Chimera-like behavior in a heterogeneous Kuramoto model: the interplay between the attractive and repulsive coupling
Nikita Frolov, Vladimir Maksimenko, Soumen Majhi, Sarbendu Rakshit, Dibakar Ghosh, Alexander Hramov
aa r X i v : . [ n li n . AO ] A ug Chimera-like behavior in a heterogeneous Kuramoto model:the interplay between the attractive and repulsive coupling
Nikita Frolov, a) Vladimir Maksimenko, b) Soumen Majhi, c) Sarbendu Rakshit, d) Dibakar Ghosh, e) and Alexander Hramov f) Neuroscience and Cognitive Technology Laboratory, Centerfor Technologies in Robotics and Mechatronics Components,Innopolis University, 420500, Innopolis, The Republic of Tatarstan,Russia Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road,Kolkata-700108, India (Dated: 18 August 2020)
Interaction within an ensemble of coupled nonlinear oscillators induces a va-riety of collective behaviors. One of the most fascinating is a chimera statewhich manifests the coexistence of spatially distinct populations of coherentand incoherent elements. Understanding of the emergent chimera behavior incontrolled experiments or real systems requires a focus on the considerationof heterogeneous network models. In this study, we explore the transitionsin a heterogeneous Kuramoto model under the monotonical increase of thecoupling strength and specifically find that this system exhibits a frequency-modulated chimera-like pattern during the explosive transition to synchro-nization. We demonstrate that this specific dynamical regime originates fromthe interplay between (the evolved) attractively and repulsively coupled sub-populations. We also show that the above mentioned chimera-like state isinduced under weakly non-local, small-world and sparse scale-free couplingand suppressed in globally coupled, strongly rewired and dense scale-free net-works due to the emergence of the large-scale connections.PACS numbers: 05.45.-a, 05.45.Xt, 87.10.-e
Synchronization phenomena in populations of interacting elements are the sub-ject of extensive research in biological, chemical, physical and social systems.The process of synchronization refers to the adjustment of rhythms of interact-ing oscillatory systems, whereas chimera states are characterized by the fasci-nating coexistence of coherent and incoherent sub-populations in networks ofcoupled oscillators. On another note, discontinuous or explosive transitions tocoherence in networks are receiving growing attention these days. The paradig-matic Kuramoto model being able to provide the most effective approach toexplain how synchronous behavior emerges in complex systems, there existssignificant attempts in exploring both chimera states and explosive transitionto synchrony. But, in most of the studies, these two processes have been stud-ied exclusively, without paying attention to a possibility in linking them. Incontrast to approaches solely concentrating on abrupt transitions to synchronyand the associated hysteresis, we here put forward the emergence of chimera-like behavior on the route to an explosive transition in networks of coupledKuramoto phase oscillators. Complex systems naturally display heterogeneityin its constituents, so in this article, we consider a heterogeneous Kuramotomodel and report a frequency-modulated chimera-like pattern during discon-tinuous transitions to coherence. We reveal that this chimera-like behavior ap-pears due to a coexistence of evolved (not induced) attractively and repulsively a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected] e) Electronic mail: [email protected] f) Electronic mail: [email protected] coupled populations of oscillators. We further establish that the uncovered typeof chimera-like state is excited under weakly non-local, small-world and sparsescale-free coupling and suppressed in globally coupled, strongly rewired anddense scale-free networks.
