Chimera states in ring-star network of Chua circuits
aa r X i v : . [ n li n . AO ] A ug Chimera states in ring-star network of Chuacircuits
Sishu Shankar Muni ∗ Astero Provata † August 6, 2020
Abstract
We investigate the emergence of amplitude and frequency chimerastates in ring-star networks consisting of identical Chua circuits connectedvia nonlocal diffusive, bidirectional coupling. We first identify single-well chimera patterns in a ring network under nonlocal coupling schemes.When a central node is added to the network, forming a ring-star net-work, the central node acts as the distributor of information, increasingthe chances of synchronization. Numerical simulations show that the ra-dial coupling strength k between the central and the peripheral nodesacts as an order parameter leading from a lower to a higher frequencydomain. The transition between the domains takes place for intermediatecoupling values, . < k < , where the frequency chimera states prevail.The transition region (width and boundaries) depends on the Chua os-cillator parameters and the network specifics. Potential applications ofstar connectivity can be found in the control of Chua networks and inother coupled chaotic dynamical systems. By adding one central nodeand without further modifications to the individual network parametersit is possible to entrain the system to lower or higher frequency domainsas desired by the particular applications. Chimera states are characterized by the coexistence of synchronous and asyn-chronous areas when identical dynamical units are coupled equivalently in somenetwork topology. Most commonly studied are the frequency chimeras, whichare distinguished by the difference in frequency of the oscillatory elements. Al-though all oscillators have the same intrinsic frequency, it is the coupling be-tween them which creates a distribution of frequencies in the network, a reason ∗ School of Fundamental Sciences, Massey University, Palmerston North, New Zealand † Institute of Nanoscience and Nanotechnology, National Center for Scientific Research"Demokritos", 15341 Athens, GreeceKeywords: Chimera states, Chua circuit, nonlocal diffusive coupling, ring-star networkE-mail addresses: S.S. Muni ( [email protected] )A. Provata ( [email protected] ) σ (see[43]), we apply an additional radial connectivity where every Chua circuit isbidirectionally connected to a central node with a variable coupling strength k .By varying the values of k and σ we can transit from a pure ring connectivity,when k = 0 and σ = 0 , to a pure star (central) connectivity, when k = 0 and σ = 0 . We investigate the prevalence of different chimera patterns in thiscomposite connectivity, which we refer to as the ring-star Chua network.In particular, in this study starting from the pure ring connectivity with non-local coupling we recover the chimera structure known as single-well chimera asshown in reference [43]. To set terminology, here we need to explain the differ-ence between multi-scroll attractors, such as the Chua attractors, and multi-levelchimeras. A multi-scroll attractor represents the phase space of a single chaoticoscillator, whose trajectory circulates in the vicinity of multiple regions, escap-ing occasionally from each of these regions to the others [32]. Chimera states,on the other hand, are formed in systems of many coupled oscillators, wheredomains of coherent and incoherent oscillators are formed. In chimera statesthe coupled oscillators can be chaotic or simpler limit cycles. When the coupledelements have complex phase space, multiple domains of synchronized elementsare formed, separated by domains where the oscillators are asynchronous. Theseare called multi-well chimeras and consist of regions (wells) of coherent elementswith constant common frequencies, separated by incoherent regions where the3scillators develop different frequencies [43]. In the present study, by turning onthe radial coupling k , we show that initially, the single-well chimera structurespersist in the ring-star network and all oscillators have specific low frequencies.As k increases, there is an abrupt transition at finite k -values, where the systempasses from the low frequency regime to a high frequency regime, passing bythe intermediate k -region where domains of high and low frequencies co-exist.The k -value where the transition occurs depends on the ring coupling strength σ . In the following, when we refer to the exchange of information between nodes i and j in the system, we mean that at a certain time t the state variables x i ( t ) and y i ( t ) of node i receive and use the values of the state variables x j ( t − ∆ t ) and y j ( t − ∆ t ) of node j in the previous time step t − ∆ t . This must not be confusedwith the notion of global information, energy or entropy exchange between thenodes as discussed in the literature of synchronization between interacting units[45, 46, 47, 48, 49, 50, 51] and more recently on synchronization in the form ofchimera states [52, 53].The transitions from asynchronous patterns to chimera states and to syn-chronized states are quantitatively studied with the help of the