Chimeras and clusters emerging from robust-chaos dynamics
aa r X i v : . [ n li n . AO ] F e b Chimeras and clusters emerging from robust-chaos dynamics
M. G. Cosenza,
1, 2
O. Alvarez-Llamoza, and A. V. Cano
4, 5 School of Physical Sciences & Nanotechnology, Universidad Yachay Tech, Urcuqu´ı, Ecuador Universidad de Los Andes, M´erida, Venezuela Grupo de Simulaci´on, Modelado, An´alisis y Accesabilidad,Universidad Cat´olica de Cuenca, Cuenca, Ecuador Institute for Integrative Biology, ETH, Zurich, Switzerland Swiss Institute of Bioinformatics, Lausanne, Switzerland (Dated: February 2021)
ABSTRACT
We show that dynamical clustering, where a system segregates into distinguishable subsets ofsynchronized elements, and chimera states, where differentiated subsets of synchronized and desyn-chronized elements coexist, can emerge in networks of globally coupled robust-chaos oscillators. Wedescribe the collective behavior of a model of globally coupled robust-chaos maps in terms of sta-tistical quantities, and characterize clusters, chimera states, synchronization, and incoherence onthe space of parameters of the system. We employ the analogy between the local dynamics of asystem of globally coupled maps with the response dynamics of a single driven map. We interpretthe occurrence of clusters and chimeras in a globally coupled system of robust-chaos maps in termsof windows of periodicity and multistability induced by a drive on the local robust-chaos map. Ourresults show that robust-chaos dynamics does not limit the formation of cluster and chimera statesin networks of coupled systems, as it had been previously conjectured.
I. INTRODUCTION
Many smooth nonlinear dynamical systems possesschaotic attractors embedded with a dense set of peri-odic orbits for any range of parameter values. Therefore,in practical systems operating in chaotic mode, a slightperturbation of a parameter may drive the system out ofchaos. Alternatively, there exist dynamical systems thatexhibit the property of robust chaos [1–6]. A chaotic at-tractor is said to be robust if, for its parameter values,there exists a neighborhood in the parameter space wherewindows of periodic orbits are absent and the chaotic at-tractor is unique [1].Robust chaos constitutes an advantageous feature inapplications that require reliable functioning in a chaoticregime, in the sense that the chaotic behavior cannot beremoved by arbitrarily small fluctuations of the systemparameters. For instance, networks of coupled maps withrobust chaos have been efficiently used in communicationand encryption algorithms [7] and they have been inves-tigated for information transfer across scales in complexsystems [8]. In addition, the existence of robust chaos al-lows for heterogeneity in the local parameters of a systemof coupled oscillators, while guaranteeing the performingof all the oscillators in a chaotic mode.On the other hand, systems possessing robust chaosmay present limitations in the types of collective behav-iors that they can achieve, in comparison with systemsdisplaying periodic windows. For example, it has beenconjectured that the phenomenon of dynamical cluster-ing in globally coupled networks (where the system seg-regates into distinguishable subsets of synchronized ele-ments) is only found when stable periodic windows arepresent in the local elements [9–11]. Recently, it has also been argued that chimera states (i.e., coexistence of sub-sets of oscillators with synchronous and asynchronous dy-namics) cannot emerge in networks of coupled oscillatorshaving robust chaotic attractors [12, 13].The phenomenon of dynamical clustering is relevant asit can provide a simple mechanism for the emergence ofdifferentiation, segregation, and ordering of elements inmany physical and biological systems [14, 15]. Clusteringhas been found in systems of globally coupled R¨ossler os-cillators [16], neural networks [17], biochemical reactions[18], and has been observed experimentally in arrays ofglobally coupled electrochemical oscillators [19] and glob-ally coupled salt-water oscillators [20]. In addition, thestudy of chimera states currently attracts much inter-est; for reviews see, [21, 22]. Chimera states have beenfound in networks of nonlocally coupled phase oscillators[23, 24], in systems with local [25–28] and global [29–34]interactions, and in networks of time-discrete maps [35–38]. These states have been investigated in a diversityof contexts [39–47]. Chimera states have been observedin experimental settings, such as populations of chemicaloscillators [48], coupled lasers [49], optical light modula-tors [50], electronic [51], and mechanical [52, 53] oscilla-tor systems. It has been shown that clustering is closelyrelated to the formation of chimera states in systems ofglobally coupled periodic oscillators [31].In this paper, we investigate the occurrence of dy-namical clustering and chimera states in systems of cou-pled robust-chaos oscillators. In Sec. II, we describe thecharacterization of synchronization, cluster and chimerastates in globally coupled systems. In Sec. III we con-sider a network of globally coupled robust-chaos mapsand show that cluster and chimera states can actuallyemerge in this system for several values of parameters.In Sec. IV, we employ the analogy between the local dy-namics of the globally coupled system with the responsedynamics of a single driven map. We interpret the oc-currence of clusters and chimeras in the globally coupledsystem in terms of windows of periodicity induced by thedrive on the local robust-chaos map. Conclusions arepresented in Sec V.
