Blinking chimeras in globally coupled rotators
BBlinking chimeras in globally coupled rotators
Richard Janis Goldschmidt,
1, 2
Arkady Pikovsky,
1, 3 and Antonio Politi Department of Physics and Astronomy, University of Potsdam, Potsdam 10623,Germany Institute of Pure and Applied Mathematics, University of Aberdeen, Aberdeen AB24 3FX,UK Department of Control Theory, Lobachevsky University Nizhny Novgorod,Nizhny Novgorod 603022, Russia (Dated: 16 July 2019)
In globally coupled ensembles of identical oscillators so-called chimera states can be ob-served. The chimera state is a symmetry-broken regime, where a subset of oscillatorsforms a cluster, a synchronized population, while the rest of the system remains a collec-tion of non-synchronized, scattered units. We describe here a blinking chimera regime inan ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It ischaracterized by a death-birth process, where a long-term stable cluster of four oscilla-tors suddenly dissolves and is very quickly reborn with a new, reshuffled configuration.We identify three different kinds of rare blinking events and give a quantitative character-ization by applying stability analysis to the long-lived chaotic state and to the short-livedregular regimes which arise when the cluster dissolves.PACS numbers: 05.45.Xt Synchronization; coupled oscillators, phase dynamics1 a r X i v : . [ n li n . C D ] J u l oupled oscillators can synchronize if the coupling is attractive, or desynchronize if the cou-pling is repulsive. This basic effect is captured by the famous Kuramoto-Sakaguchi modelof phase oscillators. However, if inertia is included, i.e. the units are rotators, more com-plex regimes in between synchrony and asynchrony can be observed. One such regime is achimera pattern, where some rotators form a fully synchronous, perfect cluster, while theothers are non-synchronized and all mutually different. In this paper we report one such achimera characterized by an interesting additional long-time dynamics. We call it a blink-ing chimera because every once in a while an event occurs where the cluster opens up andquickly closes into a new reorganized composition. This event takes place on a time scalemuch shorter than that of the very long chaotic transient that is the chimera pattern — thesystem blinks . We describe in detail how the exchange between the cluster and desynchro-nized units takes place.I. INTRODUCTION In the last years coupled oscillators proved to exhibit a very rich variety of regimes, rang-ing from perfect synchronization to extremely homogeneous asynchrony. The most intriguingregimes are the intermediate ones, especially when the oscillators spontaneously split into distinctgroups/clusters. Among them, chimera states are currently attracting a large interest. They arecharacterised by the coexistence of synchronized and desynchronized groups of identical oscil-lators which, in spite of their indistinguishability, do not all behave in the same way. The firstchimera was discovered by Kuramoto and Battogtokh in a one-dimensional medium of nonlo-cally coupled phase oscillators. Since then, many setups have been found, where symmetry isbroken, giving rise to the simultaneous presence of synchronous and asynchronous subsets (seethe reviews for both theoretical analyses and the description of experimental setups).Furthermore, examples of chimera regimes within rather small sets of oscillators have beenreported . Recent studies have revealed that chimera states can be quite complex. In particu-lar, in breathing chimeras , some oscillators join and leave the synchronous domain because ofoscillations of the governing order parameter.In the following sections we report on a different steady nonstationary chimera. We describea situation where a well-defined chimera persists for a very long time and is suddenly destroyed;2hortly afterwards, a new reshuffled and long-lived chimera reforms. Since the reshuffling eventsare rather short compared to the long chimera stages, we call this state blinking chimera . Note-worthy, in the dynamics of networks, the notion of blinking systems is well established . There,switchings in the coupling and/or network topology are imposed according to a pre-defined exter-nal protocol — for example periodic or random blinking. In our case, the blinking events are notpre-described, but appear spontaneously due to the dynamical rules.The paper is organized as follows: we introduce the model in Sec. II, and describe the phe-nomenology of blinking events in Sec. III. Next, we develop a quantitative characterization ofblinking in Sec. IV. We conclude with a discussion in Sec. V. II. THE BASIC MODEL
