Synchronizability of chaotic logistic maps in delayed complex networks
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Synchronizability of chaotic logistic maps in delayed complexnetworks
Marcelo Ponce C. , C. Masoller and Arturo C. Mart´ı Instituto de F´ısica, Facultad de Ciencias, Universidad de la Rep´ublica, Igu´a 4225, Montevideo 11400, Uruguay Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Colom 11, E-08222 Terrassa, Barcelona,Spain July 25, 2018
Abstract.
We study a network of coupled logistic maps whose interactions occur with a certain distributionof delay times. The local dynamics is chaotic in the absence of coupling and thus the network is a paradigmof a complex system. There are two regimes of synchronization, depending on the distribution of delays:when the delays are sufficiently heterogeneous the network synchronizes on a steady-state (that is unstablefor the uncoupled maps); when the delays are homogeneous, it synchronizes in a time-dependent state(that is either periodic or chaotic). Using two global indicators we quantify the synchronizability on thetwo regimes, focusing on the roles of the network connectivity and the topology. The connectivity ismeasured in terms of the average number of links per node, and we consider various topologies (scale-free,small-world, star, and nearest-neighbor with and without a central hub). With weak connectivity and weakcoupling strength, the network displays an irregular oscillatory dynamics that is largely independent ofthe topology and of the delay distribution. With heterogeneous delays, we find a threshold connectivitylevel below which the network does not synchronize, regardless of the network size. This minimum averagenumber of neighbors seems to be independent of the delay distribution. We also analyze the effect of self-feedback loops and find that they have an impact on the synchronizability of small networks with largecoupling strengths. The influence of feedback, enhancing or degrading synchronization, depends on thetopology and on the distribution of delays.
PACS.
Complex networks arise due to self-organization phenom-ena in many real systems, such as food webs, the Inter-net, social networks, genes, cells and neurons [1]. A lotof research has been devoted to understanding the collec-tive behavior emerging in complex networks [2], given theindividual dynamics of the nodes and the coupling archi-tecture. Ecological webs, for example, describe species bymeans of nodes connected by links, representing the inter-actions. The interactions can be either direct or indirectthrough intermediate species, and can be of antagonistictype, such as predation, parasitism, etc., or mutually ben-eficial, such as those involving the pollination of flowersby insects [3]. A study of the network architecture decom-pose food webs in spanning trees and loop-forming links,revealing common principles underlying the organizationof different ecosystems [4]. Cells use metabolic networks ofinteracting molecular components in processes that gener-ate mass, energy, information transfer and cell-fate speci-fication [5]. Adaptation and robustness have been shownto be consequences of the network’s connectivity and donot require the ’fine-tuning’ of parameters [6]. In a cat’s brain, functional connectivity has been studied within theframework of complex networks [7]. In the human brain,magnetic resonance imaging has been used to extract func-tional networks connecting correlated human brain sites[8]. Analysis of the resulting networks in various tasks(e.g., move a finger) has shown interesting features in thebrain, such as scale-free structure, a high clustering coef-ficient, and a small characteristic path length.Complex network are relevant not only from an aca-demical point of view but also from an applied perspec-tive. Models based on complex networks for the spread ofdiseases have identified mitigation strategies for epidemicspread. In [9] the spread of an infection was analyzed fordifferent population structures, ranging from ordered lat-tices to random graphs, and it was shown that for the moreordered structures, there was a fluctuating endemic stateof low infection. In [10] it was shown that outbreaks can becontained by a strategy of targeted vaccination combinedwith early detection, thus avoiding mass vaccination of thehole population. In communication networks, error toler-ance and attack vulnerability are key issues. In [11] it wasshown that error tolerance is not shared by all networks
Marcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... that contain redundant wiring, but is displayed only bythose with scale-free topology.The structure of a network is a key issue in determiningits functional properties. In real networks communities,or modules, associated with highly interconnected parts,have been identified [12,13]. A nice example of a com-munity structure has been unveil in networks of musicaltastes [14], having also practical applications for the de-velopment of commercial music recommendation engines.Well-characterized modules have been identified in syn-thetic gene networks [15], where positive feedback andnoise play important roles for the repression and the acti-vation of gene expression. An excitable module containingpositive and negative feedback loops has been identifiedas a key mechanism inducing transient cellular differenti-ation [16]. A method for classifying nodes into universalroles according to their intra- and inter-module connec-tions has been recently proposed [17] and applied to thestudy metabolic networks.The synchronizability of a network, or is propensityfor synchronization, is another key issue in determiningthe network functional properties [18]. Why some topolo-gies are easier to synchronize than others, is still an is-sue not fully understood. Heterogeneity in the connec-tion strengths tends to enhance synchronization [19]. Aweighting procedure based upon the global structure ofnetwork pathways has been shown to improve synchro-nizability [20]. Dynamical adaptation, where the couplingstrengths develop according to the local synchronizationof the node and its neighbors, resulting in weighted cou-pling strengths that are correlated with the topology, alsoenhances synchronizability [21]. However, heterogeneity inthe connectivity distribution can have the opposite effect:networks with homogeneous connectivity have been foundto have larger propensity for synchronization than the het-erogeneous ones [22].The speed of transmission of information among thenetwork also affects the synchronizability. Instantaneousinteractions have been studied a lot in spite of the factin many situations they are not realistic, because the in-formation propagates with a finite speed. A more realis-tic scenario considers that the links have associated delaytimes, and that the delays are the same for all the links.It has been shown that the presence of such uniform de-lays in the communications among the nodes can result inenhanced synchronizability. In [23] it was shown that, ina network of chaotic maps, it was possible to synchronizethe delayed network where the undelayed network, withinstantaneous links, did not.An even more realistic approach for complex and disor-dered systems is to consider heterogeneously distributeddelays. In previous work [24,25] we studied networks ofcoupled maps with links that have heterogenous delays,and investigated the relation of the network topology withits ability to synchronize. We found that (i) the synchro-nizability was enhanced by random delays as compared tonetworks with uniform delays, (ii) the network synchro-nized in a steady state in the presence of random delays(in contrast, with uniform delays the synchronization is in time a dependent state, [23]) and (iii) the synchroniz-ability depends mainly on the mean connectivity and israther independent of the topology.The aim of this paper is to further analyze how thesynchronizability depends on the connectivity and on thetopology, when there are delays in the links among thenodes. We consider both regular and random network topolo-gies, covering the cases of homogeneous and heterogeneousdistribution of the links. We also study the effects of cen-trality and locality on the dynamics of the array. We char-acterize the synchronizability in terms of two indicators,one that tends to zero when the networks synchronizes, re-gardless if the synchronization is in a time dependent or ina steady state, and the other that tends to zero only whenthe synchronization is in a steady state. Using these in-dicators we also analyze the impact of self-feedback links.This paper is organized as follows. Section 2 presents themodel, the different distributions of delays, and severaltopologies used. Section 3 presents the results of the sim-ulations, and, finally, Sec. 4 presents a summary and theconclusions.
We consider N logistic maps coupled as: x i ( t + 1) = (1 − ǫ ) f [ x i ( t )] + ǫb i N X j =1 η ij f [ x j ( t − τ ij )] , (1)where t is a discrete time index, i is a discrete spatialindex ( i = 1 . . . N ), f ( x ) = ax (1 − x ) is the logistic map, ǫ is the coupling strength and τ ij ≥ i th and j th nodes (the delaytimes τ ij and τ ji need not be equal). The matrix η = ( η ij )defines the connectivity of the network: η ij = η ji = 1if there is a link between the i th and j th nodes, and zerootherwise. The sum in Eq. (1) runs over the b i nodes whichare coupled to the i th node, b i = P j η ij . The normalizedpre-factor 1 /b i means that each map receives the sametotal input from its neighbors.A particularly simple solution of Eq. (1) is such thatall the maps of the network are in a fixed point of theuncoupled map, i.e., x i ( t ) = x ∀ i, (2)with x = f ( x ). While this solution exists for all delaydistributions ( τ ij ) and for all coupling topologies ( η ij ),the statistical linear stability analysis performed in [24]showed that this solution is unstable unless the distribu-tion of delays is wide enough. In the other limiting case ofall-equal delays, τ ij = τ ∀ i and j , it was shown in Ref.