II. METHODS
A global interaction in a system can be described as afield or influence acting on all the elements in the system.As a simple model of an autonomous dynamical systemsubject to a global interaction, we consider a system of N maps coupled in the form x it +1 = (1 − ǫ ) f ( x it ) + ǫh t ( x jt | j ∈ S ) , (1)where x it ( i = 1 , , . . . , N ) describes the state variable ofthe i th map in the system at discrete time t , the function f expresses the local dynamics of the maps, the function h t represents a global field that depends on the statesof the elements in a given subset S of the system, attime t , and the parameter ǫ measures the strength ofthe coupling of the maps to the field. The form of thecoupling in Eq. (1) is assumed in the commonly useddiffusive form. The function h t may not depend on allthe elements, but it must be shared by all the elementsof the system to be a global interaction.A collective state of synchronization or coherence takesplace in the system Eq. (1) when x it = x jt , ∀ i, j for asymp-totic times. A desynchronized or incoherent state corre-sponds to x it = x jt ∀ i, j for all times. Dynamical cluster-ing occurs when the system segregates into a number of K distinct clusters or subsets of elements such that elementsin given subset are synchronized among themselves. Inother words, x it = x jt = X ξt , ∀ i, j in the ξ th cluster,where X ξt denotes the value of x it in that cluster, with ξ = 1 , . . . , K . If n ξ is the number of elements belongingto the ξ th cluster, then its relative size is p ξ = n ξ /N .In general, the number of clusters, their size, and theirdynamical evolution (periodic, quasiperiodic, or chaotic)depend on the initial conditions and parameters of thesystem. A chimera state consists of the coexistence ofone or more clusters and a subset of desynchronized ele-ments. If there are K clusters, the fraction of elements inthe system belonging to clusters is p = P Kξ =1 n ξ /N while(1 − p ) is the fraction of elements in the desynchronizedsubset.In practical applications, we consider that two ele-ments i and j belong to a cluster at time t if the distancebetween their state variables, defined as d ij ( t ) = | x it − x jt | , (2)is less than a threshold value δ , i.e., if d ij < δ . The choiceof δ should be appropriate for achieving differentiation between closely evolving clusters. Then, we calculate thefraction of elements that belong to some cluster at time t as [16] p ( t ) = 1 − N N X i =1 N Y j =1 ,j = i Θ[ d ij ( t ) − δ ] , (3)where Θ( x ) = 0 for x < x ) = 1 for x ≥ p as the asymptotic time-average of p ( t ).Then, a clustered state in the system can be characterizedby the value p = 1, while an incoherent state in thesystem corresponds to p →
