Experimental investigation on the susceptibility of minimal networks to a change in topology and number of oscillators
EExperimental investigation on the susceptibility of minimal networks to a change intopology and number of oscillators
Krishna Manoj, ∗ Samadhan A. Pawar, and R. I. Sujith Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai - 600036, India (Dated: September 11, 2020)Understanding the global dynamical behaviour of a network of coupled oscillators has been atopic of immense research in many fields of science and engineering. Various factors govern theresulting dynamical behaviour of such networks, including the number of oscillators and their cou-pling schemes. Although these factors are seldom significant in large populations, a small changein them can drastically affect the global behaviour in small populations. In this paper, we performan experimental investigation on the effect of these factors on the coupled behaviour of a minimalnetwork of candle-flame oscillators. We observe that strongly coupled oscillators exhibit the globalbehaviour of in-phase synchrony and amplitude death, irrespective of the number and the topologyof oscillators. However, when they are weakly coupled, their global behaviour exhibits the inter-mittent occurrence of multiple stable states in time. In addition to states of clustering, chimera,and weak chimera, we report the experimental discovery of partial amplitude death in a network ofcandle-flame oscillators. We also show that closed-loop networks tend to hold global synchroniza-tion for longer duration as compared to open-loop networks. We believe that our results would findapplication in real-life problems such as power grids, neuronal networks, and seizure dynamics.
I. INTRODUCTION
Discovered by Huygens in the 16 th century, collec-tive interaction between oscillators has seen a flurry ofboth theoretical and experimental studies and till dateis a topic of immense interest between researchers fromaround the world [1–3]. Starting from the motions ofcoupled pendula [4, 5] and extending towards suppress-ing coronavirus spread [6], collective interaction saw anintertwine of various strata of science including physicsand biology. The mind-blowing coordination in swarmsof fishes and birds and other biological beings and the ex-hibition of various thought-provoking nonlinear dynam-ical states in natural systems are a coincidence of thisinterplay. Those dynamical states range from synchro-nization [1], clustering [7, 8], oscillation quenching [9] tosymmetry-breaking phenomena such as chimera [10–13]and weak chimera [14, 15].The occurrence of these dynamical states has beenstudied in systems with the number of oscillators rangingfrom an order of one [5, 16, 17] to thousand [18, 19]. Itis interesting to note that in a swarm, the addition orremoval of a few entities or a change in their topologicalpositions does not affect the global dynamics of the en-tire system [20]. However, these observations from largenetworks may not be applicable to networks where thenumber of oscillators is very few (i.e., minimal oscillatornetwork).Coupled behaviour of oscillators in minimalistic net-works (number of oscillators 2 to 10) has shown sev-eral interesting dynamical states, which include in-phaseand anti-phase synchrony, clustering, amplitude death,partial amplitude death, chimeras, and weak chimeras ∗ [email protected] [15, 21–23]. These networks are very susceptible to theaddition or removal of an oscillator and also to a changein the topological arrangement of oscillators in a net-work. The addition of an oscillator increases the degreesof freedom of the system, resulting in an increase in thecomplexities in the behaviour of the system.For example, in a system of coupled candle-flame os-cillators [24], when the number of oscillators is two, andthe distance between the oscillators is increased, the sys-tem shows four prominent stable behaviours such as in-phase synchronization, amplitude death, anti-phase syn-chronization, and desynchronization [25]. As the num-ber of oscillators is increased to three and located asan equilateral triangle, the system shows the presence ofmultiple stable states including in-phase synchronization,amplitude death, partial in-phase, and rotation mode[26]. When the number of oscillators is increased to fourand placed in a rectangular network, the system shows aplethora of dynamical states including in-phase synchro-nization, amplitude death, clustering, chimera, and weakchimera [23].On the other hand, the change in the topology of os-cillators in a network changes the coupling arrangementof these oscillators. For example, in a line or a ring net-work, the oscillators are locally coupled to their nearestneighbours, whereas in a star network, only the centraloscillator is coupled to all peripheral oscillators. Wick-ramasinghe and Kiss [27] studied the effect of networkstructure on the selection of self-organized patterns incoupled chemical oscillators. When six oscillators arecoupled in an extended triangular network, they observedthe presence of a partially synchronized state, where thestrongly coupled oscillators in the core of the