different syn-chronization measures [54]. The mean phase velocity [55] is a common mea-sure considered to demonstrate the presence of chimera structures. However,it frequently fails to identify them, mainly in the cases of traveling or diffusingchimeras. In those cases, there is a need for different measures to be considered,which can work as alternatives to the mean phase velocity. Alternative mea-sures analyzing the relative size of the coherent/incoherent domains, the degreeof coherence etc were considered in [54]. We establish the chimera kingdom inthe ring-star network of Chua circuits using these measures, to avoid the prob-lems of pattern displacement in space and for quantitative comparison betweenthe different chimera morphologies.The paper is organized as follows: Section 2 introduces the Chua circuitand the ring-star network topology. In a separate subsection, 2.1, the varioussynchronization measures are introduced. Section 3 discusses the simulationresults obtained in the case of Chua circuits coupled in a ring geometry forparameter values where single-well chimeras emerge. Section 4 is devoted tothe ring-star network connectivity. The deformation of the single-well chimerasas a function of the radial coupling k is discussed in this section. In section 5the transition of the system from the lower frequency to the higher frequencydomain is discussed where the frequency chimera states prevail. In all cases,alternative synchronization measures are considered for the quantitative studyof the mean phase velocity profiles. In the conclusions the main results of thisstudy are recapitulated and some challenges ahead in analyzing the ring-starnetwork are proposed. 4 Chua ring-star network model
A sketch of the ring-star network is shown in Figure 1. A number of N Chuaoscillators are connected in a ring-star network with nonlocal diffusive coupling.Oscillators are indexed as i = 1 for the central node and i = 2 , . . . , N forthe peripheral (end) nodes. The central node ( i = 1 ) is connected to all theperipherals with the same coupling strength k . Each peripheral oscillator isnonlocally connected to R nodes to its left and R nodes to its right with commoncoupling strength σ and is also linked to the central node with coupling strength k . To enforce uniformity of the end nodes, periodic boundary conditions areconsidered.The dynamical equations of the ring-star network are given by Eqs. (1) and(2). For i = 2 , . . . , N , the dynamical equations of the end nodes are given by: ˙ x i = f x + k ( x − x i ) + σ R k = i + R X k = i − R ( x k − x i ) , ˙ y i = f y + σ R k = i + R X k = i − R ( y k − y i ) , ˙ z i = f z . (1)For i = 1 (central node) the dynamical equations are: ˙ x = f x + N X j =1 k ( x j − x ) , ˙ y = f y , ˙ z = f z . (2)where f x = α ( y i − x i − ( Bx + 12 ( A − B )( | x + 1 | − | x − | ))) ,f y = x i − y i + z i ,f z = − βy i . with periodic boundary conditions: x i + N ( t ) = x i ( t ) ,y i + N ( t ) = y i ( t ) ,z i + N ( t ) = z i ( t ) . for i = 2 , , . . . , N . Following references [42, 43], we have used coupling only inthe x and y -variables and not in the z -variable of the Chua coupled elements.Similar coupling only via one variable is used in reference [56] for coupled R¨ossleroscillators. 5 N Figure 1: The Chua ring-star network. Here we consider N = 300 Chua circuitswhere the central one is labeled i = 1 and the end nodes are labeled from i = 2 , . . . , N .From the interaction scheme, it is now clear that oscillators i = 2 , . . . , N exchange information via their x − and y − variable with R neighbors symmet-rically set around i , while the central unit i = 1 exchanges information with allother units j = 2 , . . . , N (also via their x − and y − variables only). Due to theEuler integration scheme used, the variables x i ( t ) , y i ( t ) , z i ( t ) are updated usingthe values x i ( t − ∆ t ) , y i ( t − ∆ t ) , z i ( t − ∆ t ) at the previous time step, as alsostated in the Introduction.As working parameter set the following values are used throughout thisstudy: The parameters of the identical Chua circuits are set to A = − . , B = − . , α = 9 . and β = 14 . in order to keep the circuit in the oscilla-tory, double-scroll regime. The system size is set to N = 300 and the couplingrange to R = 100 . The rest of the parameters, the coupling strength σ betweenthe peripheral nodes, and the coupling strength k between the central nodeand the peripheral ones are varied to explore their influence in the networksynchronization patterns. As discussed in the Introduction, the mean phase velocity or average frequency ω is a valuable measure to quantify the synchronization of the oscillators [55].For the i -th oscillator, the mean phase velocity is denoted by ω i . For a largecomputational time interval T , ω i expresses the number of times the variable x i crosses a certain fixed constant value, say c . If the variable x i crosses theconstant c , M i times with positive slope, then the mean phase velocity of the i -th oscillator is calculated as : ω i = 2 πM i T = 2 πf i . (3)6he positive slope considered in the counting of M i in Eq. (3) is needed