The Kuramoto-Sakaguchi model (KS model from now) of globally coupled phase oscillators isa widely used system to study synchronization phenomena, see reviews . For identical units,the KS model is exactly solvable , yielding either complete synchronization (if the couplingis attractive) or an asynchronous state with vanishing order parameter (if the coupling is repul-sive). The exact solution shows that chimera states, characterized by the coexistence of fullysynchronous (identical) oscillators with asynchronous units, cannot arise.Integrability breaks up if the original model is perturbed. One popular extension of the KSmodel consists in including the effects of inertia on the oscillating units, i.e. in replacing phaseoscillators with rotators : α ¨ φ i + ˙ φ i = 1 N N (cid:88) j =1 sin ( φ j − φ i − β ) . (1)Here φ i is the instantaneous phase of the i -th rotator, N is their number, β is a constant phase shift(sometimes called frustration in the literature), and, finally, α is the (dimensionless) mass of therotators. A constant torque acts on all rotators and the equations are written in the reference framerotating with the corresponding constant frequency (thus the torque does not enter).At this point it is instructive to discuss an essential difference between oscillators and rotators.For example, pendula can perform both oscillations and rotations. When they are used in pendulumclocks and in metronoms , they operate as self-sustained oscillators. For these oscillations, aphase can be introduced, which is different from the angle variable φ in (1) and obeys a first-order (in time) equation. Thus, for coupled oscillators (and for all setups of coupled metronoms)3he usual first-order Kuramoto model is appropriate; the model “with inertia” cannot be used foroscillators, but for rotators only.Systems of coupled rotators have been widely discussed in the literature. In Refs. 15 and 26,diversity of torques has been shown to result in a hysteretic transition to synchrony. Effects ofnoise and diversity have been treated analytically and semi-analytically in Refs. 27–29. One ofthe popular applications of the rotator model of type (1) are power grids . In these ap-plications one does not consider a mean field coupling like in Eq. (1), but a network of differentproducers and consumers of electrical power, with different values of torque and different con-nectivities. Another much studied setup is that of symmetric deterministic models where chimerastates emerge as a result of symmetry breaking. This is the case of a one-dimensional mediumwith nonlocal coupling studied in Ref. 32 and of two symmetric subpopulations with differentcouplings (within and between them) considered in Ref. 33. Here, we consider a setup that iseven “more” symmetric, since all pair-wise interactions are equal to one another. The phenomenawe thus observe are entirely due to the prescribed dynamics and cannot be attributed to diversityamong the oscillators, network topology, or noise.For α → the system (1) reduces to the standard KS system, describing the behavior of(identical) phase oscillators. Therefore, for small α -values, the dynamics is expected to closelyreproduce that of the KS model. In the limit α → ∞ , the system converges to the Hamiltonianmean field model : a paradigmatic model for the study of long-range interactions in the presenceof a conservative dynamics.We expect interesting and potentially new phenomena to arise in the region where attraction andrepulsion nearly balance each other. This is indeed the parameter region where standard chimerastates are observed. More specifically, we have selected β = 0 . · π , which corresponds to aweakly repulsive coupling in the KS model, while in the KS model with inertia neither the fullysynchronous nor the splay states are stable. Additionally, in order to investigate the role of inertia,we have selected a finite and relatively large mass α = 10 . As for the number of oscillators, weassume N = 7 : it is the smallest system size for which blinking chimeras have been observed. For N < we have observed only either simple clustered or fully disordered states.4 II. PHENOMENOLOGY