trianglesynchronize easily when compared to the weakly coupledperipheral oscillators. Non-local coupling of 20 oscillatorsin a ring showed the possibility of chimera state, whileglobally coupled oscillators show the existence of clus- a r X i v : . [ n li n . AO ] S e p tering state. They also observed that chaotic oscillatorscoupled in a closed-loop topology (e.g., square, triangle,or ring) undergo faster synchrony than those coupled inan open-loop topology (e.g., linear or star).A theoretical study on Kuromoto oscillators by Ashwinand Burylko [14] showed the existence of weak chimerain the coupled behaviour of minimal closed-loop networksconsisting of 4, 6, and 10 oscillators. Later, a study con-ducted by Hart et al. [16] reported the presence of var-ious dynamical states including bare minimum chimera,global synchrony, and clusters in a network of four opto-electronic oscillators for the variation of global couplingstrength and coupling delay between the oscillators. Thestudy by Wojewoda et al. [15] on three coupled pendulauncovered the state of weak chimera in an experimen-tal system. Recently, Sharma [22] observed the states ofpartial amplitude death and phase-flip bifurcation in asystem of three theoretical time-delay coupled relay os-cillators. Although all aforementioned studies separatelyprovide insights on the role of coupling structure andnumber of oscillators on the dynamical behaviour of anetwork, none of the experimental studies so far has com-prehensively delineated the explicit dependence of theglobal behaviour of the same network on the change inthe number of oscillators, the coupling topology, and thestrength of coupling between the oscillators.In the present study, we perform an experimental in-vestigation on the coupled behaviour of a minimal net-work of candle-flame oscillators. We investigate the cou-pled behaviour of these oscillators by changing the num-ber of oscillators in a network and locating them in var-ious topological arrangements. For a given number ofoscillators, we examine two types of topological arrange-ments such as closed-loop (triangle, square, and annular)and open-loop (linear and star) networks. In a closed-loop network, the oscillators are symmetrically coupled,whereas, in an open-loop network, they are asymmetri-cally coupled. Subsequently, we characterize the dynam-ical states observed for each of these topological arrange-ments for the variation in distance between oscillators.We observe that when the distance between the os-cillators is small, the network of oscillators exhibits thestates of in-phase synchronization ( d = 0 cm) and am-plitude death ( d = 1 cm) irrespective of the numberor the topological arrangement of oscillators in the net-work. When the distance between the oscillators is large( d > II. EXPERIMENTAL SETUP
Candle-flame oscillators are one among the simplestand economical oscillators which exhibit various com-plex dynamical behaviours including synchronization[24], amplitude death [25, 26], phase-flip bifurcation [25]and, clustering, chimeras and weak chimeras [23]. Inall these studies, the candle-flame oscillators used aremade by bundling three or more candles together andlighting them to form a compound flame, which exhibitslimit cycle oscillations. However, experiments with suchcandle-flame oscillators are susceptible to uncertainitydue to various inherent disturbances. These disturbancesinclude uneven evaporation and burning of candles inan oscillator, leading to an uneven reduction in theirheights and maintenance of vertical orientation of thewick. Therefore, the manufacturing of a candle consist-ing of four wicks deems to make the experiments eas-ier and less cumbersome, by enhancing the repeatabilitywith an evenly burning oscillator having stronger oscilla-tions sustaining for a longer duration.In experiments, we make a candle-flame oscillator oflength 12 cm and a diameter of 2 cm with four wicksplaced in a square arrangement being 1 cm apart, asshown in Fig. 1. The candle-flame oscillator createdusing a single candle with four wicks exhibits limit cy-cle oscillations with nearly the same amplitude and fre-quency as the earlier used candle-flame oscillator madeup of multiple candles [25]. In order to study the ef-fect of the number of oscillators in a network and thechange in the topology of coupling between the oscilla-tors, we couple two to four such candle-flame oscillatorsand measure their dynamical response for every couplingconfiguration. The number of ways in which oscillatorscan be arranged in a network topology depends on thenumber of oscillators interacting in the system (see Fig.2). We note that the interaction of oscillators in a net-work is primarily influenced by their nearest neighbours.Therefore, we indicate the connected neighbours of anoscillator by its degree [28].In the case of two oscillators in (Fig. 2a), the onlypossible combination involves the mutual interaction be-tween both the oscillators, assigning a degree of 1 forboth the oscillators. When the number of oscillators inthe system is increased to three as presented in (Fig. 2b),two topological arrangements are possible in the system.The straight (linear) topology of the network (Fig. 2b-i)where the central oscillator, having a degree of 2, is inthe direct influence of peripheral oscillators, each hav-ing a degree of 1. In an equilateral triangular network(Fig. 2b-ii), each oscillator interacts with every other os-cillator with equal strength and, therefore, the degree of