toavoid double counting the number of periods calculated within the time interval T . The quantity f i denotes the average frequency, which differs from the meanphase velocity by a factor π . Due to this simple relation, in the following theterms “mean frequency” and “mean phase velocity” will be used interchangeably.Chimera states are characterized by the difference in frequency of the iden-tical oscillatory circuit elements. The coupling is responsible for the change infrequency in some oscillators. This is the reason why the chimera states are sonontrivial and unexpected. Different synchronization measures come to play asadditional quantitative indices when inconclusive information is conveyed by themean phase velocity. Such synchronization measures were discussed in [54]. Wecomplement our work with the computation of the relative size of the incoher-ent and coherent parts denoted by r incoh and r coh , respectively. Let us denoteby ω coh the common mean phase velocity of the coherent elements, by ω i themean phase velocity of the i -th oscillator and by N the number of oscillatorsconsidered. The quantity r coh is defined as : r coh = 1 N N X i =1 χ ( A ) (4)where χ is a step function which returns if A is positive and returns if A is negative. A is defined as: A = k ω i − ω coh k − ǫ (5)In definition (5) a small tolerance ǫ is added in order to take into account thefluctuations at the coherent level while computing r coh according to Eq. (4).Similarly, the quantity r incoh is defined as : r incoh = 1 N N X i =1 χ ( A ) (6)where A = ( ω i − ω coh − ǫ, ω coh < ω incoh ,ω coh − ω i − ǫ, ω coh > ω incoh (7)Note that in the definition (7) two cases are considered. That is because oneneeds to cover both cases: when the ω coh coincides with the minimum frequencyin the system (upper case in Eq. (7)) and when the ω coh coincides with themaximum frequency in the system (lower case in Eq. (7)). Both cases have beenrecorded in the literature on chimera states, see references [10, 54, 57]. In oursimulations, the tolerance level ǫ is fixed to be . ω , where ∆ ω = ω max − ω min is the difference between the maximum and minimum ω i values in the system. In this section, we study the case of a ring network where each element is aChua circuit nonlocally linked to its neighbors, as was proposed and studied by7hepelev et al. in reference [43]. Following [43] we consider here the formationof chimeras in the case of ring connectivity with nonlocal diffusing coupling.To avoid unnecessary complexity which arises in the parameter regions wherethe double-well chimeras prevail, we restrict ourselves in the parameter regionswhere only single-well chimera states are observed.Starting with a ring network of N = 300 Chua oscillators with nonlocaldiffusive coupling, we record single chimera states for different values of thenetwork parameters (work by Shepelev et al. [43]). As an exemplary case, inFig. 2 we record the spatial and temporal behavior of the network for couplingparameters σ = 0 . and k = 0 . Panel 2a depicts the spatial profile of thenetwork at 25 snapshots at time intervals of 40 units. This figure depicts asingle-well chimera state, where domains of alternating oscillatory propertiesare formed. Having started with initial conditions, ( x ( t = 0) , y ( t = 0) , z ( t = 0)) ,randomly distributed in the interval [0 < x ( t = 0) < , < y ( t = 0) < , < z ( t = 0) < ,the system remains always in the positive side of the axes (single-well) andall elements oscillate around the value x = 1 . , but amplitude variations areobserved in the different domains formed. For clarity, a single snapshot is pre-sented in panel 2b. The mean phase velocity, panel 2c, clearly identifies thedomains where oscillators act coherently, and the incoherent domains.In the next section, 4, we introduce a central node to the system, creatingthe ring-star network, and we investigate the system’s response by varying theradial coupling strength k . We study the system response in the case of single-well chimeras, using as coupling constant parameter σ = 0 . and variable k , asdiscussed above. a b c
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Figure 2: Ring network: Single-well chimera structures and measures for σ =0 . , k = 0 with nonlocal diffusive coupling. a) 25 snapshots of the x i -variablesat time intervals of 40 units, b) typical single snapshot of the x i -variables andc) mean phase velocities. Chua circuit parameters are A = − . , B = − . , α = 9 . , β = 14 . and network parameters are N = 300 and R = 100 .8