In this section we qualitatively describe chimera states and their blinking; a quantitative char-acterisation and a more thorough analysis is postponed to the next section.The equations of motion (1) have been simulated by implementing a standard 4th order Runge-Kutta integrator with a timestep of ∆ t = 0 . . Phases φ i and frequencies ˙ φ i are initialized bydrawing them from random distributions, φ i ∈ U (0 , π ) , and ˙ φ i ∈ N (0 , . Moreover, we haveintroduced numerical inhomogeneities on the mass α of the order ∆ α (cid:39) O (10 − ) to preventthe oscillators clumping together due to finite floating point precision. We classify the currentconfigurations by identifying clusters of oscillators in almost identical states. Two oscillatorsindexed by i and j are said to belong to the same cluster when their distance d ( i, j ) < − ,where d ( i, j ) = (cid:113) δ ( φ i − φ j ) + ( ˙ φ i − ˙ φ j ) (2)and δ = min( | φ i − φ j | , π − | φ i − φ j | ) to take into account that φ i is equivalent to φ ± π . A. Instability of the splay and the fully synchronous states
Before describing the formation of the chimera state in detail, we comment on the instabil-ity of the completely synchronous and the fully-asynchronous (splay) solutions. The completelysynchronous state is given by φ i = φ j and ˙ φ i = ˙ φ j , ∀ i, j , where all oscillators collapse into acluster. We study the stability of this cluster via the transversal Lyapunov exponent (which tells uswhether a virtual pair of oscillators would be repelled or attracted by the cluster, see Sec. IV andEq. 4 below for a discussion). The resulting second order equation for the transversal perturbation α ¨ δ + ˙ δ + cos βδ can be solved analytically. The exponents are λ , = ( − ± √ − α cos β ) / (2 α ) ,yielding for the parameters under consideration λ ≈ . and λ ≈ − . . Hence, the fullysynchronous cluster is unstable with transversal Lyapunov exponent λ ≈ . .The splay state is characterized by the frequencies ˙ φ i = 0 , ∀ i , and an equidistribution of phaseson the unit circle, φ i = i · π/N, i ∈ { , N − } , N = 7 being the number of oscillators inthe system. It is difficult to analyse the instability of this steady state analytically, but numericalexploration is straightforward: one easily observes that a small (of order − ) perturbation growsexponentially with the exponent λ ≈ . . Thus, both the fully synchronous cluster and the splaystate are unstable. 5 . Formation of a chimera state The free evolution from random initial conditions leads to the formation of a chimera state,where four oscillators clump together to form a cluster, while the three other oscillators remainisolated from one another (we indeed refer to them as to isolated units ). We denote this chimerastate as . It appears that this state is a global attractor, as in our numerical simulationswe never observed other configurations formed from random initial conditions. Chimera statesof this type have been observed in globally coupled identical phase oscillators with delay and inglobally coupled Stuart-Landau oscillators . The average time for the formation of a chimerais ≈ . · . The corresponding dynamics is chaotic, as qualitatively visible in the left panel ofFig. 1, where we plot the time series of the rotator velocities ˙ φ i . Here all units are chaotic: thosebelonging to the cluster and the isolated ones. On a more quantitative level, the correspondingLyapunov spectrum is plotted in the right panel. It is composed of 14 exponents, two of whichare indeed larger than zero, two vanish (due to invariance under time translation and under ahomogeneous shift of the phases), while all the others are negative. -0.8-0.6-0.4-0.2 0 0.2 0 50 100 150 200 250 v e l o c i t i e s time i -0.2-0.15-0.1-0.0500.05 λ ι FIG. 1. Left panel: time series of rotator velocities in the chaotic chimera state. Red, blue and green curves:isolated (not belonging to a cluster) units, black: cluster of 4 units. Right panel: the full Lyapunov spectrumin the chaotic regime (green symbols, partially overlapped with black ones). It has two positive Lyapunovexponents (thus this regime can be characterised as hyperchaos), two zero LEs (due to two invariances - withrespect to shift of time and with respect to shift of all phases), with all other exponents being negative. Onecan see a degeneracy due to the presence of a cluster: there are two groups of three equal LEs ( i = 5 , , and i = 9 , , ), which correspond to the transversal directions of the cluster, see discussion of the transversalLEs below. Other exponents (black circles) coincide with those of the reduced system (3), where the clusterconfiguration is fixed so that the system is 8-dimensional and has 8 LEs.