FIG. 1. (a) Schematic of the candle-flame oscillator. (b) A normal image of the flame luminosity and (c) a filtered CH*chemiluminescence image of a candle-flame oscillator. (d) Time series of the global heat release rate fluctuations in the flame( I ), presenting self-sustained limit cycle oscillations shown by an isolated candle-flame oscillator. (e) The amplitude spectrumcorresponding to these oscillations shows a dominant frequency of 11.46 ± d ), are kept constant in a network. The linkdistance is normalized using the radius (1 cm) of the candle-flame oscillator, as shown in Fig. 1(a). each oscillator remains 2. For the case where the numberof oscillators is increased to 4, the number of networkspossible also increases. Three such networks are investi-gated in this study, namely straight (Fig. 2c-i), star (Fig.2c-ii), and square (Fig. 2c-iii), where the degree of eachoscillator is also indicated.An acrylic platform with markings for each of the topo-logical arrangement is used to mount these oscillatorsduring experiments. The platform is placed on a tablehaving a height of 80 cm from the ground to avoid groundeffects on the dynamics of the coupled oscillators. Allexperiments are performed in a completely dark closedroom with quiescent ambient conditions. After fixing thelink distance and topological arrangement, high-speedimaging of each experiment is performed. The position ofthe camera is varied for each topology to obtain distinctflames for each oscillator in a single frame. The dynamicsproduced by the candle-flame oscillators are captured us- ing a high-speed imaging technology of iPhone7S (framerate of 240 Hz) fitted with a CH* chemiluminescence fil-ter (wavelength of 435 nm and 10 nm full width at halfmaximum) for 60 s. The filter facilitates the removal ofnoisy fluctuations associated with the black body emis-sion of the soot in the flame (see Fig. 1b) and providesinformation about the actual heat release rate fluctua-tions (indicated in blue in Fig. 1c) present in the flame[29] of a candle-flame oscillator.The instantaneous value of the global heat release ratefluctuations ( I ) is obtained by summing up the localbrightness values of the flame in a given frame (as shownin Fig. 1c) and a time series of such fluctuations is ob-tained by performing the same operation for the entirevideo. The limit cycle oscillations exhibited in the heatrelease rate by an isolated oscillator are presented in Fig.1(d) and the amplitude spectrum of these oscillations areshown in Fig. 1(e). We observe the natural frequency ofan isolated oscillator as 11.46 ± F s /N s = 240 / . III. RESULTS AND DISCUSSION
The coupled interaction between candle-flame oscil-lators in a network engenders a plethora of dynamicalstates [23–26]. We will begin our discussion on the dy-namics observed in a pair of coupled candle-flame oscilla-tors and then move on to present the various dynamicalstates observed when the number of oscillators in thenetwork is increased to three and four. As we observemultiple stable dynamical states at higher distances, wepresent the percentage occurrence of each state at a givendistance and topology, after the describing the synchro-nization properties (in terms of phase and frequency lock-ing) individual dynamical state of the network. Finally,we investigate the various dynamical behaviour observedin an annular topology consisting of five to seven candle-flame oscillators.In a network of two candle-flame oscillators (Fig. 2a),as the distance between them is varied, we observe theexistence of four dynamical states. These states includein-phase synchronization (for d ≤ . < d < . . < d ≤ . d > . d = 0 for in-phasesynchronization, d = 1 for amplitude death, d = 3 foranti-phase synchronization, and d = 4 for desynchroniza-tion) are subjected to various topological arrangementswhen the number of oscillators in the network is increasedto 3 and 4 (Fig. 2b,c). These results are consistent withthe findings in two candle-flames oscillators each consist-ing of a bundle of four candles by Manoj et al. [25].When the link distances between two candle-flame os-cillators are of d = 0 and 1, we observe dynamical statesof in-phase synchronization and amplitude death, respec-tively (Fig. 3). In the case of in-phase synchronization,all oscillators reach their corresponding maximum andminimum amplitudes simultaneously; thus, exhibiting aphase-shift of nearly 0 deg between their oscillations (Fig.3a). A uniform and nearly identical motion of all oscilla-tors in the network is observed during in-phase synchro-nization. We observe that the variation in the numberof oscillators or a change in the network topology fora fixed number of oscillators, for d = 0, does not dis-turb the existence of in-phase synchronization observedin candle-flame oscillators (as seen in Fig. 3b). On theother hand, the