Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations
CConnecting minimal chimeras and fully asymmetric chaotic attractors throughequivariant pitchfork bifurcations
Sindre W. Haugland and Katharina Krischer ∗ Physics Department, Nonequilibrium Chemical Physics,Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany (Dated: February 23, 2021)Highly symmetric networks can exhibit partly symmetry-broken states, including clusters andchimera states, i.e., states of coexisting synchronized and unsynchronized elements. We addressthe S permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover aninterconnected web of differently symmetric solutions. Among these are chaotic − − minimalchimeras that arise from − − periodic solutions in a period-doubling cascade, as well as fullyasymmetric chaotic states arising similarly from periodic − − − solutions. A backbone ofequivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariantpitchforks of chaotic attractors. Symmetrically coupled identical units do not alwaysbehave identically. Instead, they may conglomerate intoseveral distinct internally synchronized clusters [1–4]. Oronly some of them may synchronize, while the othersremain unclustered [1, 5, 6], forming a so-called chimerastate [7]. The latter phenomenon in particular has gar-nered a lot of attention over the last two decades and hasbeen identified in a wide range of theoretical models andexperimental setups [8–10].A relatively recent question related to chimera statesis that of small or minimal chimeras, identified in ensem-bles of only a few [11], usually four [12–16], sometimesthree [17, 18] oscillators. Some of these, like some of themacroscopic chimeras previously identified [1, 5, 19–21],even occur in the case of global coupling, that is, when allthe oscillators are coupled equally to each other [15, 16].The corresponding models have in common that they donot consist of pure phase oscillators, but of oscillatorswith several components.In this Letter, we use a model of four globally coupledcomplex-valued oscillators to demonstrate how chaoticminimal chimera states arise as a consequence of the step-wise breaking of the symmetry of the underlying equations.Our analysis uncovers a network of interconnected bifurca-tion curves, wherein the chimeras are embedded togetherwith other partially synchronized states. The backboneof this network is a sequence of equivariant pitchforkbifurcations [22], on the one side of which two of the oscil-lators are clustered, while on the other, none of them are.Along this equivariant pitchfork sequence, several otherbifurcations affect the symmetries and the periodicitiesof the stable solutions on either side, culminating in twointerconnected period-doubling cascades to chaos. In thechaotic regime of this cascade, equivariant pitchforks ofchaotic attractors govern the transition between chimerasand asymmetric chaos, allowing for full-circle transitionsvia one route from regular motion to chaos and back againvia a different route.Our model is an ensemble of N Stuart-Landau oscil-lators W k ∈ C , k = 1 , . . . , N , with nonlinear global cou- pling [19]: 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 h . . . i = 1 /N P Nk =1 . . . denotes ensemble averageswhile c and ν are real parameters. The Stuart-Landauoscillator itself is a generic model for a system close toa Hopf bifurcation, that is, to the onset of self-sustainedoscillations [23]. Because the oscillators are identical andonly affect each other through the mean quantities h W i and h| W | W i , Eq. (1) is S N -equivariant: If W ( t ) ∈ C N isa solution, then so is γ W ( t ) ∀ γ ∈ S N , where S N , denotedthe symmetric group , is the group of all permutations ofthe N oscillators [22]. When taking the ensemble averageof the equations (1), we further find that h d W k d t i = dd t h W i = − iν h W i ⇒ h W i = η e − iνt , (2)which implies that h W i is confined to simple harmonicmotion with frequency ν . Its amplitude η ∈ R is implic-itly set by choosing the initial condition and thus consti-tutes an additional parameter. Eq. (2) also implies thatthere always exists a fully synchronized periodic solution W k = h W i = η e − iνt ∀ k . For N = 1 , this is clearly theonly solution possible, while for N > , it has been shownto lose stability and give rise to different two-cluster solu-tions [24]: This happens either in an equivariant