I. INTRODUCTION
Network science provides a universal language to create relevant models and understandthe behavior of complex systems . Among diverse dynamical phenomena, i.e., synchro-nization, adaptation, clustering, etc. performed by the complex network models, chimerastate is one of the most intriguing types of collective behavior. Originally, it implies thecoexistence of coherent and incoherent populations in a symmetrically coupled ensemble ofidentical nonlinear oscillators .For almost two decades from its discovery , many aspects of this specific dynamical regimewere explored in detail. Specifically, chimera patterns were demonstrated to be a universalphenomenon for the models of different nature, including phase oscillators , oscillators withinertia , chaotic systems , biological neurons based on the Hodgkin-Huxley , FitzHugh-Nagumo , Hindmarch-Rose models. Several remarkable fundamental effects suchas coherence-resonance chimera and virtual chimera were discovered in the last fewyears. Chimeras were also shown to be robust against the topology and reported in globallycoupled , hierarchical , scale-free and small-world networks , multilayer andmultiscale networks , and even hypergraphs . For a long time observed only in the modelsystems, chimera patterns were experimentally verified in the mechanical , chemical ,and optical setups.The chimera behavior is still closely studied as it fits the dynamics of various real-lifesystems, i.e., social and biological systems, power grids etc. Special interest ispaid to the application of chimeras in neuroscience , since spatio-temporal coherence is acornerstone of the normal and pathological brain activity . Earlier, chimera patternswere observed in animals’ neural networks . In humans, such forms of the brain activityas epileptic seizures , Parkinson’s and Alzheimer’s disease , bump states , cognitivefunctions and resting-state are shown to perform the pronounced properties of chimerabehavior.However, the approach to more realistic models requires consideration of non-homogeneousensembles since the condition of elements’ identity is hardly fulfilled in the real networks.Several studies addressed the problem of network heterogeneity in the context of chimerabehavior. Specifically, bifurcation analysis of the Kuramoto network with heterogeneousintrinsic frequencies was performed by Laing . Based on the results of numericaland analytical treatment, the author concluded that chimera is robust to such type ofheterogeneity. Nkomo et al. demonstrated the chimera state in the ensemble of heteroge-neous Belousov-Zhabotinski oscillators both numerically and experimentally. Several worksreported that the chimera state could be induced in the presence of phase-lag heterogene-ity . Chimera state was also explored in networks with irregular topology . On theother hand, intense research efforts have also been made in order to study mechanisms thatlead to discontinuous or explosive transition to synchrony .Despite the above discussed extensive studies on chimera behavior, even simply con-structed complex networks still hide unexpected aspects of this phenomenon due to hetero-geneity of its elements. In this paper, we report the emergence of a frequency modulatedchimera-like behavior in a non-homogeneous Kuramoto model during an explosive transi-tion of the networked system to a certain level of coherence. We argue that the uncoveredchimera-like behavior occurs in weakly non-local, small-world (SW) and sparse scale-free(SF) coupling. We demonstrate that it originates from the self-organization of the entireensemble into attractively and repulsively coupled populations. II. MATHEMATICAL MODEL
We consider a network of N number of phase oscillators, in which the dynamics of eachnode is represented by the following form of the Kuramoto equation:˙ φ i = ω i + λR i N X l =1 A il sin( φ l − φ i ) ,R i = 1 k i (cid:12)(cid:12)(cid:12)(cid:12) N X l =1 A il e j φ l (cid:12)(cid:12)(cid:12)(cid:12) , (1)where φ i , ω i and k i are the phase, natural frequency and the degree of the i th Kuramotooscillator respectively, also j = √−