state of amplitude death is characterizedby simultaneous quenching of oscillations in both the os-cillators, where all the oscillators theoretically reach ahomogenous steady state, but in experiments, they showminimal noisy fluctuations around the mean value of zero(Fig. 3c). Similar to the behaviour of candle-flame os-cillators at d = 0, the existence of amplitude death isobserved for d = 1 in all oscillators irrespective of theirnumber and the network topology (not presented herefor brevity). Thus, the presence of stronger coupling be-tween the oscillators at small link distances ( d = 1, 2)and the existence of high stability of the dynamical statesmight be a plausible reason for the occurrence of theseinvariant states in a network of candle-flame oscillators.However, this observation is not true for the case wherethe link distances between oscillators are larger than 2(i.e., d > FIG. 3. Time series of the global heat release rate fluctua-tions ( I and I ) corresponding to (a) in-phase synchroniza-tion ( d = 0) and (c) amplitude death ( d = 1) obtained fromcoupled pair of oscillators. (b) Snapshots of the candle-flameoscillators corresponding to different network topologies, asdiscussed in Fig. 2 for link distances that correspond to in-phase synchronization ( d = 0). complexity of coupled dynamics exhibited by the net-work of candle-flame oscillators. We witness the emer-gence of several symmetry-breaking states (as shown inFigs. 4 to 6) where the network dynamics tend to exhibitmultiple stable states of coupled oscillations for a speci-fied distance and topology of oscillators (Fig. 7). Notethat each dynamical state is observed for a minimum of100 oscillatory cycles and, therefore, we do not considerthem as transient dynamics. The coupled dynamics ofoscillators gradually shifts from one dynamical state intoanother with time. This transition happens either via atransient change in the frequency of a few oscillators ora momentary quenching of a few oscillators, observed forapproximately 3 to 5 s (a maximum of 50 cycles), to ad-just their dynamics to achieve the subsequent dynamicalstate. Furthermore, we do not specify the distances cor-responding to each dynamical state discussed in Figs. 4 FIG. 4. (I) Temporal variation of the relative phase (∆Φ)between a pair of oscillators and (II) the dominant frequencies( f ) of all oscillators for the dynamical states of (a) cluster-ing, (b) weak chimera, (c) desynchronization, and (d) rotatingclusters observed in a network of three coupled candle-flameoscillators. to 6, as multiple dynamical states are observed for a givendistance and vice versa (multiple distances for which weobserve a given dynamical state). An overall descriptionof the occurrence of these states is summarized in Fig. 7.To characterize the coupled dynamics exhibited by anetwork of three and four candle-flame oscillators, weplot the temporal variation of the instantaneous phasedifference between the pair of oscillators. The absolutevalue of the relative phase (wrapped between -180 degto 180 deg) is obtained after applying the Hilbert trans-formation [1] on the signal, as shown in Fig. 4(I). Thesynchronization characteristics of oscillator pairs in a net-work decide the global behaviour of the network. A barchart depicting the dominant frequency of each oscillatorat a given state is shown in Fig. 4(II).When the number of oscillators in the network is three,we observe four possible states of coupled dynamics asthe topology and the distance between the oscillators ina network are varied (Fig. 4a-d). In Fig. 4(a), we plotthe dynamical features of the state of clustering of oscil-lators, where three oscillators exhibit an equal frequencyand maintain a constant phase difference between eachother. During clustering, the phase difference betweenthe oscillator pairs {
1, 2 } , {
2, 3 } , and {
3, 1 } are nearly84 deg, 152 deg, and 68 deg, respectively (Fig. 4a-I),and all oscillators exhibit a dominant frequency of 11.03Hz (Fig. 4a-II). We further notice that the occurrenceof the clustering state in a network topology depends onthe degree of each oscillator. This type of clustering iswitnessed in a triangular (closed-loop) network, wherewe do not observe the phase shift between the oscillatorsat 0 or 180 deg (usually observed in the literature [7]).The unstable nature of maintaining 0 or 180 deg due to FIG. 5. (I) The temporal variation in the relative phase (∆Φ)between a pair of candle-flame oscillators and (II) the domi-nant frequency ( f ) of each oscillator for the states of (a), (b)clustering, (c) chimera, and (d) weak chimera observed in anetwork of four coupled candle-flame oscillators. the closed-loop arrangement with three oscillators is apossible reason behind such a state of clustering.In Fig. 4(b), we show the dynamical behaviour of thestate of weak chimera [14] observed in a three-oscillatornetwork. The state is characterized by the presence ofa pair of frequency synchronized oscillators, which aredesynchronized with the third oscillator due to a differ-ence in the frequency. During weak chimera (Fig. 4b),the oscillator pair {