pitchforkbifurcation, producing separate clusters that continue tocircle the origin with frequency ν at different fixed am-plitudes, or in an equivariant secondary Hopf bifurcation,producing separate modulated-amplitude clusters thathenceforth oscillate with two superposed frequencies ν and ω H . For large N , each of these cluster states has pre-viously been deduced to somehow produce its own kindof chimera states, labeled type-I and type-II chimeras,respectively [25]. Most recently, the concrete path frommodulated-amplitude two-cluster states to macroscopictype-II chimeras was uncovered [26]. a r X i v : . [ n li n . AO ] F e b S × S o Z S × Z S S Z Z { e } { e } (a) (b) (c) (d)(e) (f) (g) (h)FIG. 1. Complex-plane trajectories of different N = 4 solutions. Each color corresponds to an oscillator, lines to trajectoriesand filled circles to current positions. Labels in the upper right denote interchange symmetries of the oscillators. All solutionsare viewed in the co-rotating frame of the ensemble average h W i . (a) − period-1 solution for c = − . and η = 0 . .(b) − − period-2 solution emerging from (a). c = − . and η = 0 . . (c) Less symmetric − − period-2 solution emergingfrom (b). c = − . and η = 0 . . (d) Chaotic − − solution for c = − . and η = 0 . . (e) − − − rotating-waveperiod-2 solution emerging from (a). c = − . and η = 0 . . (f) Less symmetric − − − period-2 solution mediating between(b) and (e). c = − . and η = 0 . . (g) Fully asymmetric period-2 solution mediating between (c) and (f). c = − . and η = 0 . . (h) Fully asymmetric chaotic solution for c = − . and η = 0 . . ν = 0 . throughout. Here, we restrict ourselves to the case N = 4 , and againfocus on modulated-amplitude dynamics with two com-ponent oscillations. We also keep ν = 0 . fixed. Becausethe ensemble average h W i = η e − iνt is independent ofthe individual oscillator dynamics, it describes an alwaysaccessible rotating frame of reference, wherein the valueof each oscillator is always given by the following relation: W k = η e − iνt (1 + w k ) ⇒ w k = W k η − e iνt − , (3)where w k is the value of the oscillator W k in the co-rotating frame and P k w k = 0 . Viewing the dynamics inthe frame of h W i means that two-frequency quasiperiodicmotion becomes a simple limit cycle. A − originallymodulated-amplitude solution with two oscillators in eachof two clusters thus looks as in Fig. 1 a. The cross de-notes the position of the ensemble average, which in theco-rotating frame is always at the origin. Because thetwo clusters of the − solution are of equal size, their re-spective displacement from the origin is always equal andopposite. Furthermore, they follow the same trajectorywith a mutual phase shift of half their common period.The symmetry (or isotropy subgroup [27]) of the − solution in the rotating frame is S × S o Z , where S denotes the permutations of the oscillators within eithercluster, operations that do not change the current solution.If we interchange the clusters themselves, this does change the instantaneous positions of the individual oscillators.Yet, it projects each oscillator exactly onto the the pointwhere it would normally be half a period later. Combiningthis permutation with a time shift of half a period thusalso leaves the system state unchanged. We denote thiskind of spatiotemporal component symmetry by Z , thesymmetry Z n of a cyclic group of n = 2 elements.The permutation symmetries S combine by means ofa direct product “ × ” because their operations commute.In contrast, interchanging the oscillators of the first clus-ter and then interchanging the clusters is not the sameoperation as first interchanging the clusters and then in-terchanging the oscillators of the first cluster. (Of course,the two-cluster solution in Fig. 1 a remains the same ineither case, but the mapping from old to new oscillatorindices does not.) Therefore, S × S and Z combinewith a so-called semidirect product ” o “ [22].In total, the − solution in Fig. 1 a is invariant under | S | · | S | · | Z | = 2 · · (4)different symmetry operations (including the identity),where | Γ | denotes the number of elements in the group Γ .The fully synchronized solution w k = 0 ∀ k is invariantunder a total of | S | = 4! = 24 permutations. Thosepermutations in S that are not contained in S × S o Z transform the current − solution into one out of twodistinct, but equivalent