incrementedsimulation in Fig. 3 a is continued further, the ensembleat some point is thrown onto a − − − four-clustersolution. This − − − is actually even co-stable with allsteps of the cascade from the − − state onward, as canbe deduced from the dashed line in Fig. 2 c, where the − − − solution is stable below the dashed green pitch-fork bifurcation line of the − − state. Analogously,the N = 32 c -incremented simulation in SupplementaryFig. 1, jumps to a − − − solution shortly to theright of the depicted c interval.For η = 0 . , the situation is different, as no N/ − N/ -derived four-cluster solutions like the − − − solu-tion are stable beyond the end of an equivalent cluster-splitting cascade. This is illustrated in SupplementaryFig. 4, where the ensemble, when c -incremented fromthe − state at c = − . , first assumes a − − solution and then goes through a sequence of bifurcationssimilar to that in Supplementary Fig. 1. As indicated bythe cluster sizes in the lower part of the figure, it at somepoint ends up in a − − (7 × configuration. From thereon, it jumps to a − − − solution. This solution is partof a the different somewhat longer-lasting cascade via the − − solution that is co-stable with the cascade viathe − − solution that the ensemble originally wentthrough here. Even further upward in c , no N/ − N/ -derived solutions are stable anymore, and the dynamicsconsequently become fully incoherent. (Whereas it mightappear from the lower part of the figure as if there stillexist various non-trivial clusters N i > for c ≈ − . ,in particular of sizes N = 2 , , these are actually onlyprecision-dependent approaches to ruins of former clus-ters, similar to those we observe in Fig. 4 of the mainarticle.)Before looking closer at those incoherent dynamics, wetrace an entirely different path that also lead to them.This path starts at a − − − − solution for whichthe maxima of Re( w k ) are shown in the leftmost partof Supplementary Fig. 5 a. The trajectories of the clus-ters and single oscillators of this solution in the rotatingframe of h W i are shown in Supplementary Fig. 6 a-b.At c ≈ − . , an equivariant torus bifurcation intro-duces a tertiary frequency to the dynamics and explodesthe cluster of into a ring of single oscillators. This is a r X i v : . [ n li n . AO ] J a n reflected in the broadening of the maxima in Supplemen-tary Fig. 5 a into continuous bands. The new solution isdepicted in Supplementary Fig. 6 c-d.After a sufficient further increase in c the differentvariants of the emergent − (30 × solution collide in asymmetry-increasing bifurcation, leaving only single os-cillators. (While again, Supplementary Fig. 5 b appearsto detect several clusters N i > , these, like in Supple-mentary Fig. 4 b, are also just cluster ruins.) Initially,however, the resultant fully incoherent attractor is stillco-stable with the cascades the ensemble goes throughin Supplementary Fig. 4, as evident from the fact thatthe extent of the c axis in the two figures is the same.This only changes shortly before we reach c = − . .At this value, the incoherent dynamics incorporate at-tractor ruins of both the formerly stable − (30 × inSupplementary Fig. 6¸-d and the − -derived solutionsin Supplementary Fig. 4. In Supplementary Fig. 6 e, theformer is mirrored by the large almost circular excursionsof single oscillators away from the origin. The latter ismirrored by the two more oval blue-green loops similarto those in the solutions in Fig. 2 of the main article. PATH FROM − SOLUTION TO CHIMERA STATE
In Fig. 5 of the main article, the bifurcations from a − two-cluster solution to a − (5 × chimera stateare traced exactly. For N = 200 , initializing a − two-cluster solution at the same parameter values as inthe N = 20 case and gradually increasing c provokes acluster-breaking cascade as well. This can be seen in Sup-plementary Fig. 7, which depicts two such c -incrementedsimulations with different ∆ c .In the more coarsely incremented simulation in the up-per half of the figure, the cluster of clearly splits intoa cluster of and a cluster of . The more finely incre-mented simulation in the lower half is initialized in theresultant − − configuration. A subsequent break-up of the cluster of into a cluster of and a clusterof occurs in both simulations. In the lower half of thefigure, we can also distinguish a break-up of the clusterof into a cluster of and a cluster of , as well as thelater collapse of the cluster of five.Further increases in c causes the ensemble to jumpbetween various parallel multi-cluster solution branchesthat all have in common that the large cluster of oscillators remains intact. Eventually, the last of thesemulti-cluster solutions undergoes