1. For further simplicity, let us introduce the notationfor the effective frequency of the i th oscillator as f i = ˙ φ i . The parameter λ is the overallcoupling strength. The matrix A = [ A il ] is the underlying graph adjacency. In the case ofregular and SW coupling, it is generated using the Watts-Strogatz (WS) algorithm with k nearest neighbors (in each side of a one-dimensional ring) and the probability p of addinga shortcut in a given row . The SF adjacency matrix is generated using Barab´asi-Albert(BA) algorithm with the growing parameter m . R i represents the local order parameterand evaluates the degree of coherence in the neighborhood of the i th element. It contributesadiabatically to the coupling term and provides the mechanism for explosive synchroniza-tion. The values of ω i are uniformly distributed over the range [ ω − ∆2 , ω + ∆2 ], where ω is the central frequency and ∆ is the width of the frequency range.To quantify the network’s coherence, we use the averaged global order parameter as R = 1 N ( t max − t trans ) Z t max t trans (cid:12)(cid:12)(cid:12) N X l =1 e j φ l ( t ) (cid:12)(cid:12)(cid:12) dt, (2)where t trans and t max respectively denote the transient time and maximal simulation time.Moreover, we illustrate the collective behavior of the Kuramoto model using the meaneffective frequency h f i i defined by time averaging instantaneous effective frequency f i ( t )after the transient process. III. RESULTS
Specifically, we consider the dynamical network (1) consisting of N = 100 oscillators.The value of the central frequency is fixed at ω = 10. The network model simulation isconducted using the Runge-Kutta method of order 5(4) implemented in the DifferentialEquation Solver for Julia programming language . To control the accuracy of the numericalintegration, we use the adaptive time-stepping with relative tolerance parameter equal to10 − , maximal simulation time t max = 2000 and transient time t trans = 1500. A. Observation of the chimera-like behavior
Depending on the level of heterogeneity, i.e., the width ∆ of the natural frequency dis-tribution, we observe different transitions to coherence in a Kuramoto model under theadiabatically increasing coupling strength λ (Fig 1). Obviously, an ensemble with a homo-geneous frequency distribution, i.e., for ∆ = 0, the coupled system (1) undergoes a smoothtransition to coherence at very small values of the coupling strength. The introduction ofheterogeneity in the considered network system (cf. Eq. (1)) leads to the explosive transitionto coherence. Here, the incoherence for the values of coupling strength below the criticalpoint λ cr , is supported by the low degree of local synchrony R i that reduces the value ofcoupling term in Eq. (1). Interestingly, a heterogeneous Kuramoto model does not converge λ Δ R incoherent coherent Δ =0.5 λ R Δ =1.0 Δ =1.5 (b)(a) FIG. 1. (a) Averaged global order parameter R versus the coupling strength λ in the non-locallycoupled network of N = 100 oscillators with p = 0 . k = 10 for different values of naturalfrequency distribution width: ∆ = 0 . . . λ, ∆)parameter plane for the global order parameter R , color bar represents its variation. to a global frequency-locking ( π -state) immediately after the explosive transition. Instead,we find a finite-size plateau, where the Kuramoto model exhibits a partially coherent statewith the averaged order parameter R ≈ .
7. As seen in Fig 1(a), the way of transition doesnot depend on the degree of heterogeneity ∆. Notable, that in the case of higher values of∆, the transition occurs at the greater values of the critical coupling strength λ cr and it isfollowed by a wider ‘partially coherent’ plateau.To illustrate the effect of the coupling strength λ on this chimera-like state for continuousvariation of ∆, we plot the global order parameter R in the ( λ, ∆) parameter plane inFig. 1(b). The region between the dashed white and black lines reflects the existence ofchimera-like state. However, the yellow and black regions respectively correspond to thecoherent and incoherent states. The figure explicitly demonstrates the interval of λ forwhich chimera-like state emerges. Interestingly, this interval that supports the chimera-likestate improves considerably as ∆ increases. Beyond certain values of the coupling strength λ (depending on the width ∆), the coupled Kuramoto oscillators undergoes the coherentstate and persists further.Let us now take a close look at the transitions in the considered Kuramoto model. With-out any loss of generality, we fix ∆ = 1 . λ , in terms of the averaged global order parameter R (cf.Fig. 2a) and the distribution of mean effective frequencies h f i i (cf. Fig. 2b). It is seen thateven at λ = 0 .