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Figure 3: Ring-star network: Single-well chimera structures and measures for σ = 0 . , k = 0 . with nonlocal diffusive coupling. a) 25 snapshots of the x i -variables at time intervals of 40 units, b) typical single snapshot of the x i -variables and c) mean phase velocities. The arrow in panel c) indicates themean phase velocity of the central node. All other parameters as in Fig.2. We now extend the Chua ring network model by adding a central node in thecenter of the ring which is linked equally to all the external nodes. We considerthe mixed dynamics of the system and record its transition between chimerastates and the full synchronization regimes. Simulations were carried out byconsidering random initial conditions for x, y, z state variables in the interval [0 , , as in previous section.To investigate the influence of the central node in the Chua ring networkwe gradually vary the coupling strength k between the central node and theperipheral ones. All parameters of the identical Chua circuits are kept to theworking parameter set, while the network contains N = 300 nodes and eachChua oscillator is connected to R = 100 neighbors to the left and R = 100 neighbors to the right. The ring coupling strength is fixed to σ = 0 . . [Thepure-ring network briefly discussed in the previous section is equivalent to thecase of k = 0 (no central node), while the ring-star network is realized when k = 0 .]The central node of the network plays a double role: First it receives “infor-mation” from the peripheral nodes and integrates it forming its own dynamics.Second, it redistributes the obtained information to the peripheral nodes, insuch a way that each peripheral node receives information about the average(mean-field) dynamics of the ensemble of all peripheral nodes. Therefore, thecentral node acts as a modulator of the local dynamics using information overthe ensemble dynamics. Based on this view of the system, we ask the question:How does the strength of the central coupling k influences the distribution of in-formation and the overall synchronization properties of the network? To answerthis question we performed numerical simulations using the same parameters as9
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Figure 4: Ring-star network: Double-well chimera structures and measuresfor σ = 0 . , k = 1 with nonlocal diffusive coupling. a) 25 snapshots of the x i − variable at time intervals of 40 units, b) typical single snapshot of the x i − variable and c) mean phase velocities. The arrow in panel c) indicates themean phase velocity of the central node. All other parameters as in Fig.2.
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Figure 5: Ring-star network: Double-well chimera structures and measures for σ = 0 . , k = 1 . with nonlocal diffusive coupling. a) 25 snapshots of the x i − variables at time intervals of 40 units, b) typical single snapshot of the x i − variables and c) mean phase velocities. The arrow in panel c) indicates themean phase velocity of the central node. All other parameters as in Fig.2.10 b c
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Figure 6: Ring-star network: Double-well chimera structures and measures for σ = 0 . , k = 2 . with nonlocal diffusive coupling. a) 25 snapshots of the x i − variables at time intervals of 40 units, b) typical single snapshot of the x i − variables and c) mean phase velocities. The arrow in panel c) indicates themean phase velocity of the central node. All other parameters as in Fig.2.in the case of the single-well chimera and varied the central coupling parameterin the range ≤ k ≤ .We provide below some examples of the modifications which take place whenthe central coupling strength k becomes nonzero. When a small deviation isapplied leading from the ring network, k = 0 , to the ring-star network, k = 0 . ,as in Fig. 3, we observe that the single-well chimera persists: the x state variablekeeps oscillating in the positive part of the axis and does not transverse below x = 0 . For this low k − value, all oscillators have very similar frequencies, ascan be seen in Fig. 3c. The green arrow in panel (c) points to the mean phasevelocity ω of the central node, which is slightly lower than the rest of theelements. Increasing the radial strength to k = 1 in Fig. 4 and k = 1 . inFig. 5, the single-well chimeras change to double-well ones. In both cases twodomains of oscillators are formed: one domain where the x -variables oscillatein the positive axis and one in the negative axis. Related to the mean phasevelocity values, in the case of k = 1 , Fig. 4c, all peripheral nodes have thesame mean phase velocity, while the central node has increased its mean phasevelocity, ω ∼ . , indicating a tendency of the system to transit to higherfrequencies (see position of the green arrow in Fig. 4c). Increasing further the k -values, see the case k = 1 . in Fig. 5c, the nodes occupying the transitionregions, between the negative and the positive x − value domains, follow thecentral node (see position of the green arrow) and attain in their turn highermean phase velocities. With a further increase in the radial coupling strength k , for example k = 2 . , we observe that the x state variables still traverse boththe positive and negative part of the axis as in Fig. 6. Furthermore, in panel (c)we observe that all mean-phase velocities have increased as compared to lowervalues of k .In the next section, sec. 5, we study how the transition from the the single11o double-well chimera takes place as we gradually increase the radial couplingstrength k . We analyze here the dynamics of the ring-star Chua network as the couplingstrength k (the ray coupling