6n the spectrum, we also notice two triples of identical negative exponents. As confirmedbelow (see Sec. IV), they account for the transversal stability of the 4-cluster. The remainingeight Lyapunov exponents (see the filled black circles in Fig. 1) contribute to the dimension ofthe underlying attractor. By virtue of the Kaplan-Yorke formula, D KY ≈ . represents an upperbound to the information dimension.No other configurations have been observed in the system (1) for the same parameter values —neither fully synchronous states, nor chimera-type configurations with 2, 3, 5, or 6 elements in thecluster. C. Blinking of chimera
The regime described above is observed on a relatively long time scale, but it is notthe asymptotic one. Over very long time scales, one observes a picture like in Fig. 2: rare eventslead to a reshuffiling of the cluster composition, with some oscillators leaving the cluster andothers joining. These reshufflings are observed continuously and they are apparently independentrandom events following a Poisson process with a rate ≈ . · − (corresponding to a mean time ≈ . · between the events). This follows from an exponential distribution of inter-event timeintervals, presented in Fig. 3.The observed pattern of reshuffling events shows that the chimera state is not stationary, butblinking. Below we provide further illustrations of this blinking, and characterise it more quanti-tatively. D. Destruction and re-formation of a 4-1-1-1 chimera state
Here we describe in detail, mostly qualitatively, what happens during the blinking (reshuffling)events. In fact, we have found that three different reshuffling scenarios can happen; they arepresented in Figs. 4, 5, and 6. We refer to these cases as to A, B, and C. We first describe scenarioA with reference to Fig. 4; it will then be straightforward to explain also the other two scenarios.In the two panels of Fig. 4 we show distances between oscillators, defined according to Eq. (2),as a function of time (time offset is chosen arbitrarily at some instant around 1000 units prior thestart of reshuffling). In this event, the initial 4-cluster configuration contains the units 4, 5, 6, and7, while the reshuffled 4-cluster contains the units 2, 4, 6, and 7. Accordingly, in the top panel we7 × × × time o s c ill a t o r (a) (b) (c) (d) (e) FIG. 2. Pattern of the oscillators belonging to the four-oscillator cluster (marked by black squares whichare seen as bold lines) versus time. One can clearly see five epochs with different cluster compositions. Inepoch (a), oscillators 1, 2, 3, and 5 belong to the four-cluster. In epoch (b), oscillators 2, 3, 4, and 5. In (c),oscillators 1, 2, 3, and 7. In (d), oscillators 1, 2, 3, and 4. And finally in epoch (e), oscillators 1, 2, 3, and 6belong to the cluster. p r ob ( d T > t i m e ) time FIG. 3. Distribution of time intervals between different reshuffling events depict distances from unit 4, which belongs to the cluster both prior and after reshuffling (in thesame way, we always choose the reference unit as belonging to the cluster prior and after the eventin top panels of Figs. 5,6). In the bottom panel we show distances between all the three pairs ofunits not belonging to the cluster prior to reshuffling.In the top panel of Fig. 4, one can appreciate the presence of a chaotic state for t (cid:46) . It is followed by a regular regime during which the isolated unit 2 comes rather close tothe 4-cluster. In this regime, the 4-cluster is unstable and starts dissolving. Meanwhile, the units 1and 3 come close to each other. The dissolution of the 4-cluster, accompanied by the appearance8 − − − − − − − − − − − − − − − − D i s t a n ce s t oo s c ill a t o r λ s = 0 . λ r = − . D i s t a n ce s Time t (1, 2)(1, 3)(2, 3) λ c = − . λ g = 0 . FIG. 4. Reshuffling of a chimera state: scenario A. Top panel: distances of all oscillators fromoscillator 4. Isolated oscillators at the beginning are 1, 2, 3, distances between them are depicted in thebottom panel. Around t ≈ , a regular regime emerges from the chaotic state, with oneisolated oscillator (here unit 2). Simultaneously, the 4-cluster begins to disintegrate (with a rate λ s ) andtwo of the initially isolated units come close and form a 2-cluster state (here units 1,3). (bottom panel, rate λ c ). Disintegration ends at t ≈ in a chaotic state, where the units 2, 4, 6, 7 form a not so perfect4-cluster (mutual distances are ≈ − ), while the units 1, 3 form a 2-cluster (see bottom panel), and unit5 is isolated. The 2-cluster begins