[23]that the network synchronizes isochronally, in a spatiallyhomogeneous time-dependent state: x i ( t ) = x ( t ) ∀ i, (3)with x ( t + 1) = (1 − ǫ ) f [ x ( t )] + ǫf [ x ( t − τ )]. arcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... 3 We consider delays distributed as: τ ij = τ + near( cξ ),where c is a parameter that allows varying the width ofthe delay distribution; ξ is Gaussian distributed with zeromean and standard deviation one; near denotes the near-est integer. Depending on τ and c the distribution of de-lays is truncated to avoid negative delays.The synchonizability of the network depends on both,the mean delay and the width of the delay distribution[24]. The larger the value of τ , the larger the dispersionof the delays has to be, for the network to synchronize inthe steady state (see Fig. 3 of [24]). A similar observationwas recently reported in [32], for an integro-differentialequation describing the collective dynamics of a neuralnetwork with distributed signal delays. An interesting in-terpretation of both observations is provided by the workof Morgado et al. [33]: a certain degree of randomnessin a network (due to random delays, random connectiv-ity, or even random initial conditions) results in additiveand/or multiplicative noise terms in an ”effective” single-node equation of motion for x i ( t + 1), expressed in termsof x i ( t ) and nonlinear feedback memory terms, that forthe Logistic map are of polynomial type in x i ( t − n ) with1 < n < t . In [33] it was found that the right memoryprofile can create optimal conditions for synchronization,in other words, an optimal memory range enhances a net-work propensity for synchronization.Since our aim is to study the roles of the topologyand of the connectivity, we keep fixed the distribution ofdelays, defined by the parameters τ and c . However, wecompare the results obtained for distributed delays ( c = 0)with those obtained for all-equal delays ( c = 0) and withthose obtained for no delays ( τ = 0, c = 0). The connectivity of a network is measured in terms of theaverage number of links per node, h b i = 1 N N X i =1 b i . (4)We consider the five topologies, three of them are reg-ular networks where the links are distributed determinis-tically among the nodes, while the other two are heteroge-nous networks where the links are distributed stochasti-cally, with given rules. The networks are:(i) a nearest-neighbor network (referred to as NN net-work) with periodic boundary conditions where each node i is linked to its neighboring nodes i ± , i ± , ..., i ± K ,with K an integer. The number of neighbors is the samefor all the nodes, and h b i = 2 K .(ii) the NN network with the addition of a central nodeconnected to all other nodes (referred to as ST network).In this case, N − K + 1 links and one nodehas N − h b i = [(2 K +1)( N − N − /N =(2 K + 2)( N − /N . (iii) a star-type network where there are K centralnodes that are connected to all other nodes and N − K peripheric nodes that are connected only to the centralones (referred to as KA). In this case, K nodes have N − N − K nodes have K links. Thus, h b i =[ K ( N −
1) + ( N − K ) K ] /N . This network has a pure star-type structure and is centrally organized , in contrast toNN, that is locally organized .(iv) a small-world (SW) network constructed accordingto the Newman and Watts algorithm [26].(v) a scale-free (SF) network constructed according tothe Barabasi and Albert method [27].Recently, the influence of feedback loops on the syn-chronization of complex systems has received attention.In networks of globally coupled units (self-sustained os-cillators or maps) delayed feedback in the mean field can,depending on the feedback strength and on the delay time,enhance or suppress synchronization [28]. In a system com-posed by two phase oscillators with instantaneous mutualcoupling, a delayed feedback loop in each oscillator en-hances synchronizability if the coupling strength is nottoo strong, while degrades synchronizability if the cou-pling exceeds a threshold value [29]. In small networks ofmutually delayed-coupled oscillators, which typically donot exhibit stable isochronal synchronization, by includ-ing delayed feedback loops to the nodes, the oscillatorsbecome isochronally synchronized [30,31]. To analyze therelevance of feedback loops we consider two situations: thediagonal elements of the coupling matrix, η ii , are all setequal to 1, or are all set equal to 0. In this way, in eachnode a feedback loop is included, or is forbidden. The de-lays of the feedback loops, τ ii , have the same distributionas the delays of the mutual interactions, τ ij . To characterize the degree of synchronization and to dis-tinguish between steady state and time dependent syn-chronization, we use the