02 effective frequencies remain uniformly distributed over the ensemble andare almost unchanged with respect to the initial distribution of natural frequencies, so that h f i i ≈ ω i , i = 1 , , . . . , N . While coupling strength approaches the critical value of explosivetransition λ cr = 0 . ω . After the critical explosive transition at λ cr = 0 . N coh undergoes the abrupt frequency-locking, so that h f i i ≈ ω for all i ∈ N coh . At the same time, a group of oscillators N inc remains desynchro-nized, i.e., |h f i i − ω | >> i ∈ N inc . Thus, the balance between the heterogeneityof natural frequencies and the coupling strength, which is insufficient to provide a globalcoherence, supports a partially coherent state in a non-homogeneous Kuramoto ensemble.However, the sharp increase of the network’s coherence boosts faster convergence of theremaining part of oscillators to a globally frequency-locked state at λ cr = 0 . R on the coupling strength λ has two peaks in the area, where the network exhibits partiallycoherent state. It reflects the switching between two distinct regimes of partial coherence.Let us consider the latter in detail by tracking the network’s behavior at points A ( λ =0 . λ = 0 . h f i i (top row) along with the corresponding space-time plots color-coded by the instantaneous effective frequency f i (bottom row). We find R λ λ < f i > (a)(b)incoherent coherent ( π -state) partially coherent A B Δ = 1.00.030.03 10.59.510.0 f i ( t ) λ=0.048 i t , < f i > λ=0.051 i t , < f i > (c) (d) FIG. 2. (a) Averaged global order parameter R and (b) the distribution of mean effective frequencies h f i i versus the coupling strength λ in the heterogeneous non-locally coupled network ( p = 0 . k = 10 and ∆ = 1 . h f i i profiles (top) and the space-time plots of the instantaneous effective frequency f i (bottom) for the different values of the coupling strength λ corresponding to points A and B: (c) λ = 0 . λ = 0 . that both partially coherent states that occurred after the critical transition represent aspecific form of a frequency-modulated ‘chimera-like’ behavior. Specifically, we observe thecoexistence of two distinct clusters: a larger one that is frequency-locked and follow a smoothcoherent spatiotemporal profile, however the smaller one evolves in a drifting-like manner.Here, we intentionally refer this regime to as a ‘chimera-like’ behavior, since it differs fromthe classical definition of the ‘chimera’ mostly because we here consider a heterogeneousensemble of phase oscillators. Also, the traditional chimera state implies coherence in termsof the phase-locking, instead of the frequency-locking reported here. Despite that, we stillobserve the relevant feature of chimera behavior in the uncovered network dynamics, i.e.,the coexistence of spatially dissociated groups of coherent and incoherent network elements,that gives us a fair basis to determine the uncovered phenomenon as a chimera-like state.Interestingly, the observed chimera-like regimes are not stationary – the incoherent clusterappears and collapses in time. The way of evolution in time determines the differencebetween these partially coherent states. The regime at λ = 0 .
048 formed after the criticaltransition and presented in Fig. 2c is characterized by the fast and irregular burst-likeoscillations of the incoherent cluster. On the contrary, an increase of the coupling strength λ switches the chimera-like regime to slow and periodic oscillations (Fig. 2d). < f i > i < f i > < f i > < f i > < f i > < f i > c i < f i > < f i > < f i > < f i > w i c i w i w i w i w i λ=0.02λ=0.045λ=0.048λ=0.051λ=0.06(a)(b)(c)(d)(e) i -0.5 0.0 0.5 c i -0.5 0.0 0.5 c i i -0.5 0.0 0.5 c i -0.5 0.0 0.5 c i i -0.5 0.0 0.5 c i -0.5 0.0 0.5 c i i -0.5 0.0 0.5 c i -0.5 0.0 0.5 c i repulsive attractive repulsive attractiverepulsive attractive repulsive attractiverepulsive attractive repulsive attractiverepulsive attractive repulsive attractiverepulsive attractive repulsive attractive FIG. 3. Illustration of the mechanism underlying the chimera-like pattern formation in the non-locally coupled network ( p = 0 . k = 10) with heterogeneous natural frequency distribution(∆ = 1 . h f i i profile (left column), its correspondence to thecoupling term c i (middle column) and the natural frequency ω i versus the coupling term c i (rightcolumn) for different values of the coupling strength λ : (a) λ = 0 .