to disintegrate with a rate λ g ; this disintegration ends around t ≈ .From this moment on, the new 4-cluster becomes more stable, the mutual distances reduce with the rate λ r .At the end of the event, at t ≈ , a reshuffled configuration appears. of a 2-cluster emerging from the isolated units, continues until t ≈ . Afterwards, the 2-clusterdissolves and the dynamics again become chaotic. Around t ≈ , no clusters are observed.Four units are close to each other, although they do not fulfill our criterion for the definition ofa cluster. However, they start approaching each other and a new 4-cluster eventually forms in9 − − − − − − − − − − − − − − − − D i s t a n ce s t oo s c ill a t o r λ s = 0 . λ r = − . D i s t a n ce s Time t (2, 3)(2, 5)(3, 5) FIG. 5. Reshuffling of a chimera state: scenario B. Top panel: distances of all oscillators fromoscillator 4. Isolated oscillators at the beginning are 2, 3, 5; distances between them are depicted in thebottom panel. The process is qualitatively similar to that of Fig. 4, but here in the regular stage around t ≈ two isolated oscillators (2 and 5) are both close to each other and to the still existing cluster; theyeventually join the new 4-cluster. the same way as observed when starting from random initial conditions (cf. Section III B above).Typically, the unit which was already close to the 4-cluster around t ≈ , joins the novelcluster, “exchanging” with one unit that leaves the cluster. However, in some cases the temporary2-cluster does not dissolve but enters the novel 4-cluster, exchanging with two units therein.The blinking event shown in Fig. 5 is quite similar to that of Fig. 4, with the following differ-ences: (i) now, during the regular state arising at the beginning of reshuffling (around t ≈ ),not one, but two isolated units orbit close to the 4-cluster (here units 2 and 5); (ii) the disintegra-tion of the 4-cluster is much faster than in Fig. 4; (iii) the two isolated units that were close to the4-cluster join the new cluster, so there is always a ↔ exchange.10 − − − − − − − − − − − − − − − − D i s t a n ce s t oo s c ill a t o r λ s = 0 . λ r = − . D i s t a n ce s Time t (2, 3)(2, 5)(3, 5) FIG. 6. Reshuffling of a chimera state: scenario C. Top panel: distances of all oscillators fromoscillator 1. Isolated oscillators at the beginning are 2,3,5; distances between them are depicted in thebottom panel. At the end units 1,2,4,5 belong to the 4-cluster and units 3,6,7 are isolated.
The blinking event in Fig. 6 is different from those in Figs. 4, 5: here there is no formationof a temporary 2-cluster. The break-up of the 4-cluster (at (cid:46) t (cid:46) ) is faster than incase Fig. 4, but slower than in case Fig. 5. At the end of this process, all the units are separated,and a new 4-cluster begins to form, with a ↔ exchange. Sometimes we observed that all 3previously isolated units join the new cluster, so that also a ↔ exchange is possible. On theother hand, since the formation of a new cluster is a statistical process, it can happen that the finalconfiguration is formed of the same units as the initial one.11 V. QUANTIFYING BLINKING EVENTS
The above description of the reshuffling process suggests the existence of regular (nonchaotic)temporary stages. To resolve them, we proceed by performing simulations with four units glued together to enforce the presence of a 4-cluster at all times. Such a state (and, more generally, anyclustered state) can be modelled also with reduced equations, the identical elements of a clusterbeing described by just one set of variables ( ϕ, ˙ ϕ ) . Generally, if N rotators build K clusters ofsizes n , n , . . . , n K , with (cid:80) k n k = N , then the equations (1) can be rewritten as a smaller reducedsystem of K equations α ¨ ϕ k + ˙ ϕ k = Im (cid:0) Z e − iϕ k − iβ (cid:1) , Z = K (cid:88) k =1 n k N e iϕ k . (3)In our case of a chimera, we have K = 4 with n = 4 , n = n = n = 1 . The reducedsystem (3) has the same dynamics as the original system (1) so long as the 4-cluster persists.In fact, simulations of the chimera state with Eqs. (3) reveal that this regime isnothing but a very long pseudo-stationary chaotic transient . The dynamics, eventually, collapsesonto one of the following four attractors A: a quasiperiodic state, characterized by a 2-cluster, while the isolated oscillator orbitsclose to the 4-cluster; B: a quasiperiodic state such that the 2-cluster orbits close to the 4-cluster; C: a periodic state, where three units remain asymptotically isolated; D: a