1, 2 } are anti-phase synchronized andtheir synchronization frequency is 11.99 Hz, whereas theoscillator 3 is desynchronized with the pair {
1, 2 } as it ex-hibits a frequency of 10.39 Hz. We also observe the pres-ence of complete desynchrony in the system of three os-cillators which can be observed from the presence of threedifferent frequencies and the phase-drifting behaviour ofthe relative phase between each pair of oscillators (Fig.4c-I). The oscillators 1, 2, and 3 exhibit three differentfrequencies which are 11.84 Hz, 10.64 Hz, and 11.99 Hz,respectively (Fig. 4c-II).A novel type of clustering dynamics observed in a net-work of three mutually coupled oscillators is called ro-tating clusters, where we observe the temporal switch-ing from one type of cluster to another type of cluster.To elaborate, the system exhibiting a particular form ofclustering transitions into another form of clustering intime. Both the forms of clustering observed in a rotatingclustered state need not have identical frequencies (Fig.4d-II). Here, we observe two types of clustering: in thefirst type, oscillators 2 and 3 are in-phase synchronizedand they are anti-phase synchronized with the oscillator1. This state of clustering is marked in Fig. 4(d) as Cluster , having a frequency of 11.28 Hz as marked inviolet in Fig. 4(d-II). On the other hand, in the secondtype of clustering, oscillators 1 and 2 are in-phase syn-chronized, and they are anti-phase synchronized with theoscillator 3. This type of clustering is marked in blue as Cluster having a frequency of 11.08 Hz in Fig. 4(d).Such a phenomenon is predominantly observed in thestraight configuration, where the oscillator in the centrehas a degree of two and that on the edges have a degreeof one.In the case of a network of four oscillators, we primarilyobserve two types of clustered states. In the first type ofthe clustered state (Fig. 5a), we observe a pair of clustershaving two oscillators each. The oscillator pairs {
1, 3 } and {
2, 4 } are in-phase synchronized and between theseclusters, we observe anti-phase synchronization. All fouroscillators in the clustering state show a dominant fre-quency of 12.31 Hz. Conversely, in the other type of theclustered state (Fig. 5b), one cluster consisting of threeoscillators (1, 2, 4) which is anti-phase synchronized withanother cluster with a single oscillator (3). Here, everyoscillator in the network exhibits a frequency of 12.09 Hz.Due to the presence of desynchrony in the system ofcandle-flame oscillators, we observe the occurrence ofsymmetry-breaking phenomena such as chimera (Fig. 5c)and weak chimera (Fig. 5d). During the state of chimera,we notice the coexistence of a synchronized pair and adesynchronized pair of oscillators in a network of four os-cillators [16]. In the state of chimera, the oscillator pair {
2, 3 } are synchronized and oscillate at a frequency of11.51 Hz (Fig. 5c-I). The other oscillators 1 and 4 hav-ing frequencies of 11.03 Hz and 10.87 Hz, respectively,are desynchronized with each other and also with thesynchronized pair of oscillators (Fig. 5c-II). During theoccurrence of weak chimera (Fig. 5d), oscillators {
1, 2,and 3 } are frequency synchronized having equal frequen-cies of 11.69 Hz, while oscillator 4 is desynchronized withall three oscillators and has a frequency of 11.09 Hz.Apart from the aforementioned either oscillatory orcompletely quenched (amplitude death) states of coupleddynamics, we also witness the first observation of a dy-namical state called partial amplitude death (PAD) ina mutually coupled network of four candle-flame oscilla-tors. Partial amplitude death is characterized by the co-existence of nearly quenched states and oscillatory (limitcycle) states in a system of coupled oscillators [22, 30].We observe two variants of PAD states in our system. Inthe first type of PAD (Fig. 6a) observed in the straightconfiguration of four oscillators (Fig. 2c-i), the outer os-cillators (1 and 4) are in the quenched state and the innertwo oscillators (2 and 3) are in the oscillatory state. Toelaborate, we observe that the oscillators on either edgehaving a degree of 1 are quenched, while the other two os-cillators in the middle having a degree of 2 are in a stateof desynchronized oscillation. In another form of PAD(Fig. 6b) observed in the star configuration of four oscil-lators (Fig. 2c-iii), the central oscillator 1 exhibits limitcycle oscillations and the other oscillators (2, 3, and 4)that surround the central oscillator are in the quenched FIG. 6. (a), (b) The time series of the global heat releaserate fluctuations ( I ) observed during two different variants ofpartial amplitude death states observed in the network of fourcoupled candle-flame oscillators. state. Here, the three oscillators on the periphery havinga degree of 1 are quenched and that at the centre havinga degree of 3 remains in an oscillatory state. We believethat a system with the coexistence of oscillators havinga degree of 1 and a higher degree is pertinent for the ex-hibition of PAD. The oscillator having degree 1 in suchsystems would be quenched while the oscillators havinga higher degree would remain oscillatory at a particularstrength of the coupling between