solution variants, whose overalldynamics are the same as those in Fig. 1 a, but whereinnot the yellow, but either the blue or the green oscillatorare clustered with the red one. All equivalent solutionvariants together form the so-called group orbit [27]. Asthe equations themselves are S -equivariant, the size ofany solution’s isotropy subgroup multiplied by the size ofits group orbit is always | S | = 24 .If c is sufficiently increased for suitable values of η ,the − solution loses stability in an equivariant period-doubling bifurcation. For η ∈ [0 . , . , this produces astable − − solution like the one in Fig. 1 b. Thereby,either of the clusters of two is split, and the resultant singleoscillators henceforth undergo the same kind of period-2motion, with double the period of the remaining intactcluster. The mutual phase shift of the single oscillatorsis exactly half their common period, ensuring the overallsymmetry S × Z . As either of the two clusters can bethe one to split up, two S × Z variants emerge fromeach variant of the − solution.For appropriate values of η , additionally increasing c breaks the Z symmetry by differentiating the trajecto-ries of the single oscillators. Because of the constraint P k w k = 0 , this also means that the trajectory of theintact cluster becomes period-2, as shown in Fig. 1 c, butthe overall periodicity of the ensemble remains the same.As either of the single oscillators can be the one to starttraveling along the largest loop in the complex plane,this change is an equivariant pitchfork bifurcation, with | S × Z | / | S | = 2 equivalent S variants emerging fromeach S × Z variant.Notably, the − − solution in Fig. 1 b is not the onlyoutcome of the equivariant period-doubling of the − solution. Below η = 0 . , increasing c instead causes thebreak-up of both clusters, and a stable − − − solutionis produced. Here, the four single oscillators all proceedalong the same trajectory, with consecutive phase shiftsof a quarter period, as shown in Fig. 1 e. The symmetryis thus a cyclic Z symmetry.Comparing the S × Z and the Z solution, we realizethat the former might be turned into the latter if itsremaining cluster were to split and the resultant singleoscillators to become period-2. If η is decreased fromthe region where the S × Z solution is stable, such asplit does indeed occur, and the solution in Fig. 1 f isproduced. It symmetry is Z because only the operationof exchanging both the blue and the green, and the red andthe yellow oscillator, respectively, followed by a time shiftof half a period, leaves the current solution unchanged.As η is further decreased, the less pronouncedly period-2trajectory of the blue and the green oscillator becomes evermore similar to that of the red and the yellow oscillator,until we reach the Z solution. Supplementary Fig. 1details which Z variants mediate between the two S × Z and two Z variants emerging from the same − variant.Similarly, the S solution in Fig. 1 c may also lose its remaining symmetry in an equivariant pitchfork affectingthe remaining cluster. This results in the fully asymmetricsolution in Fig. 1 g. From here on, the loops of the blueand the green, and the red and the yellow oscillator,respectively, may also become more similar again, and forsuitable values of c and η , the fully asymmetric solutionbecomes a Z solution like that in Fig. 1 f. How differentfully asymmetric variants mediate between the different S and Z variants is shown in Supplementary Fig. 2.In parameter space, the stable regions of these solutionsare connected as shown in Fig. 2 a, where the equivariantperiod-doubling of the − solution is denoted by theleftmost blue line and the equivariant pitchfork bifurca-tions between the different period-2 solutions by the greenlines. Additional features of this bifurcation diagram arethe torus bifurcations shown in red as well as the collec-tion of blue period-doublings in the upper right. Eachof the former simply adds a tertiary frequency to the ad-joining period-2 solution, causing the system to undergo T quasiperiodic motion overall, that is, two-frequencyquasiperiodic motion in the frame of h W i . A little furtherto the right, this motion also becomes unstable and theensemble jumps to a multistable − solution.The parameter region of the extra period-doublings isshown in greater detail in Fig. 2 c. Here, it becomes ap-parent that these are