02; (b) λ = 0 . λ = 0 . λ = 0 . λ = 0 .
06. Blue and red colors highlight the attractive and repulsive couplingareas respectively in the middle and right columns.
B. Birth of chimera-like state: Mechanism
To understand the mechanism of the birth of chimera-like behavior in a heterogeneousKuramoto model let us rewrite the model Eq. (1) in the following form:˙ φ i = f i = ω i + c i ,c i = λR i N X l =1 A il sin( φ l − φ i ) , (3)where we introduce a notion called mean coupling term c i associated with the i th element’scoupling term in the governing Kuramoto equation.It is clear from the modified Eq. (3) that frequency-locking h f i i = Ω, where Ω is a mean-field frequency, implies ω i + h c i i = Ω, i = 1 , , ..., N . In the case of uniform natural frequencydistribution, Ω ≈ ω and, therefore, coupling term should provide the compensation of thedifference between the central and natural frequencies of the i th oscillator h c i i ≈ ω − ω i . k =40 2 k =20 2 k =14 π -stateTW-stateincreasing λ (2 k = 12)decreasing λ (2 k = 12) R λ R (a) (c) π πφ i ( t ) φ i φ i π π π π i t , π-stateTW-state (d) (e) t , i incoherent coherent (b) k R λ λ FIG. 4. (a) Averaged global order parameter R versus the coupling strength λ in the non-locallycoupled network of N = 100 oscillators with p = 0 . . k ≥
14: 2 k = 14 (black); 2 k = 20 (green); 2 k = 40 (red). Shadings highlight therespective areas of partially coherent chimera-like regimes. (b) Phase diagram in the ( λ, k ) pa-rameter plane for the global order parameter R , color bar represents its variation. (c) 2 k = 12(exemplary illustration of the network dynamics in the case of 2 k < λ (forward transition resulting in traveling wave (TW) state and red linecorresponds to decreasing λ (backward transition resulting in π -state). Illustration of the TW (d)and frequency-locked π -state (e) for λ = 0 .
2: instantaneous phase φ i profiles at t = t max (top) andtheir space-time plots (bottom). Fig. 3a shows that in the case of weak coupling strength λ = 0 .
02, mean coupling term h c i i remains approximately at the zero-level supported by the low values of local coherence R i .After the critical transition at λ = 0 .
045 (cf. Fig. 3b), the above described compensatorymechanism is explosively induced – elements with ω i < ω become attractively coupled( c i >
0) and those with ω i > ω become repulsively coupled ( c i < h f i i → Ω for all i such that c i > h f i i ≈ ω + ∆ / λ > .
054 demonstrating the expected linear relation between thenatural frequency ω i and the mean coupling term h c i i (cf. Fig. 3e). C. Influence of the network topology
Above we have considered the formation of the chimera-like state in a heterogeneous non-locally coupled network with fixed topological properties ( p = 0 . k = 10). Now, let usanalyze the influence of the network topology on the transitions in the considered networkmodel.First, we explore how the number of the nearest neighbors k affects the route to coherencein the regular non-locally coupled Kuramoto network (cf. Fig. 4a,b). Here, the previouslyconsidered network topology corresponds to a green curve. The increase of the nearestneighbors k (2 k ≥
40, red curve) suppresses the emergence of a partially coherent state. Asthe coupling term c i summarizes the influence from all elements coupled to the i th one, anincrease of k gains the coupling term c i . Besides, each element interacts with a larger groupof neighboring oscillators, that counteracts the network’s heterogeneity and contributes tothe emergence of the first-order transition. Thus, strong interaction within the large groupof elements leads to the explosive transition directly from the incoherent to a globallyfrequency-locked state in the absence of the intermediate partially-coherent state. On thecontrary, the decrease of k (black curve) promotes weaker interaction between networkelements and makes it of a more local kind. These factors strengthen the influence of thenetwork’s heterogeneity, slow down the transition to coherence and support the partiallycoherent state in a wider range of λ .For the values of k presented in Fig. 4a, the observed transitions are reversible, i.e.,the system undergoes the same transitions in both forward (increasing λ ) and backward(decreasing λ ) directions. Interestingly, the transition becomes irreversible with a furtherdecrease of k , specifically for 2 k <
14 (cf. Fig. 4c). The forward transition results in atraveling-wave (TW) solution, whose phase profile and space-time plot are presented inFig. 4d. In turn, during the backward transition, the network converges to a more stablefrequency-locked ( π -state) at the high values of coupling strength (cf. Fig. 4e). We suppose,that for 2 k <