oscillators.Having discussed all the dynamical states observed ina network of coupled candle-flame oscillators individuallyin Figs. 3 to 6, we now move our attention to categoriz-ing the occurrence of these states in networks of 3 and4 oscillators when the link distances between the oscil-lators are d = 3 and 4 (Fig. 7a,b, respectively). Asmentioned previously, when the number of oscillators ina network is greater than 2 and the distance betweenoscillators is larger ( d > d = 3 (Fig. 7a), we observe the oc-currence of three dynamical states: clustering, chimera,and weak chimera. The percentage of occurrence of thesestates for the configuration is 42%, 27%, and 31%, re-spectively. Such description can be extended to othernetwork topologies. The maximum standard deviationof the percentage occurrence of each dynamical state isapproximately 13%.From Fig. 7, we observe that for a closed-loop net-work (e.g., square network containing equal degree forall oscillators), one stable dynamical state of coupled os-cillators dominates over the other states. For example, FIG. 7. Percentage occurrence of different dynamical statesin a network of coupled candle-flame oscillators for a givennumber of oscillators (specified in braces) arranged in differentconfigurations when the distance between the oscillators is(a) d = 3 and (b) d = 4. Colour indicates the percentage ofoccurrence of a particular state in a given network topology. for a square network of 4 oscillators with d = 3 (Fig.7a), we see the singular dominance of clustering in itsglobal behaviour. Similarly, for the same network at d =4 (Fig. 7b), chimera state dominates with 57 % than thestates of clustering (28 %) and in-phase synchronization(15 %). However, such behaviour of oscillators is not asprominent in the case of a closed-loop triangular network.Furthermore, we observe that the number of stable statesobserved for a given number of oscillators is lesser for aclosed-loop network as compared to open-loop networks.For example, in the case of three oscillators having a linkdistance of d = 4, we observe that the straight config-uration displays four dynamical states, whereas the tri-angular configuration exhibits only two (Fig. 7b). Wecan also note that the global synchronization between alloscillators (state of in-phase synchronization) is observedonly for closed-loop networks. We further note that thePAD states are observed only for open-loop topologies(straight and star configuration) with four oscillators.Similarly, we observe the existence of rotating clustersonly in the open-loop topology (straight configuration)with three oscillators.As we increase the link distance without changing thetopology of the oscillators, we observe the increased ex-istence of desynchrony in the system (as oscillators havedifferent frequencies). To elaborate, for a given topologi-cal arrangement (say, the triangular network with 3 oscil-lators), as we increase the link distance from d = 3 (Fig.7a) to 4 (Fig. 7b), we observe an increase in the existenceof states of weak chimera and a decrease in the occurrenceof synchronized states such as in-phase synchronizationand clustering. Moreover, as the number of oscillators( N ) is increased from 3 to 4, we observe the emergenceof states such as PAD and chimera along with the dis-appearance of states such as rotating clusters. We can extend this argument and conjecture that as the num-ber of oscillators is increased further ( N >
N >
4. In a networkof four oscillators, we observe the occurrence of chimerastate in every topology except for the square topology ata distance of d = 3 (Fig. 7), where the highly stable na-ture of the clustering state restricts the oscillators fromoscillating at different frequencies. The state of chimerain open-loop networks is observed to alternate betweenstates of clustering and in-phase synchronization, calledalternating chimera [23].Having discussed the various dynamical behaviour ex-hibited by networks of candle-flame oscillators consistingof two to four oscillators placed in different topologies, wenext move on to investigating the global dynamics of anannular (regular) network in detail (see Fig. 8). In thiscase, the effect of an increase in the number of oscillatorsfrom 5 to 7 at a fixed link distance of d = 3 is investigated.In the network, as the oscillators are locally coupled totheir nearest neighbours, all oscillators possess a degreeof 2. From Fig. 7, we understood that closed-loop topolo-gies tend to exhibit increased synchrony and stability ascompared to open-loop topologies. In annular networksof oscillators, we primarily observe the states of clustering(CL), weak chimera (WC), and chimera (CH). For net-works having more than four oscillators, the dynamicalstate consisting of two or more frequency synchronizedgroups (each group having oscillators with identical fre-quencies) with one or more oscillators having differentfrequencies in a group is categorized as weak chimera.Conversely, the state of chimera is manifested as the co-existence of a single group consisting of two or more syn-chronized oscillators with two or more desynchronizedoscillators [4, 32]. Note that chimera state is a