really two adjacent period-doublingcascades. The starting point for the upper cascade is the S solution in Fig. 1 c while that of the lower one is thefully asymmetric solution in Fig. 1 g. Unlike the period-doubling in the left of Fig. 2 a, these period-doublingbifurcations do not affect the symmetry of the solution,but simply double the period of every oscillator. Thus,if we start at the S period-2 solution and increase c ordecrease η appropriately, we first obtain an S period-4solution, then an S period-8 solution, and so on, untilwe finally end up at the chaotic S solution in Fig. 1 d.This is a minimal chimera state for which the underlyingchaotic attractor is not symmetric under the exchangeof the two unsynchronized oscillators [29]. If we start atthe asymmetric period-2 solution in Fig. 1 g and increase η , the system undergoes a period-doubling cascade to achaotic state in which the trajectories of all four oscillatorsdiffer. See Fig. 1 h.Let us now consider the region of the bifurcation dia-gram in Fig. 2 c where the two period-doubling cascadesmeet. Above, we saw that each variant of the S period-2 solution bifurcates in an equivariant pitchfork to twodistinct, but equivalent variants of the fully asymmetricperiod-2 solution. Similarly, the region where S period-4solutions are stable borders on the region where fullyasymmetric period-4 solutions are. Every S period- n solution also undergoes an equivariant pitchfork bifurca-tion, and in these equivariant pitchforks, two equivalentvariants of the asymmetric period- n solutions emergefrom each variant of the S -symmetric period- n solution.Continuing along the juncture of the two period- S × S oZ S × Z Z Z S { e } S { e } (a) (b)(c) (d)FIG. 2. (a) Bifurcation diagram showing the regions of stability of the solutions in Fig. 1 and the bifurcations between them.PD = period-doubling. PF = pitchfork. ν = 0 . . (b) Maxima of Re( w k ) for each oscillator k = 1 , . . . , as η is graduallyincreased along the vertical black line at c = − . in (a). At η = 0 . , the blue and green oscillator merge in a pitchforkbifurcation of chaotic attractors. (c) Magnification of the region within the gray rectangle in (a). (d) Upper part of (b) showingthe increase in symmetry in greater detail. The bifurcation lines in (a) and (c) were calculated using Auto07p [28]. doubling cascades, the succession of equivariant pitchforksfinally reaches the chaotic domain. If initializing the en-semble in the asymmetric period-2 solution at c = − . and η = 0 . and slowly increasing η along the verticalblack line in Fig. 2 c, we thus observe the developmentdepicted in Fig. 2 b and d. Here, each oscillator is rep-resented by the maxima reached by its real part Re( w k ) .Initially, each oscillator reaches either two or three distinctmaxima. At η = 0 . , the number of these maxima aredoubled, at η = 0 . , they are doubled again, and at η = 0 . again. More distinct period-doublings cannotbe discerned, but the overall resemblance to the classicFeigenbaum period-doubling cascade remains clear.The crucial point of the development happens at η = 0 . , where the current fully asymmetric chaotic so-lution merges with one of its own variants – here: the onein which the blue and green oscillator are interchanged –to form a single variant of a − − chaotic chimera state.Continuing further upward in η takes us full circle fromthis chimera backwards through the S period-doublingcascade to the S period-2 solution in Fig. 1 c and the other regular solutions.In summary, we have uncovered the interconnectedbifurcation structure of differently symmetric and differ-ently periodic solutions arising in an S -symmetric system.Embedded in this structure are minimal chaotic chimerastates. The bifurcation-theoretical route to these mini-mal chimera states differs from that of the macroscopicchimeras found in the same model for larger ensemblesizes N [26], a trait they incidentally share with the other-wise unrelated minimal chimeras found in the two-groupsmodel of phase oscillators [12, 30]. Our minimal chimerasstrongly resemble the so-called asymmetric chimera statespreviously reported in Stuart-Landau oscillators with lin-ear global coupling [16], whose context of other solutions isnot as extensively charted. Notably, our bifurcation struc-ture contains a juncture of two period-doubling cascades,leading to equivariant pitchfork bifurcations of chaoticattractors and rendering the chimera states reachable bymeans of two different routes in parameter space. Towhat extent this is a universal property of S -symmetricsystems remains an interesting question.The authors thank Felix P. Kemeth for fruitful discus-sions. Financial support from the Studienstiftung desdeutschen Volkes and the Deutsche Forschungsgemein-schaft, project KR1189/18 “Chimera States and Beyond”,is gratefully acknowledged. ∗ [email protected].