0. The values p min < p < p min is the minimumcluster size to be taken into consideration.A synchronization state corresponds to the presence ofa single cluster of size N and has also the value p = 1. Todistinguish a synchronization state from a multiclusterstate, we calculate the asymptotic time-average h σ i as h σ i = 1 T − τ T X t = τ σ t , (4)where τ is the number of discarded transients, T is a suf-ficiently large time, and σ t is the instantaneous standarddeviation of the distribution of state variables defined by σ t = " N N X i =1 ( x it − ¯ x t ) / , (5)where ¯ x t = 1 N N X j =1 x jt . (6)Statistically, a synchronization state is characterizedby the values h σ i = 0 and p = 1, while a cluster statecorresponds to h σ i > p = 1. Chimera states arecharacterized by h σ i > p min < p <
1, and desyn-chronization is described by h σ i > p < p min . In thispaper we set δ = 10 − and p min = 0 . h σ i and p , not onthe spatial location of synchronized and desynchronizeddomains. III. RESULTS AND DISCUSSIONChimeras and clusters in globally coupled robustchaos maps
Let us consider a network of globally coupled mapsdescribed by the equations [14] x it +1 = (1 − ǫ ) f ( x it ) + ǫN N X j =1 f ( x jt ) , (7)where the global interaction function is the mean field ofthe system, h t = 1 N N X j =1 f ( x jt ) . (8)As local dynamics exhibiting robust chaos, we considerthe following smooth, unimodal map defined on the in-terval x ∈ [0 ,
1] [54], x t +1 = f ( x t ) = 1 − b (1 − x t ) x t − b / . (9)which is chaotic with no periodic windows on the param-eter interval b ∈ [0 , λ = ln 2.The bifurcation diagram of map Eq. (9) in Figure 1 showsthe absence of periodicity in the interval b ∈ [0 , . . . .
81 0 0 . . . . x t b FIG. 1. Bifurcation diagram of the map Eq. (9) as a functionof the parameter b . Figure 2 shows the asymptotic temporal evolution ofthe states of the system Eqs. (7) and (9), for different val-ues of parameters. Since the system is globally coupled,there is no natural spatial ordering. For visualizationpurposes, the indexes i are ordered at time t = 10 suchthat i < j if x it < x jt and kept fixed afterwards. Thevalues of the states x it are represented by distinct colorcoding; two elements i, j share the same color if x it = x jt .A desynchronized state is displayed in Fig. 2(a) and acomplete synchronization state occurs in Fig. 2(d), whilea chimera state and a two-cluster state are visualized inFigs. 2(b) and 2(c), respectively. FIG. 2. Asymptotic evolution of the states x i (horizontal axis)as a function of time (vertical axis) for the system Eqs. (7)and(9) with size N = 100 and fixed b = 0 .
5, for different val-ues of the coupling parameter. Random initial conditions areuniformly distributed in the interval [0 , transients, 100 iterates t are displayed. Ordering of themap indexes is explaining in the text. Color code: two ele-ments i, j share the same color if x it = x jt . (a) Incoherent ordesynchronized state, ǫ = 0 .
15. (b) Chimera state, ǫ = 0 . ǫ = 0 .