subset ofweak chimera [33].For link distances of d = 0 and 1, we observe the stateof in-phase synchronization and amplitude death, respec-tively, irrespective of the number of oscillators present inthe annular networks. This is similar to the observationof these dynamical states in other network topologies dis-cussed in Fig. 3. When the link distance between theoscillators is d = 3, the neighbouring oscillators have atendency to exhibit anti-phase synchrony. According toDange et al. [34], anti-phase synchrony is observed be-tween the oscillators at d = 3 due to the alternate shed-ding of vortices from neighboring oscillators. If the num-ber of oscillators in a network is even, there exists a globalsynchrony, in the form of clustering, between the oscilla-tors, where the neighbouring oscillators are locked at 180degrees of phase shift and alternate oscillators are lockedat 0 degrees of phase shift. Hence, global synchrony ismaintained in an annular network at this value of d only FIG. 8. (a)-(c) The effect of an increase in the number of oscillators on the global dynamics of an annular network consisting offive, six, and seven oscillators, respectively. (I) Schematic of annular network topology for different oscillators, (II) bar charts ofdominant frequencies of oscillations corresponding to different dynamical states, namely clustering (CL), weak chimera (WC),and chimera (CH), and (III) percentage occurrence of these dynamical states in a given experiment for these annular networks.The numbers indicated in (a) is a reference number for each oscillator. Roman numerals in (II) indicate the different variantsof the states of CL, WC and CH. if the number of oscillators is even and not when it is odd.During the state of clustering, the network separates intotwo clusters consisting of equal number of oscillators. Forexample, in a square network of four oscillators, the stateof clustering is observed with the formation of two clus-ters, each having two oscillators (Fig. 5a). Similarly,for an annular network of six oscillators (see Fig. 8b),we observe the formation of two clusters, each consistingof three oscillators. In both the cases discussed above,the formation of two clusters occurs such that adjacentoscillators belong to different clusters.In contrast, in annular networks having odd number ofoscillators with
N >
3, we do not observe the existenceof clustering states; however, we observe the occurrenceof only chimera and weak chimera states. These statesoccur in various forms which are referred to as variantsof the given dynamical state. As the number of oscilla-tors in an annular topology is increased, we observe anincrease in the number of variants of weak chimera andchimera states exhibited by these oscillators (Fig. 8).Here, the number of oscillators in each frequency syn-chronized group or the number of frequency synchronizedgroups is different in each variant of the weak chimerastate. For example, we observe the existence of two vari- ants of weak chimera in a network with five oscillators(Fig. 8a-II-i and Fig. 8a-II-ii) and four variants in thenetwork with seven oscillators (Fig. 8c-II-i to Fig. 8c-II-iv). Similarly, we observe single variant of chimera state(Fig. 8a-II-iii) in a network with five oscillators and twovariants (Fig. 8c-II-v and Fig. 8c-II-vi) in a network withseven oscillators. Thus, in an annular network of sevenoscillators, we observe that the chimera states have agreater number of desynchronized oscillators (Fig. 8c-II-vi), as opposed to the annular network with five oscil-lators which predominantly has synchronized oscillators(Fig. 8a-II-iii). We also observe an increase in the per-centage occurrence of chimera states as the number ofoscillators is increased from five to seven (compare Fig.8a-III and 8c-III) which, in turn, points towards the in-crease in the stability of these symmetry-breaking statesin larger oscillator networks.
IV. CONCLUSION
To summarize, in this paper, we investigate the depen-dency of parameters such as the number of oscillators,coupling topology, and the strength of interaction on theglobal behaviour of a minimal network of limit cycle os-cillators. Towards this purpose, we perform an exper-imental investigation on candle-flame oscillators, wherethe number of oscillators in a network is increased from2 to 4, the oscillators are coupled in closed-loop (triangleand square) or open-loop (straight and star) networksand the strength of coupling between the oscillators isdecreased by increasing the distance between them.When the oscillators are very close to each other, thecoupling strength between them is very high. As a result,global dynamical behaviours exhibited by these oscilla-tors, i.e., the state of in-phase synchronization ( d = 0)and amplitude death ( d = 1), are highly stable (sus-tained for longer duration). We also observe that theoccurrence of these dynamical states is independent ofother parameters, such as the number of oscillators andthe network topologies considered in the study. How-ever, as the distance between the oscillators is increased( d > d = 3, the state of clustering is exhibited with theformation of two clusters, with adjacent