[1] K. Kaneko, Clustering, coding, switching, hierarchicalordering, and control in a network of chaotic elements,Phys. D , 137 (1990).[2] D. Golomb, D. Hansel, B. Shraiman, and H. Sompolinsky,Clustering in globally coupled phase oscillators, Phys. Rev.A , 3516 (1992).[3] V. Hakim and W. J. Rappel, Dynamics of the globallycoupled complex Ginzburg-Landau equation, Phys. Rev.A , R7347 (1992).[4] K. Okuda, Variety and generality of clustering in globallycoupled oscillators, Phys. D , 424 (1993).[5] N. Nakagawa and Y. Kuramoto, Collective Chaos in aPopulation of Globally Coupled Oscillators, Prog. Theor.Phys. , 313 (1993).[6] Y. Kuramoto and D. Battogtokh, Coexistence of Co-herence and Incoherence in Nonlocally Coupled PhaseOscillators, Nonlinear Phenom. Complex Syst. , 380(2002).[7] D. M. Abrams and S. H. Strogatz, Chimera states forcoupled oscillators, Phys. Rev. Lett. , 174102 (2004).[8] M. J. Panaggio and D. M. Abrams, Chimera states: Co-existence of coherence and incoherence in networks ofcoupled oscillators, Nonlinearity , R67 (2015).[9] E. Schöll, Synchronization patterns and chimera states incomplex networks: Interplay of topology and dynamics,Eur. Phys. J. Spec. Top. , 891 (2016).[10] O. E. Omel’chenko, The mathematics behind chimerastates, Nonlinearity , R121 (2018).[11] P. Ashwin and O. Burylko, Weak chimeras in minimalnetworks of coupled phase oscillators, Chaos , 013106(2015).[12] 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).[13] 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).[14] D. Dudkowski, J. Grabski, J. Wojewoda, P. Perlikowski,Y. Maistrenko, and T. Kapitaniak, Experimental multi-stable states for small network of coupled pendula, Sci. Rep. , 29833 (2016).[15] A. Röhm, F. Böhm, and K. Lüdge, Small chimera stateswithout multistability in a globally delay-coupled networkof four lasers, Phys. Rev. E , 1 (2016).[16] F. P. Kemeth, S. W. Haugland, and K. Krischer, Sym-metries of Chimera States, Phys. Rev. Lett. , 214101(2018).[17] J. Wojewoda, K. Czolczynski, Y. Maistrenko, and T. Kap-itaniak, The smallest chimera state for coupled pendula,Sci. Rep. , 34329 (2016).[18] Y. Maistrenko, S. Brezetsky, P. Jaros, R. Levchenko, andT. Kapitaniak, Smallest chimera states, Phys. Rev. E ,010203 (2017).[19] L. Schmidt, K. Schönleber, K. Krischer, and V. García-Morales, Coexistence of synchrony and incoherence inoscillatory media under nonlinear global coupling, Chaos , 013102 (2014).[20] G. C. Sethia and A. Sen, Chimera states: The existencecriteria revisited, Phys. Rev. Lett. , 144101 (2014).[21] F. Böhm, A. Zakharova, E. Schöll, and K. Lüdge,Amplitude-phase coupling drives chimera states in glob-ally coupled laser networks, Phys. Rev. E , 040901(2015).[22] J. Moehlis and E. Knobloch, Equivariant bifurcation the-ory, Scholarpedia , 2511 (2007).[23] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu-lence , Springer Series in Synergetics, Vol. 19 (SpringerBerlin Heidelberg, Berlin, Heidelberg, 1984).[24] L. Schmidt and K. Krischer, Two-cluster solutions in anensemble of generic limit-cycle oscillators with periodicself-forcing via the mean-field, Phys. Rev. E , 042911(2014).[25] L. Schmidt and K. Krischer, Clustering as a prerequisitefor chimera states in globally coupled systems, Phys. Rev.Lett. , 034101 (2015).[26] S. W. Haugland and K. Krischer, A hierarchy of co-existence patterns mediating between low- and high-dimensional dynamics in highly symmetric systems,arXiv:2101.10242 (2021).[27] R. B. Hoyle, Pattern Formation An Introduction to Meth-ods , 1st ed. (Cambridge University Press, Cambridge,2006).[28] E. Doedel and B. Oldeman,