strength) increases between ≤ k ≤ . The ringcoupling strength is fixed as σ = 0 . . We study the behavior of the meanphase velocity of the central node, ω central , of the coherent oscillators, ω coh , andof the “leader” incoherent oscillator, ω leader . From Fig. 5 in the previous section,we record in the incoherent regions a continuous distribution of frequenciesand not a single one. In these regions we call “leader” the oscillator whichdemonstrates the maximum mean phase velocity, which, for this reason is called ω leader . Note that there can be more than one leaders in the system, one foreach incoherent region, as Fig. 5 indicates. In a way, the leader oscillators canbe considered as the ones which lead the deviations from coherence, while thedifference ∆ ω = ω leader − ω coh is indicative of the total incoherence in the system.In Fig. 7 we plot the mean phase velocities of the central node ω central (bluecolor), of the coherent nodes ω coh (green color), and of the leader incoherentnodes ω leader (red color). Initial conditions were chosen randomly in all sim-ulations within the positive interval, [0 , , for the x, y, z state variables. Thisfigure indicates the presence of a phase transition taking place in the parameterregion . ≤ k trans ≤ . In particular, for small values of the radial coupling, k ≤ . , all oscillators present similar ω -values, around ω ∼ . In this region( k ≤ . ), the central oscillator, i = 1 , has the smallest frequency, slightly belowthe coherent ones, while the leaders have frequencies slightly above the coherent.As k increases above 0.5 the abrupt transition occurs. First the central nodesdouble (almost) their mean phase velocity which becomes close to 1.6, while therest of the oscillators remain close to the values ω ∼ . This behavior holdsin the intermediate coupling region, . ≤ k ≤ . (for the parameter values σ = 0 . , N = 300 oscillators and R = 100 neighbors). Above this transitionregion, and for k > , the coherent and incoherent nodes also increase graduallytheir mean phase velocities, which also attain values around ω ∼ . . In partic-ular, for values . < k < . , the central node frequency is located betweenthe coherent and the leader ones. When k reaches strengths > . the oscillatorregions stabilize and the system attains constant frequencies, ω central ∼ . , ω coh ∼ . and ω leader ∼ . independent of k . The above discussion tells usthat the frequency “chimera kingdom” characterized by considerable differencesin the frequencies between coherent and incoherent domains is established for k -parameter values in the transition region . < k < . .Figure 8 shows the ring-star network in action. We represent the central(blue) node, coherent (green) and incoherent (red) nodes for k = 1 , σ = 0 . in the ring-star network with different colors. As we see from the figure, the12 eader coh central
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Figure 7: Mean phase velocities as a function of the coupling strength k . Themean phase velocities of the coherent nodes, ω coh , are marked in green color,of the central node, ω central , is marked in blue color, while the maximum meanphase velocity in the incoherent regions, ω leader , is marked in red color. Allother parameters as in Fig.2.coherent nodes are not isolated but form clusters in the ring. At the sameinstant the incoherent nodes are most abundant.The transition shown in Fig. 7 is corroborated by the plots depicting ω central − ω coh and ω leader − ω coh in Figs. 9 and 10, respectively.Namely, in Fig. 9 we note that the values ω central − ω coh remain slightly below0, for k < . indicating that the central node has constantly lower frequencythan the coherent nodes. Above k > , the same behavior persists. In theintermediate region, . < k < . , first the central node acquires abruptly veryhigh frequencies for . < k < , and later on its frequency gradually decreasesto stabilize slightly below the frequency of the coherent nodes. We recall that,in all cases, the central node has the same characteristics (parameters) as allother nodes in the system. Concerning the leader nodes, in Fig. 10 the values ω leader − ω coh remain above 0 for k < . , demonstrating a gradual increase as afunction of k in this region. They also demonstrate a delay in the transition withrespect to the central node. The leaders are entrained to transit for k ∼ . while the coherent ones reach the higher frequency domain after k ∼ . . Above k > , the leaders keep a constant small mean phase velocity difference fromthe coherent nodes of the order ∼ . .13igure 8: Coherence circle plot. The coherent nodes are marked with green,incoherent nodes are marked with red color. The number of nodes depictedhere is N = 300 . The central node is marked with blue color. The couplingstrengths are fixed as: k = 1 and σ = 0 . . All other parameters as in Fig.2.To study further the inhomogeneity of frequencies in the system we com-pute the ratio of coherent, r coh and incoherent nodes r incoh as a function of k , excluding the central one. The ratios of coherent and incoherent elementswere calculated using Eqs. (4) and (6) and the results are