following indicators, σ = 1 N h X i ( x i − h x i s ) i t (5) σ ′ = 1 N h X i ( x i − x ) i t , (6)where h . i s denotes a space average over the nodes of thenetwork, h . i t denotes a time average, and x is the fixedpoint of the uncoupled logistic map, x = f ( x ). σ = 0if and only if x i = x j ∀ i , j , while σ ′ = 0 if and only if x i = x ∀ i . Thus, σ ′ allows to distinguish synchroniza-tion in the steady state from synchronization in an timedependent state. In the former case, both σ and σ ′ arezero, in the latter case, only σ = 0. In this section we present the results of the simulations.We consider a network of N = 200 logistic maps with Marcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... a = 4 interacting with different topologies, as describedabove. The parameters of the Gaussian delay distributionare τ = 5 and c = 2. Similar results are found for othervalues of τ and c , with c large enough [24]. To asses therole of distributed delays, we compare with two limits: (i)instantaneous coupling, τ ij = 0 ∀ i , j , and (ii) homoge-neous delays, τ ij = τ ∀ i , j ( c = 0).The simulations start from a random initial configu-ration, with x i (0) randomly distributed in [0,1], and themaps evolve initially without coupling, during a time in-terval 0 < t < max ( τ ij ), because the integration of de-layed equations requires the knowledge of the past state ofthe system over a time interval given by the maximum de-lay. After that, the coupling is turned on. We neglect tran-sient effects disregarding a few thousands of iterations.Networks of coupled elements usually show multista-bility: different initial conditions lead to different finalstates, and delayed coupling tends to increase the numberof coexisting states [34,35]. Multistability is also enhancedwhen the local dynamics of the uncoupled maps presentstwo or more competing attractors [36], which does not oc-cur for the logistic map with a = 4. When the couplingis weak and the network is not synchronized, we observea large diversity of dynamical clustered states. However,in this ”weak coupling regime” the global synchronizationindicators, σ and σ ′ , depend mainly on ǫ and only in asmall extend on the initial conditions, the distribution ofdelays, the network topology and the connectivity (withthe exception of all-even delays, for them there is a syn-chronization ”island” that will be discussed below). In thefollowing, the plots of σ and σ ′ are done by averagingover 10 different states generated from randomly choseninitial conditions. The only exception is Fig. 6, that isdone with just one initial condition (that is the same forall ǫ ).In the first section we characterize the network propen-sity for synchronization in terms of the indicator σ ; in thesecond section, in terms of σ ′ . In these two sections thenetwork does not contain self-feedback links ( η ii = 0 ∀ i ).In the last section we asses the role of feedback links, bystudying the same networks (with the same delay distri-butions and topologies) but with a feedback link in eachnode ( η ii = 1 ∀ i ). σ Figures 1 and 2 display color-coded plots of the synchro-nization indicator σ as a function of the coupling strength, ǫ , and the average number of neighbors, h b i , for the dif-ferent topologies and delay distributions.Figure 1 displays results for the regular topologies (NN:top row, ST: central row, KA: bottom row) and the threedistributions of delays considered (no-delays: left column,homogeneous delays; central column, and distributed de-lays: right column). Figure 2 displays results for the het-erogeneous networks (SW: top row, SF: bottom row) andthe same delay distributions.We observe a general trend towards synchronizationwhen increasing the coupling strength and the average ε < b > Fig. 1. (Color online) Plot of σ in the parameter space (cou-pling strength, average number of neighbors) for three net-works with regular topologies: nearest-neighbors (NN), toprow, nearest-neighbors with central node (ST), middle row,and bottom row K-to-all (KA). The delays are: zero (left col-umn), homogeneous ( τ ij = 5 ∀ i and j , central column), andheterogeneous ( τ = 5, c = 2, right column). ε < b > Fig. 2. (Color online) Plot of the order parameter σ , Eq.5in the parameter space (coupling strength, average number ofneighbors) for two heterogeneous networks of 200 nodes: SW(small-world, top row), SF (scale-free, bottom row). The de-lays are as in Fig. 1: no-delays (left column), homogeneous de-lays (center column), and heterogeneous delays (right column).Other parameters as Fig. 1.arcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... 5 number of links. However we note some important differ-ences depending on the topology and on the delays.