periodic state.Upon performing 1000 simulations of the reduced model, starting from random initial con-ditions, we found that roughly 54%, 31%, 13%, and 2% converge to attractor A, C, B, and D,respectively. We have used the notations A, B, C, because these regular states correspond to thescenarios A, B, and C discussed in section III D. As for the probability to observe the various sce-narios in the original blinking dynamics, they are approximately equal to the above reported rates,the major difference being that the state D has been never observed in the original system (1),presumably because the 4-cluster is destroyed prior to the formation of the 3-cluster.The reason why the attractors of the reduced model are not seen as such in the simulationsof the global system is that the 4-cluster is transversally unstable in all of the above scenarios.12ransversal Lyapunov exponents can be determined by perturbing only the variables of the cluster ( ϕ k , ˙ ϕ k ) , while leaving the mean field Z unchanged (cf. the general discussion of transversalLyapunov exponents in Ref. 38). So the linear equation for a deviation ( δ k , ˙ δ k ) from cluster k reads α ¨ δ k + ˙ δ k = − δ k Re [ Ze − iβ − iϕ k ] . (4)In practice, the above equation, solved together with Eq. (3), tells us whether a virtual pair ofoscillators leaving cluster k would be either attracted or repelled by the cluster. As Eq. (4) istwo-dimensional, it yields two transversal Lyapunov exponents; we are interested in the maximalone that can be computed in the usual way by virtue of the Benettin algorithm , i.e. by regularlyrenormalizing the vector ( δ k , ˙ δ k ) and averaging the logarithm of the norm.The implementation of this approach during the pseudo-stationary transient evolution of the configuration yields two values which coincide with the two triples visible in the gen-eral Lyapunov spectrum plotted in Fig. 1 and confirm that the 4-cluster is stable on average. In thecase of the above mentioned four attractors, we instead find that the largest transversal Lyapunovexponent is: λ At ≈ . [for case A]; λ Bt ≈ . [B]; λ Ct ≈ . [C]; λ Dt ≈ . [D], thusconfirming the instability of the 4-cluster.Regular attractors do not only qualitatively correspond to the initial states of the blinking eventsdescribed in Section III D, but give also a quantitative description of the disintegration of the 4-cluster: the growth rates of the inter-oscillator distances in Figs. 4, 5, 6 correspond to the valuesof the transversal Lyapunov exponents for cases A-C.Additional insight into cases A and B can be obtained from the computation of the transversalLyapunov exponents of the 2-cluster in the corresponding configuration. In both cases itsvalue is ≈ − . . This quantity describes the rate λ c with which 2-cluster is formed, cf. Figs. 4and 5.The formation of a new 4-cluster from a non-clustered chaotic regime is a statistical event.However, its late stage, where the cluster is basically formed and the units are progressively ap-proaching each other, can be again compared with the transversal Lyapunov exponents. In thiscontext, the stable transversal exponent λ t ≈ − . of the 4-cluster in the chaotic regime (Fig. 1)is relevant, as it gives approximately the convergence rate λ r in Figs. 4, 5, 6.Finally, the rate of disintegration of the two-cluster λ g (bottom panel of Fig. 4) can be explainedas follows: this regime corresponds to a temporary chaotic state in the fixed configuration .Calculations of the transversal Lyapunov exponent of the 2-cluster here are not reliable, as this13egime quite soon ends in one of the A, B, D states. However, if one starts the configuration from random initial conditions, in many runs one observes that during the initial stagesthe transversal Lyapunov exponent of cluster 4 fluctuates around zero, while the transversal LEof cluster 2 is ≈ . . This value corresponds to the rate λ g of disintegration of the 2-cluster inFig. 4.Summarizing, we explained the origin of blinking events via an interplay of two propertiesof the system: structural, i.e. the composition of clusters, and dynamical, i.e. the complexity ofthe dynamics and the resulting stability characteristics of clusters. During long epochs a chaoticregime with stable clustering is observed. However, after a long but finite time, chaos issucceeded by a regular regime, and this triggers a blinking event: first, the big cluster becomes un-stable and dissolves, and then from a chaotic disordered state a new chimera state with a reshuffledcomposition emerges. V. DISCUSSION AND CONCLUSIONS