depicted in Figs. 11and 12, respectively. We have a similar picture of the transition from small tolarge frequency values with increasing the strength k of radial coupling. Forsmall ( k < . ) and large ( k > ) radial coupling strengths the ratio of co-herent elements stays low, r coh ∼ . and the ratio of incoherent stays large, r incoh ∼ . − r coh . In the intermediate region, . < k < a transitive be-havior is recorded, where the coherent ratio increases, while the incoherent onedecreases. This reorganization of the system taking place in the intermediate k regions where the frequency chimera states prevail, marks the passage from thelower to the higher frequency domain.As a general conclusion, in the transition between low and high frequencies,first the central node makes the jump to the higher frequencies at k ∼ . entraining the rest of the nodes. Following the central node the incoherent14 -0.1 -0.2 Figure 9: The difference ω central − ω coh as a function of the radial couplingstrength k . Ring coupling strength σ = 0 . and other parameters as in Fig. 4.nodes are entrained. Their leaders make the transition at k ∼ , while thecoherent nodes attain the transition at k ∼ . . Note that the present valuesare indicative and hold for the working parameter set. For different parameters ( σ, N, R ) , the transition values as well as the transition regions are expected tovary depending on these parameters. We have studied a ring network of Chua circuits, equipped with a central nodewhich serves to redistribute to the peripheral nodes information about the meanfield state of all nodes. For small values of the radial coupling strengths single-well amplitude chimeras are observed. At intermediate radial couplings a tran-sition region is observed where the frequency chimeras prevail with a large dif-ference in the frequencies between coherent and incoherent nodes. This regionmediates the transition between the lower and higher frequency domains. Forlarge radial coupling strengths, the system attains the higher frequency domainand keeps constant mean phase velocities and ratios of coherent to incoherentnodes, independent of the radial coupling range. The frequency chimera king-dom is established for the intermediate radial couplings k -values, as evidencedby the plots of all different synchronization measures.The above results have potential applications in the control of Chua net-15 Figure 10: The difference ω leader − ω coh as a function of the radial couplingstrength k . Ring coupling strength σ = 0 . and other parameters as in Fig. 4.works as well as other coupled chaotic dynamical systems. By just adding onecentral node, identical to all peripheral ones, and without further modifica-tions to the individual oscillators or to the network parameters, it is possible toentrain the system to lower or higher frequency domains as desired by the par-ticular applications by only adjusting the radial coupling. We must stress herethat the transition described above is an example of transitions taking place innonequilibrium systems (nonequilibrium transitions); the Chua system (1) is acharacteristic example of such systems since it presents chaotic, nonconservativedynamics [32, 33, 34, 39, 44]. This transition cannot be directly related to theknown phase transitions in equilibrium systems at criticality, such as the Isingmodel phase transitions (see reference [58]).For future studies, it would be interesting to understand how the dynamicsof the Chua network changes with different coupling forms such as conjugatecoupling or mean-field coupling, or by strengthening the role of the central nodeand endowing it with interactions to the peripheral nodes using all three x , y and z variables. Transitions in different network types may also be consideredas, for example, in the case of a 2D lattice of Chua oscillators equipped with acentral element, or extensions to Chua circuits in multilayer arrangements.A different study concerns the connection between star and ring-star net-works. In order to achieve a complete star network (central node connected toperipheral nodes and no connection in between the peripheral nodes), mathe-16 Figure 11: Ratio of coherent elements as a function of the central coupling range k . All other parameters are as in Fig. 2.
0 0.5 1 1.5 2 2.5 3 3.5 4 00.20.40.60.81 i n c o h Figure 12: Ratio of incoherent elements as a function of the central couplingrange k . All other parameters are as in Fig. 2.17atically we need to fix a finite value of k while letting σ → . It would beinteresting to account for chimera states and potential transitions in this limiting σ case and to investigate the link to the phenomenon of Remote Synchronization(RS) [56, 59], a nontrivial phenomenon in star networks, where the peripheraloscillators synchronize (without being directly linked), while the central, relaynode remains asynchronous. Acknowledgements
This work was supported by computational time granted from the Greek Re-search & Technology Network (GRNET) in the National HPC facility - ARIS -under project CoBrain4 (project ID: PR007011).
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