Auto 07p: Continuation andbifurcation software for ordinary differential equations ,Tech. Rep. (Concordia University and McGill HPC Centre,Montreal, Canada, 2019).[29] P. Chossat and M. Golubitsky, Symmetry-increasing bi-furcation of chaotic attractors, Phys. D , 423 (1988).[30] D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley,Solvable model for chimera states of coupled oscillators,Phys. Rev. Lett. , 84103 (2008). upplemental material: Connecting minimal chimeras and fully asymmetric chaoticattractors through equivariant pitchfork bifurcations Sindre W. Haugland and Katharina Krischer ∗ Physics Department, Nonequilibrium Chemical Physics,Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany (Dated: February 23, 2021) a r X i v : . [ n li n . AO ] F e b FIG. 1. Overview of the two variants of the S × Z (left column) and Z (right column) solutions emerging from one variant ofthe − solution, as well as the four variants of the intermediate Z solution connecting these. Solutions are represented by thetime series of the real part of each oscillator in the rotating frame. Black arrows indicate which Z variants emerge from each ofthe variants of the more symmetric solutions. Notably, the two Z variants emerging from each S × Z variant do not merge intothe same Z variant. The states shown here are all descended from the − variant wherein the red and the yellow, and thegreen and the blue oscillator are clustered, respectively. The situation is analogous for each of the two other variants of the − solution. FIG. 2. Overview of the two variants of the S (left column) and Z (right column) solutions emerging from one variant of the S × Z solution, as well as the four variants of the fully asymmetric state mediating between these. Solutions are representedby the time series of the real part of each oscillator in the rotating frame. Black arrows indicate which asymmetric variantsemerge from which variants of the more symmetric solutions. Notably, the two totally asymmetric variants emerging from each S variant do not merge into the same Z variant. The variants shown here are all descended from that variant of the S × Z solution wherein the green and blue oscillator are clustered, that is, the upper left variant in Fig. 1. The situation is analogousfor the states emerging from each of the five other variants of the S2
Auto 07p: Continuation andbifurcation software for ordinary differential equations ,Tech. Rep. (Concordia University and McGill HPC Centre,Montreal, Canada, 2019).[29] P. Chossat and M. Golubitsky, Symmetry-increasing bi-furcation of chaotic attractors, Phys. D , 423 (1988).[30] D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley,Solvable model for chimera states of coupled oscillators,Phys. Rev. Lett. , 84103 (2008). upplemental material: Connecting minimal chimeras and fully asymmetric chaoticattractors through equivariant pitchfork bifurcations Sindre W. Haugland and Katharina Krischer ∗ Physics Department, Nonequilibrium Chemical Physics,Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany (Dated: February 23, 2021) a r X i v : . [ n li n . AO ] F e b FIG. 1. Overview of the two variants of the S × Z (left column) and Z (right column) solutions emerging from one variant ofthe − solution, as well as the four variants of the intermediate Z solution connecting these. Solutions are represented by thetime series of the real part of each oscillator in the rotating frame. Black arrows indicate which Z variants emerge from each ofthe variants of the more symmetric solutions. Notably, the two Z variants emerging from each S × Z variant do not merge intothe same Z variant. The states shown here are all descended from the − variant wherein the red and the yellow, and thegreen and the blue oscillator are clustered, respectively. The situation is analogous for each of the two other variants of the − solution. FIG. 2. Overview of the two variants of the S (left column) and Z (right column) solutions emerging from one variant of the S × Z solution, as well as the four variants of the fully asymmetric state mediating between these. Solutions are representedby the time series of the real part of each oscillator in the rotating frame. Black arrows indicate which asymmetric variantsemerge from which variants of the more symmetric solutions. Notably, the two totally asymmetric variants emerging from each S variant do not merge into the same Z variant. The variants shown here are all descended from that variant of the S × Z solution wherein the green and blue oscillator are clustered, that is, the upper left variant in Fig. 1. The situation is analogousfor the states emerging from each of the five other variants of the S2 × Z2