oscillators being allotted in different clusters. Conversely, we observe onlyweak chimera and chimera states in networks with oddnumber of oscillators and the stability of chimera statesincreases as the number of oscillators in the network isincreased.Thus, the present experimental study highlightsthat coupled behaviour of limit cycle oscillators inminimal networks depends on the number of oscillators,the coupling strength between the oscillators, andthe coupling structure/topology of these oscillators.Further investigation through mathematical models ornumerical simulations of the system is needed to deepenthe physical understanding on the dependence of thenetwork topology on the coupled interaction of candleflame oscillators. We strongly believe that these resultson topological dependence can be extended to othersystems such as power grids [35], neuronal networks[36], vortex interactions in turbulence [37], and seizuredynamics [38]. Stability of national power grids andelectrical grids is highly dependent on the synchrony invarious critical infrastructure [35]. Healthy brains havesparse connectivity, whereas epileptic brain has richconnectivity with a modular structure which plays a rolein the functional organization of the brain cells [36, 38].Furthermore, the interaction of vortices in a turbulentflow governs its global dynamics [37]. ACKNOWLEDGMENTS
We acknowledge the assistance of Mr. Thilakaraj andMr. Anand (IIT Madras) in the development and con-struction of our experimental setup with deep gratitude.We are greaterful to Mr. Dave (ISSER Tirupati) andMr. Kadokawa (Hokkaido University) for their help inexperiments. This work is supported by ONRG, USA(Contract Manager: Dr. R. Kolar, Grant No.: N62909-18-1-2061). [1] A. Pikovsky, M. Rosenblum, and J. Kurths,
Synchroniza-tion: A Universal Concept in Nonlinear Sciences , Vol. 12(Cambridge University Press, 2003).[2] S. Strogatz,
Sync: The emerging science of spontaneousorder (Penguin UK, 2004).[3] Y. Kuramoto,
Chaos and Statistical Methods: Proceed-ings of the Sixth Kyoto Summer Institute, Kyoto, JapanSeptember 12–15, 1983 , Vol. 24 (Springer Science & Busi-ness Media, 2012).[4] E. A. Martens, S. Thutupalli, A. Fourri`ere, and O. Hal-latschek, Proceedings of the National Academy of Sci-ences , 10563 (2013).[5] T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski,and Y. Maistrenko, Scientific Reports , 6379 (2014).[6] P. V. Savi, M. A. Savi, and B. Borges, arXiv preprintarXiv:2004.03495 (2020).[7] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Nature Communications , 1(2014).[8] K. Premalatha, V. Chandrasekar, M. Senthilvelan, andM. Lakshmanan, Chaos , 033110 (2018).[9] G. Saxena, A. Prasad, and R. Ramaswamy, Physics Re-ports , 205 (2012).[10] D. M. Abrams and S. H. Strogatz, Physical Review Let-ters , 174102 (2004).[11] J. H. Sheeba, V. Chandrasekar, and M. Lakshmanan,Physical Review E , 055203 (2009).[12] E. Omelchenko, M. Wolfrum, and Y. L. Maistrenko,Physical Review E , 065201 (2010).[13] M. Shanahan, Chaos , 013108 (2010).[14] P. Ashwin and O. Burylko, Chaos , 013106 (2015).[15] J. Wojewoda, K. Czolczynski, Y. Maistrenko, andT. Kapitaniak, Scientific Reports , 34329 (2016).[16] J. D. Hart, K. Bansal, T. E. Murphy, and R. Roy, Chaos , 094801 (2016).[17] F. P. Kemeth, S. W. Haugland, and K. Krischer, Phys-ical Review Letters , 214101 (2018).[18] Y. Kuramoto and D. Battogtokh, arXiv preprint cond-mat/0210694 (2002).[19] A. M. Hagerstrom, T. E. Murphy, R. Roy, P. H¨ovel,I. Omelchenko, and E. Sch¨oll, Nature Physics , 658(2012).[20] A. Arenas, A. D´ıaz-Guilera, J. Kurths, Y. Moreno, andC. Zhou, Physics Reports , 93 (2008).[21] Y. Maistrenko, S. Brezetsky, P. Jaros, R. Levchenko, andT. Kapitaniak, Physical Review E , 010203 (2017).[22] A. Sharma, Physics Letters A , 1865 (2019).[23] K. Manoj, S. A. Pawar, S. Dange, S. Mondal, R. I. Sujith,E. Surovyatkina, and J. Kurths, Physical Review E ,062204 (2019).[24] H. Kitahata, J. Taguchi, M. Nagayama, T. Sakurai,Y. Ikura, A. Osa, Y. Sumino, M. Tanaka, E. Yokoyama,and H. Miike, Journal of Physical Chemistry A , 8164(2009).[25] K. Manoj, S. A. Pawar, and R. I. Sujith, Scientific Re-ports , 11626 (2018).[26] K. Okamoto, A. Kijima, Y. Umeno, and H. Shima, Sci-entific Reports , 36145 (2016). [27] M. Wickramasinghe and I. Z. Kiss, PloS One , e80586(2013).[28] A.-L. Barab´asi et al. , Network Science (Cambridge uni-versity press, 2016).[29] Y. Hardalupas and M. Orain, Combustion and Flame , 188 (2004).[30] F. M. Atay, Physica D: Nonlinear Phenomena , 1(2003).[31] M. Wolfrum and E. Omelchenko, Physical Review E ,015201 (2011).[32] M. R. Tinsley, S. Nkomo, and K. Showalter, NaturePhysics , 662 (2012).[33] C. Bick, M. Sebek, and I. Z. Kiss, Physical Review Let-ters , 168301 (2017).[34] S. Dange, S. A. Pawar, K. Manoj, and R. I. Sujith, AIPAdvances , 015119 (2019).[35] F. D¨orfler, M. Chertkov, and F. Bullo, Proceedings ofthe National Academy of Sciences , 2005 (2013).[36] E. D. Herzog, Nature Reviews Neuroscience , 790(2007).[37] K. Taira, A. G. Nair, and S. L. Brunton, Journal of FluidMechanics (2016).[38] M. Chavez, M. Valencia, V. Navarro, V. Latora, andJ. Martinerie, Physical Review Letters104