14, the network topology exhibits pronounced local coupling properties, sothe collective dynamics represent the interaction of locally coupled populations. Such non-homogeneity of interaction in combination with the initial heterogeneity of the networkelements promote the phase lags between local interacting groups. The latter provides theconvergence to a TW-solution during the forward transition under the slowly increasingcoupling strength λ . During the backward transition, high coupling strength λ forces thenetwork to switch to a globally frequency-locked π -state. The decrease of λ causes a smoothdesynchronization of the ensemble and two solutions – TW and π -state – meet at thebifurcation point at λ = 0 . p lowers the critical value of the coupling strength λ cr providing the explosive transition and smooths the area of the partially coherent state(black and green curves for p = 0 . p = 0 .
25, respectively). In the limit case of p = 1 . π -state) at λ = 0 . m <
12) (Fig. 6). For the dense coupling m ≥
12, only an explosive transition is observed.Taken together, these results demonstrate that the detected chimera-like behavior in aheterogeneous Kuramoto model could be suppressed (i) by the increase of the neighborhoodin the case of non-local coupling, (ii) by a strong rewiring in the SW network and (iii) bygrowing a densely coupled SF graph. We argue that these ways share a similar mechanismbased on the establishment of the long-scale coupling between the network elements. Thus, R λ Rp = 1 p = 0.25 p = 0 λ incoherent coherent (b)(a) p FIG. 5. (a) Averaged global order parameter R versus the coupling strength λ in the non-locallycoupled network of N = 100 oscillators with 2 k = 20 and ∆ = 1 . p = 0 . p = 0 .
25 (green); p = 1 . λ, p ) parameter planefor the global order parameter R , color bar represents its variation. incoherent coherent λ m λ R m =12 m =10 m =8 R (b)(a) FIG. 6. (a) Averaged global order parameter R versus the coupling strength λ in the non-locallycoupled network of N = 100 oscillators with ∆ = 1 . m : m = 8 (black); m = 10 (green); m = 12 (red). Shadings highlight the respective areas ofpartially coherent chimera-like regimes. (b) Phase diagram in the ( λ, m ) parameter plane for theglobal order parameter R , color bar represents its variation. the effect of initial heterogeneity of network oscillators could be annihilated by expanding thecoupling area for each element, that provides the dominance of the attractive mechanisms. IV. CONCLUSION
To summarize, we have considered the transitions in a heterogeneous Kuramoto model,where the natural frequencies of its elements are chosen from a uniform distribution. Con-sistent with the earlier studies , we have demonstrated that non-homogeneity of theinteracting oscillators does not ruin the emergent chimera state. Moreover, it contributesto a specific type of frequency-modulated chimera-like behavior in which frequency-lockedpopulation coexists with a non-frequency-locked one. Interestingly, the observed chimera-like pattern is not stationary – depending on coupling strength a non-frequency-lockedpopulation appears and collapses in time either regularly or not. This is due to the origin ofthe chimera-like behavior. Specifically, we have shown that the interaction within the ini-tially heterogeneous ensemble of phase oscillators leads to the splitting into the attractivelyand repulsively coupled groups. While the attractively coupled elements rapidly converge toa frequency-locked state, the repulsively coupled population tends to counteract the globalfrequency-locking, thus forming an unstable incoherent cluster.Importantly, the uncovered chimera-like state has been observed in non-locally coupled,small-world and sparsely connected scale-free networks. On the contrary, in globally cou-0pled networks, networks with completely random rewiring and densely connected scale-freenetworks, the ensemble undergoes the direct transition from the incoherent state to a globalfrequency-locking. We conclude that in the latter networks, the emergence of large-scaleconnections contributes to the dominance of the attractive coupling by influencing excita-tory on a larger group of oscillators. We also hypothesize that this mechanism could beused in the real-world networks exhibiting strong rewiring of links, for example, brain neu-ral networks, to overcome the inherent heterogeneity of its elements and suppress partiallycoherent states.