In this paper we reported on a novel state of blinking chimera in a small system of seven iden-tical rotators (phase oscillators with inertia). The asymptotically stationary regime consists of asequence of long epochs each characterized by a (temporary) chaotic chimera state with four os-cillators synchronized into a single cluster, and three isolated ones. Such regimes are separatedby relatively short reshuffling events when the composition of the synchronous cluster is recon-figured. These rare events, which we call blinking events , appear to be distributed according to aPoisson process with a very small, but finite rate. There are three types of such events (see theabove described scenarios A, B, and C); all of them are characterized by a regular (either periodicor quasiperiodic — depending on the scenario) dynamics.Altogether, the chaotic chimera state is not an attractor, but rather a transient chaotic state,eventually ending in a temporary regular dynamics. The emergence of long chaotic transientsis a well known phenomenon in nonlinear dynamics : they typically arise because of “holes”in phase space, where the trajectory suddenly jumps out of the pseudo-attractor. Identifying thespecific conditions for these events to occur is not an easy task: we leave it to future investigations.What makes the regime discussed in this paper different from standard chaotic transients is thatonce the temporary chaotic state is over, another equivalent such regime emerges. In fact, the threetypes of exit events all lead to unstable attractors. In practice, during the blinking event, the cluster14s transversally unstable and it thereby starts disintegrating, leading to a short-lived non-chimerastage. A new chimera configuration of the type finally forms, due to the transversalstability of this regime.In order to clarify the quantitative properties of these processes, we explored the dynamics ofrestricted systems with fixed cluster compositions. This allowed us to calculate, via time averag-ing, the transversal Lyapunov exponents governing the stability of the clusters, without destroyingthe clusters themselves. The fixed configuration allowed us to determine the basic un-stable transversal Lyapunov exponents governing disintegration of the main 4-cluster. In somecases, an intermediate 2-cluster is formed, with this formation being governed by the stable Lya-punov exponent of the 2-cluster in the fixed configuration. Finally, the rate of formation ofthe new 4-cluster from the unclustered chaotic state is governed by the stable transversal Lyapunovexponent of the 4-cluster in the chaotic transient state of the configuration.From the general viewpoint of topology of the dynamics in the phase space of the system, theblinking chimera can be described as follows: there is an invariant manifold where the states offour oscillators coincide, while three differ. This 8-dimensional manifold is attractive for a setof large measures in the original 14-dimensional phase space, but is not a global attractor. Onthis invariant set a chaotic transient (chaotic saddle) sets in, characterized by a very long lifetime:this regime corresponds to a chaotic chimera. Eventually, generic trajectories leave the chaoticsaddle and approach one of the sets (A, B, C) all characterized by a regular dynamics. On theseregular sets the 8-dimensional manifold is transversally unstable, so that trajectories leave it (thechaotic chimera is destroyed), but then enter again the domain of attraction of the chaotic 8-dimensional saddle, and a new, reshuffled chimera is established. Noteworthy, intermittent chaoticchimeras have been reported for a two-population setup . However, no reshuffling, and thus alsono blinking, was observed therein.Preliminary simulations suggest that this phenomenon is not peculiar of the parameters selectedin this paper. However, in our simulations we never observed blinking chimera for less thanseven units. Therefore, in this short communication we restricted ourselves to a description of theminimal blinking chimera and do not discuss other possible dynamical states of this system.15 CKNOWLEDGMENTS
We thank M. Rosenblum and Yu. Maistrenko for useful discussions. This work has been fundedby the EUs Horizon 2020 research and innovation programme under the Marie Sklodowska CurieGrant Agreement No. 642563. A.Pik. acknowledges support of the Russian Sceince Foundation(Grant 17-12-01534).
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