39. (d) Synchronization, ǫ = 0 . Figure 3 shows the collective states arising in the sys-tem Eqs. (7) and (9) on the space of parameters ( ǫ, b ),characterized through the quantities p and h σ i . Labelsindicate the regions where these behaviors occur: CS:complete synchronization; C: cluster states; Q: chimerastates, and D: desynchronization. QDCCS bǫ . . . . . . . . . . FIG. 3. Phase diagram on the plane ( ǫ, b ) for the autonomoussystem Eqs. (7) and (9) with size N = 500. For each datapoint, the quantities p and h σ i are obtained by averaging over50 realizations of random initial conditions x i uniformly dis-tributed in the interval [0 , The linear stability analysis for the complete synchro-nization state in the globally coupled system Eq. (7)shows that this state is stable if the following conditionis satisfied [14], | (1 − ǫ ) e λ | < , (10)where λ is the Lyapunov exponent for the local map f ( x ).For the map Eq. (9), we obtain that the completely syn-chronized state is stable for 1 / < ǫ < /
2, which agreeswith the numerical characterization for this state per-formed in Fig. 3. Figure 3 reveals that both cluster andchimera states can arise in globally coupled map net-works for appropriate values of parameters, even whenthe individual maps lack periodic windows. Clusters andchimera states regions occur adjacent to each other foran intermediate range of values of the coupling parame-ter ǫ on the phase diagram of Fig. 3. In fact, chimerasand clusters are closely related collective states in sys-tems subject to global interactions [30]. Chimera statesappear to mediate between dynamical clustering and in-coherence.Multicluster chimera states are also possible in systemsof globally coupled robust chaos maps. As an illustration,consider the smooth unimodal map [55], f ( x t ) = sin ( r arcsin( √ x t )) , (11)defined on the interval x t ∈ [0 ,
1] for parameter values r >
1. Figure (4) shows the bifurcation diagram of theiterates of map Eq. (11) as a function of the parameter r . The dynamics of the map displays robust chaos withno periodic windows for r >
1. The Lyapunov exponentis λ = ln r [55]. FIG. 4. Bifurcation diagram of the map Eq. (11) as a functionof the parameter r . Figure (5) shows the temporal evolution of the statesof the globally coupled system Eqs. (7) with the localmap Eq. (11), for different values of parameters. Achimera state with multiple clusters occurs in Fig. 5(a),while a two-cluster state is shown in Fig. 5(b). Mul-tichimera states or multiheaded chimeras (coexistence ofmultiple localized domains of incoherence and coherence)have been reported in systems with long-range interac-tions [56]. However, those states are not equivalent to a chimera state with multiple clusters in a globally cou-pled system, such as Fig. 5(a), where there is no notionof locality. (b)(a)
FIG. 5. Asymptotic states x i (horizontal axis) as a function oftime (vertical axis) for the system Eqs. (7) with size N = 100and local map Eq. (11), for different values of parameters.Initial conditions and ordering of the maps are similar to thosein Fig. 2. Color code: two elements i, j share the same colorif x it = x jt . (a) Chimera state with two clusters, r = 3, ǫ =0 . r = 3, ǫ = 0 . Dynamics of clusters and chimera states with globalinteractions
Consider a chimera state consisting of K clusters anda desynchronized subset in the system of globally cou-pled maps Eq. (1). The dynamics of this state can bedescribed by the equations X ξt +1 = (1 − ǫ ) f ( X ξt ) + ǫh t , ξ = 1 , . . . , K,x jt +1 = (1 − ǫ ) f ( x jt ) + ǫh t , j = 1 , . . . , (1 − p ) N. (12)The mean field Eq. (8) in a chimera state can be ex-pressed as the sum of two contributions h t = h C + h I , (13)where h C = K X ξ =1 p ξ f ( X ξt ) , (14) h I = 1 N (1 − p ) N X j =1 f ( x jt ) . (15)The term h C is the contribution to the mean field corre-sponding to elements belonging to clusters, whereas h I is the average of the states of the elements belonging tothe incoherent subset.Figure 6 shows the temporal behavior of both contri-butions h C and h I in a chimera state for the globallycoupled autonomous system Eqs. (7) and (9). The timeevolution of the cluster contribution h C is chaotic, simi-lar to that of the local map Eq. (9), but h C has a smalleramplitude. In general, the form of h C can be approxi-mated as h C ≈ Af ( y t ), where A < h I fluctuates about a mean value with a smalldispersion, corresponding to the superposition of the dy-namics of many incoherent chaotic elements. Thus, forthe given parameter values, the incoherent contributionto the mean field for a large system size can be expressedapproximately as a constant; i. e., h I ≈ k . FIG. 6. Cluster h C (green line) and incoherent h I (red line)contributions to the mean field of the system Eqs. (7) and (9),as functions of time. Fixed parameters b = 0 . ǫ = 0 .