ACKNOWLEDGMENTS
This work has been supported by the Russian Foundation for Basic Research (Grant No.19-52-45026) and the Department of Science and Technology, Government of India (Projectno. INT/RUS/RFBR/360).
V. DATA AVAILABILITY
All numerical experiments with a heterogeneous Kuramoto model are described in thepaper and can be reproduced without additional information.
REFERENCES S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Physics Reports , 175 (2006). D. M. Abrams and S. H. Strogatz, Physical Review Letters , 174102 (2004). Y. Kuramoto and D. Battogtokh, Nonlinear Phenomena in Complex Systems , 380 (2002). M. J. Panaggio and D. M. Abrams, Nonlinearity , R67 (2015). S. Olmi, Chaos: An Interdisciplinary Journal of Nonlinear Science , 123125 (2015). P. Jaros, Y. Maistrenko, and T. Kapitaniak, Physical Review E , 022907 (2015). I. Omelchenko, Y. Maistrenko, P. H¨ovel, and E. Sch¨oll, Physical Review Letters , 234102 (2011). S. A. Bogomolov, A. V. Slepnev, G. I. Strelkova, E. Sch¨oll, and V. S. Anishchenko, Communications inNonlinear Science and Numerical Simulation , 25 (2017). A. Andreev, N. Frolov, A. Pisarchik, and A. Hramov, Physical Review E , 022224 (2019). I. Omelchenko, A. Provata, J. Hizanidis, E. Sch¨oll, and P. H¨ovel, Physical Review E , 022917 (2015). I. A. Shepelev, T. E. Vadivasova, A. Bukh, G. Strelkova, and V. Anishchenko, Physics Letters A ,1398 (2017). S. Guo et al., Chaos, Solitons & Fractals , 394 (2018). J. Hizanidis, V. G. Kanas, A. Bezerianos, and T. Bountis, International Journal of Bifurcation and Chaos , 1450030 (2014). B. K. Bera, D. Ghosh, and M. Lakshmanan, Physical Review E , 012205 (2016). N. Semenova, A. Zakharova, V. Anishchenko, and E. Sch¨oll, Physical Review Letters , 014102 (2016). L. Larger, B. Penkovsky, and Y. Maistrenko, Physical Review Letters , 054103 (2013). L. Larger, B. Penkovsky, and Y. Maistrenko, Nature Communications , 1 (2015). A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Physical Review Letters , 144103 (2014). S. Ulonska, I. Omelchenko, A. Zakharova, and E. Sch¨oll, Chaos: An Interdisciplinary Journal of NonlinearScience , 094825 (2016). Y. Zhu, Z. Zheng, and J. Yang, Physical Review E , 022914 (2014). A. Rothkegel and K. Lehnertz, New Journal of Physics , 055006 (2014). J. Hizanidis, N. E. Kouvaris, G. Zamora-L´opez, A. D´ıaz-Guilera, and C. G. Antonopoulos, ScientificReports , 19845 (2016). V. A. Maksimenko et al., Physical Review E , 052205 (2016). S. Ghosh and S. Jalan, International Journal of Bifurcation and Chaos , 1650120 (2016). S. Ghosh, A. Zakharova, and S. Jalan, Chaos, Solitons & Fractals , 56 (2018). N. S. Frolov et al., Physical Review E , 022320 (2018). V. V. Makarov et al., Communications in Nonlinear Science and Numerical Simulation , 118 (2019). B. K. Bera, S. Rakshit, D. Ghosh, and J. Kurths, Chaos: An Interdisciplinary Journal of NonlinearScience , 053115 (2019). T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, and Y. Maistrenko, Scientific Reports , 