a symmetry-increasingbifurcation, creating a − (50 × chimera state.For even greater values of c , the large cluster absorbssome of the single oscillators, increasing the imbalancebetween synchronized and single oscillators. Ultimately,the large cluster also collapses, leading to a state of fullincoherence. ab Supplementary Fig. Cluster-splitting cascade and ensuing bifurcations for N = 32 . a , All occurring maximaof the rotating-frame real parts Re( w k ) of all clusters and single oscillators against c as c is gradually increased at a rate of ∆ c = 2 · − every time steps for N = 32 , ν = 0 . and η = 0 . . Oscillators are colored by the clusters to which theybelong in the cluster-splitting cascade. Initially, there are two clusters of , reaching a single maximum shown in green. At c ≈ − . , one of these clusters splits up into a cluster of nine, shown in blue, and a cluster of seven, shown in purple, thatboth are period-2. At c ≈ − . , the cluster of seven splits up into a cluster of four (purple) and a cluster of three (yellow).At c ≈ − . , the cluster of three splits up into a cluster of two (yellow) and a single oscillator (red). When the cluster oftwo is destroyed, the two resulting single oscillators are shown in red and black. At higher c values, additional single oscillatorsretain the color of the cluster to which they belonged in the − − − − − solution. b , Cluster sizes at each value of c during the c -incremented simulation in a . Calculations are based on the cross correlations of trajectories (in the non-rotatingframe) over the last time steps of simulation at each c value and a threshold of ε = 10 − . See Methods section. ab Supplementary Fig. First cluster-splitting bifurcations for N = 256 . a , All occurring maxima of the real parts ofall clusters and single oscillators against c as c is gradually increased at a rate of ∆ c = 10 − every · time steps for N = 256 , ν = 0 . and η = 0 . . The simulation is initialized in the − − solution at c = − . . Oscillators arecolored by the clusters to which they belong in the final − − − − solution: In the leftmost part of the figure, themaxima of the cluster of are shown in green, the maxima of the two clusters of in purple. At c ≈ − . , one ofthese clusters splits up into a cluster of (blue) and a cluster of (red). At c ≈ − . , the cluster of splits up intoa cluster of (red) and a cluster of (yellow). b , Cluster sizes at each value of c during the c -incremented simulation in a . Calculations are based on the cross correlations of trajectories (in the non-rotating frame) over the last time steps ofsimulation at each c value and a threshold of ε = 10 − . See Methods section. The vertical axis scales logarithmically. abc Supplementary Fig. Continued cluster-splitting cascade and ensuing bifurcations for N = 256 . a , All occurringmaxima of the real parts of all clusters and single oscillators against c as c is gradually increased at a rate of ∆ c = 2 · − every · time steps for N = 256 , ν = 0 . and η = 0 . . The simulation is initialized in the − − − − solutionfound at c = − . in Supplementary Fig. 2. In the leftmost part of the figure, the maxima of the clusters of , , , and are shown in green, purple, blue, red and yellow, respectively. At c ≈ − . , the cluster of splits up into acluster of eight (yellow) and a cluster of seven (black). At c ≈ − . , the cluster of seven splits up into clusters of fourand three (in black and pink, respectively). At c ≈ − . , the cluster of three splits up into a cluster of two (pink) anda single oscillator (grey), manifest only in the appearance of additional grey dots and easier seen in c . At higher c values,additional single oscillators retain the color of the cluster to which they belonged in the − − − − − − − solution. b , Magnified view of the maxima . < max n [Re( w k )] < . in a . c , Cluster sizes at each value of c during the simulationin a,b . Calculations are based on the last time steps of simulation at each c value and a threshold of ε = 10 − . SeeMethods section. The vertical axis scales logarithmically. ab Supplementary Fig. Cluster-splitting cascade and ensuing bifurcations for N = 32 and η = 0 . , startingat − . a , All occurring maxima of the real parts Re( w k ) of all clusters and single oscillators against c as c is graduallyincreased at a rate of ∆ c = 2 · − every time steps. Oscillators are colored by the clusters to which they belong in thecluster-splitting cascade taking up most of the depicted range of c : Initially, there are two clusters of , reaching a singlemaximum shown in green. At c ≈ − . , one of these clusters splits up into a cluster of nine (blue) and a cluster of seven(purple), that both are period-2. At c ≈ − . , the cluster of seven splits up into a cluster of four (purple) and a cluster ofthree (yellow). At c ≈ − . , the cluster of three splits up into a cluster of two (yellow) and a single oscillator (red). Whenthe cluster of two is destroyed, the two resulting single oscillators are shown in red and black. At higher c values, additionalsingle oscillators retain the color of the cluster to which they belonged in the − − − − − solution. At c ≈ − . ,the ensemble jumps to the − − − solution. At c ≈ − . , the cluster of is destroyed and there are henceforth onlysingle oscillators. In this irregular regime, the maxima of Re( w k ) reach as far down as max[Re( w k )] = − . . These maximahave been cut off for a clearer view of the dynamics at lower c values. This same final state is also reached from the verydifferent starting point in Supplementary Figure 5. b , Cluster sizes at each value of c during the c -incremented simulationin a . Calculations are based on cross correlations of trajectories (in the non-rotating frame) over the last time steps ofsimulation at each c value and a threshold of ε = 10 − . See Methods section. ab Supplementary Fig. Bifurcations encountered for N = 32 and η = 0 . , when starting at − − − − and increasing c . a , All occurring maxima of the real parts Re( w k ) of all clusters and single oscillators against c as c isgradually increased at a rate of ∆ c = 2 · − every time steps for N = 32 , ν = 0 . and η = 0 . . The initial state is a − − − − quasiperiodic solution, with maxima of the cluster of shown in blue, those of the cluster of two in grey andthose of the three single oscillators, which all pursue the same solution, in red. At c ≈ − . , a torus bifurcation breaksthe cluster of into single oscillators, and broadens the formerly discrete maxima into continuous ranges. At c ≈ − . ,this solution undergoes a symmetry-increasing collision to only single oscillators. This final state is also reached from the verydifferent starting point in Supplementary Figure 5. b , Cluster sizes at each value of c during the c -incremented simulationin a . The further c is increased away from the symmetry-increasing bifurcation, the less distinct do the ruins of the formerordered state become and the less likely are many of the oscillators to cluster at any given time. Calculations are based oncross correlations of trajectories (in the non-rotating frame) over the last time steps of simulation at each c value and athreshold of ε = 10 − . See Methods section. a bc de f Supplementary Fig. Solutions encountered along the simulation in Supplementary Fig. 5. a , Complex-planeportrait of the − − − − solution at c = − . . The cluster of two is shown in blue, the cluster of two in grey and thethree single oscillators in red, green and yellow, respectively. In order not to obscure its trajectory, the blue dot marking theinstantaneous location of the cluster of is smaller than times as large as the dots marking the single oscillators. b , Timeseries of the real part of each cluster and oscillator in a . c,d , Like a,b for the − (30 × state found at c = − . , pastthe torus bifurcation in Supplementary Figure 5. Here, the single oscillators from the former cluster of are all still plottedin blue. e,f , Solution without permanent clusters at c = − . . Here, eight different colors are arbitrarily used to depictthe N = 32 different oscillator trajectories, with four oscillators of each color. Thus, line segments of the same color do notnecessarily belong to the same oscillator. In the depicted interval, the ensemble has moved both close to ruins of the former − − − − branch and to ruins of the − branch, as indicated by the two blue-green loops similar to those of thesolutions in Fig. 2 of the main article. ab Supplementary Fig. Cluster sizes recorded along the path from a − solution to chimera state. a , Occurringcluster sizes among N = 200 oscillators as c is gradually incremented by ∆ c = 2 · − every time steps at ν = 0 . and η = 0 . . Two oscillators are counted as being in the same cluster if their correlation distance in the interval from t = 2000 to t = 2100 at the relevant value of c is less than − . Note that the vertical axis scales logarithmically. b , Analogue to a , butfor a shorter c interval and with a smaller increment ∆ c = 5 · − and longer simulation time of time steps at each c value. In both a and b , the initial cluster splittings are accompanied by transient many-cluster solutions that are not allowedto subside before the number of clusters is calculated. This causes the temporary increase in recorded small clusters at thebeginning of each step of the cascade. Some of the multi-cluster solutions at higher values of c2