19, andsize N = 10 . The dynamics of the globally coupled system Eqs. (1),where each map is subject to a feedback field h t , can becompared to that of a replica system of maps subject toa global external drive g ( y t ) in the form x it +1 = (1 − ǫ ) f ( x it ) + ǫg ( y t ) ,y t +1 = g ( y t ) . (16)It has been shown that an analogy between the au-tonomous system Eq. (1) and the driven system Eq. (16)can be established when the time evolution of the field h t is identical to that of the function g ( y t ) [9]. Then,the drive-response dynamics at the local level in bothsystems are similar, and therefore their correspondingemerging collective states can be equivalent for some ap-propriate parameter values and initial conditions. In par-ticular, chimera or cluster states in the system Eq. (16)should be induced by an external drive function of theform g ( y t ) = Af ( y t ) + k , with A , k constants, that im-itates the mean field h t . The realization of these statesdepends on the parameters A and k of the drive, and onthe coupling strength ǫ .Figure 7 shows the temporal evolution of the statesof the driven system Eqs. (16) with local map Eq. (9),for some values of parameters. A chimera state with asingle cluster takes place in Fig. 7(a) for parameter values( ǫ, b ), where chimera states also occur in the autonomoussystem Eqs. (7) and (9), as seen in the corresponding phase diagram of Fig. 3. Figure 7(b) shows a two-clusterstate for values ( ǫ, b ) located in the region correspondingto clustered states in Fig. 3. The dynamics of the drivensystem Eqs. (16) displays multistability; depending oninitial conditions, chimeras with different partitions maybe induced for given parameters values ( ǫ, b ) in the regionlabeled Q in Fig. 3. Similarly, different initial conditionsmay produce cluster states with different partition sizesfor fixed parameter values in region C of Fig. 3. (b)(a) FIG. 7. Asymptotic evolution of the states x i (horizontalaxis) as a function of time (vertical axis) for the driven systemEq. (16) with size N = 100 and local map Eq. (9), for differ-ent values of the coupling parameter. Fixed values: A = 0 . k = 0 . b = 0 .
5. Random initial conditions x i are uniformlydistributed in the interval [0 , tran-sients, 100 iterates t are displayed. Ordering of the maps issimilar to that in Fig. 2. Color code: two elements i, j sharethe same color if x it = x jt . (a) Chimera state; ǫ = 0 . ǫ = 0 . The system Eqs. (16) can be considered as N realiza-tions for different initial conditions of a single driven map x t +1 = (1 − ǫ ) f ( x t ) + ǫg ( y t ) ,y t +1 = g ( y t ) . (17)Analogously, each local map in the globally coupled sys-tem Eqs. (7) can be seen as subject to a field h t thateventually induces a collective state. Clustering in glob-ally coupled systems of identical elements has been at-tributed to the existence of periodic windows in the localdynamics [10]. On the other hand, clustering is consid-ered a prerequisite for the occurrence of chimera statesin globally coupled systems [31]. Thus, to elucidate theorigin of clusters and chimeras in system Eqs. (7) withlocal robust chaos, one can explore the response dynam-ics of the single driven map Eq. (17) with a function ofthe form g ( y t ) = Af ( y t ) + k and f having robust chaos.Then, if periodic windows are induced by the drive on asingle map, one may expect that clusters and chimerasshould arise in a globally coupled system of those maps.Even a trivial function g can modify the dynamics of adriven robust chaos map in Eq. (17) to produce periodicwindows. Figure 8(a) shows the bifurcation diagram of x t in Eq. (17) versus ǫ for the map f given by Eq. (9)with g ( y t ) →
0, which is equivalent to a rescaling of f .Periodic windows typical of unimodal maps appear inthe rescaled map x t +1 = (1 − ǫ ) f ( x t ). In general, thedriven map Eq. (17) represents a rescaling of the robust-chaos map f that acquires periodic windows. Similarly,the periodic cluster states arising in the globally coupledsystem Eqs. (7) and (9) are a consequence of the win-dows of periodicity induced locally by the mean field h t ,in analogy to the periodic windows created by an exter-nal drive g acting on a single map Eq. (9). Differentinitial conditions may lead to different out-of-phase or-bits with diverse partitions that appear as clusters in theglobally coupled system. A synchronization state in thesystem Eqs. (7) and (9) can be associated to the fixedpoint interval of the bifurcation diagram of Fig. 8(a),while a desynchronization state in the globally coupledsystem is a manifestation of a chaotic regime as seen inFig. 8(a). Nontrivial forms of the driving function cangive rise to multistable behavior besides periodic win-dows. For example, we have verified that a drive such as g ( y t ) = 0 . f ( y t ) + 0 . FIG. 8. Bifurcation diagrams of the driven map x t +1 = (1 − ǫ ) f ( x t ) in Eq. (17) as a function of ǫ for different robust chaosmaps f . (a) f ( x ) = − b (1 − x ) x − b / with b = 0 .