6379(2014). J. Wojewoda, K. Czolczynski, Y. Maistrenko, and T. Kapitaniak, Scientific Reports , 1 (2016). M. R. Tinsley, S. Nkomo, and K. Showalter, Nature Physics , 662 (2012). A. M. Hagerstrom et al., Nature Physics , 658 (2012). J. C. Gonz´alez-Avella, M. G. Cosenza, and M. San Miguel, Physica A: Statistical Mechanics and itsApplications , 24 (2014). J. Hizanidis et al., Physical Review E , 012915 (2015). P. S. Dutta and T. Banerjee, Physical Review E , 042919 (2015). T. Banerjee, P. S. Dutta, A. Zakharova, and E. Sch¨oll, Physical Review E , 032206 (2016). S. Kundu, S. Majhi, P. Muruganandam, and D. Ghosh, The European Physical Journal Special Topics , 983 (2018). S. K. Dana, S. Saha, and N. Bairagi, Frontiers in Applied Mathematics and Statistics , 15 (2019). A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Nature Physics , 191 (2013). L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Nature Communications ,1 (2014). S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Physics of Life Reviews , 100 (2019). P. Fries, Neuron , 220 (2015). P. J. Uhlhaas and W. Singer, Neuron , 155 (2006). M. Santos et al., Chaos, Solitons & Fractals , 86 (2017). R. G. Andrzejak, C. Rummel, F. Mormann, and K. Schindler, Scientific Reports , 23000 (2016). P. R. Protachevicz et al., Frontiers in Computational Neuroscience (2019). J. C. Coninck et al., Physica A: Statistical Mechanics and its Applications , 124475 (2020). A. Roxin, N. Brunel, and D. Hansel, Physical Review Letters , 238103 (2005). C. R. Laing, Physica D: Nonlinear Phenomena , 1960 (2011). K. Bansal et al., Science Advances , eaau8535 (2019). L. Kang, C. Tian, S. Huo, and Z. Liu, Scientific Reports , 1 (2019). C. R. Laing, Chaos: An Interdisciplinary Journal of Nonlinear Science , 013113 (2009). C. R. Laing, Physica D: Nonlinear Phenomena , 1569 (2009). S. Nkomo, M. R. Tinsley, and K. Showalter, Chaos: An Interdisciplinary Journal of Nonlinear Science , 094826 (2016). Y. Zhu, Z. Zheng, and J. Yang, EPL (Europhysics Letters) , 10007 (2013). E. A. Martens, C. Bick, and M. J. Panaggio, Chaos: An Interdisciplinary Journal of Nonlinear Science , 094819 (2016). C.-U. Choe, R.-S. Kim, and J.-S. Ri, Physical Review E , 032224 (2017). S. Majhi, M. Perc, and D. Ghosh, Chaos: An Interdisciplinary Journal of Nonlinear Science , 073109(2017). B. Li and D. Saad, Chaos: An Interdisciplinary Journal of Nonlinear Science , 043109 (2017). J. G´omez-Gardenes, S. G´omez, A. Arenas, and Y. Moreno, Physical Review Letters , 128701 (2011). X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, Physical Review Letters , 038701 (2015). A. D. Kachhvah and S. Jalan, New Journal of Physics , 015006 (2019). S. Jalan, V. Rathore, A. D. Kachhvah, and A. Yadav, Physical Review E , 062305 (2019). D. J. Watts and S. H. Strogatz, Nature , 440 (1998). A.-L. Barab´asi and R. Albert, science , 509 (1999). C. Tsitouras, Computers & Mathematics with Applications , 770 (2011). C. Rackauckas and Q. Nie, Journal of Open Research Software5