5. (b) f ( x ) =sin ( r arcsin( √ x )) with r = 3. (c) f ( x ) = ln | x | . These results suggest that the emergence of cluster andchimera states in a globally coupled system of robust-chaos maps can be inferred from the occurrence of peri-odic windows in the response dynamics of a single mapsubject to an appropriate drive, as a function of param-eters. Figure 8(b) shows the corresponding bifurcationdiagram of x t +1 = (1 − ǫ ) f ( x t ) versus ǫ for the map f given by Eq. (11) which also has robust chaos. Again,we see the emergence of periodic windows as the cou- pling parameter is varied. A globally coupled system ofthese maps also shows clusters and chimera states, as il-lustrated in Fig. 5. Figure 8(c) presents the bifurcationdiagram of x t +1 = (1 − ǫ ) f ( x t ) versus ǫ for the logarith-mic map f = a + ln | x | , which possesses robust chaoson the parameter interval a ∈ [ − ,
1] and its dynamicsis unbounded [2]. In contrast to Figs. 8(a)-(b), no peri-odic windows appear on the dynamics of the driven mapEq. (17) which remains unbounded; only chaotic orbitsand a fixed point attractor appear. As a consequence,clusters and chimera states should not be expected ina globally coupled system of logarithmic maps. In fact,only synchronization and nontrivial collective behaviorhave been observed in such a system [57].
IV. CONCLUSIONS
Networks of globally coupled identical oscillators areamong the simplest symmetric spatiotemporal systemsthat can display clustering and chimera behavior. Previ-ous works have conjectured that these phenomena can-not occur when the local oscillators possess robust-chaosattractors [9–13]. We have shown that the presence ofglobal interactions can indeed allow for emergence of bothcluster and chimera states in systems of coupled robust-chaos maps. Chimeras appear as partially ordered statesbetween synchronization or clustering and incoherent be-havior. We have found that chimera states are associatedto the formation of clusters in these systems, a featurethat has been observed in other globally coupled systems[31].The existence of intrinsic periodic windows in the dy-namics of local oscillators, such as in logistic maps, is notessential for the emergence of clusters with periodic be-havior in a globally coupled system of those oscillators.Windows of periodicity and multistability can be inducedin the dynamical response of a robust-chaos map subjectto an appropriate external forcing. Because of the anal-ogy between a single driven map and the local dynamicsof a globally coupled map system, the global interactionfield h t can also induce periodic windows and multista-bility on local robust-chaos maps. Those are the essen-tial ingredients for the occurrence of cluster and chimerastates in globally coupled systems. Since clustering isa prerequisite for chimeras [31], a single driven robust-chaos map that develops periodic windows on some rangeof parameters allows us to infer that a globally cou-pled system of such maps shall also exhibit cluster andchimera states on some range of parameters. Conversely,a robust-chaos map, such as the logarithmic or anothersingular map, that does not give rise to periodic windowswhen subject to a drive, implies that a system of glob-ally coupled logarithmic or singular maps do not showclusters nor chimera states.Further extensions of this work include the investi-gation of chimera states in networks of globally cou-pled continuous-time dynamical systems possessing ro-bust chaos or hyperbolic chaotic attractors, the study ofinteracting populations of robust-chaos elements, and therole of the range of interaction in a network of dynamicalrobust-chaos units. ACKNOWLEDGMENT
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