(i) Heterogeneous delays : if h b i is large enough, the syn-chronizability does not dependent on the topology, notethe remarkable similarity of the five panels in the rightcolumns of Figs. 1 and 2, for h b i & No-delays and homogeneous delays (left and centralcolumns in Figs. 1 and 2): the topology makes a signifi-cant difference in regular networks (Fig. 1), and is less im-portant in heterogenous networks (Fig. 2), although somedifferences can be observed for small h b i ( h b i . Regular networks: in Fig. 1, comparing the pan-els in the right and central columns we notice that thenetwork KA (bottom row) is the one with better propen-sity to synchronization, while the network NN (top row) isthe one exhibiting poorer synchronizability. We also noticethat ST (middle row) has better synchronizability thanNN when h b i is small, but there are no significant differ-ences between them for large enough h b i .We note that for larger values of h b i (not shown in Fig.1 because we focus on the weak connectivity region) theNN network eventually synchronizes. Synchronization oc-curs for ǫ above a certain value, ǫ ∗ , that decreases with in-creasing h b i , in good agreement with the results of Ref.[37](see in particular Fig. 4(b) of [37], where the variable α plays the role of h b i here; increasing h b i resulting in a tran-sition from local to global coupling).(iv) Heterogeneous networks: in Fig. 2 we do not ob-serve a significant difference: the panels in the top andbottom rows are similar, at least for h b i > ǫ ∼ σ ′ In order to investigate in more detail the two synchroniza-tion regimes (time-dependent and fixed-point synchroniza-tion) now we consider the synchronization indicator σ ′ ,Eq. (6), that tends to zero only when the network syn-chronizes in the fixed point. Figures 3 and 4 display color-coded plots of σ ′ as a function of the coupling strength, ǫ , and the average number of neighbors, h b i , for the samedelay distributions and network topologies as Figs. 2 and3 respectively.In the case of heterogenous delays, again we notice a re-markable similarity in the five panels in the right columnsof Figs. 3 and 4 for h b i large enough, confirming that thesynchronizability of the network is largely independent ofits topology. However, we note that for the topology KA(bottom panel, right column in Fig. 3) the network lossessynchrony if the coupling is too strong ( ǫ ≈ ε < b > Fig. 3. (Color online) Plot of the order parameter σ ′ , Eq. (6)in the parameter space (coupling strength, average number ofneighbors) for three networks with regular topologies: nearest-neighbors (NN), top row, nearest-neighbors with central node(ST), middle row, and nearest-neighbors with two centralnodes (KA). Parameters are as in Fig. 1. Regular networks: comparing the panels in the rightand central columns of Fig. 3, we can notice that the net-work KA (bottom row) is the one with poorer synchroniz-ability in the steady-state.
Heterogenous networks: in Fig. 4 is observed that theSW and SF topologies have very similar propensity forsynchronization in the steady-state.It can be noticed that, as the average number of linksper node and the coupling strength increase, the networkswith heterogenous delays tend to synchronize in the steadystate, while the networks with no-delays or with homoge-nous delays do not [in Figs. 3 and 4, compare the leftand central columns with the right column]. This ten-dency is particularly clear in heterogeneous networks, Fig.4, where in the left and in the central columns σ ′ is largefor h b i ≥
15 and ǫ ≥ . σ ′ is zero orvery small in that region.The “synchronization island” discussed in the previoussection is not observed in Figs. 3 and 4 because in this ”is-land” the network synchronizes in a time-dependent statethat has σ ′ different from zero. In the presence of heterogeneous delays, it can be expectedthat the average number of neighbors has to be above acertain value for the distribution of delays to play an effec-tive role. This is indeed shown in Fig. 5, where we plot thesynchronization indicators σ and σ ′ in the plane ( c/τ , h b i ) for various network sizes and the SW topology. σ and σ ′ are here plotted on a logarithmic scale to revealthe following features of the synchronization transition: Marcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... ε < b > Fig. 4. (Color online) Plot of σ ′ in the parameter space (cou-pling strength, average number of neighbors) for three hetero-geneous networks: SW (small-world, top row), RN (randomnetwork, middle row), SF (scale-free, bottom row); and thedifferent delay distributions; instantaneous (left column), ho-mogeneous delays (center column), and heterogeneous delays(right column). Parameters are as in Fig. 1. for connectivity below h b i ≈
10, the network does not syn-chronize, regardless of the network size and of the width ofthe delay distribution. Similar results are found for othervalues of ǫ and τ . For h b i ≥
10, as c increases and the de-lays become more heterogeneously distributed there is asharp transition to isochronous synchronization, reveled inthe plot of σ (left column), which shows a sharp boundarybetween the dark and light regions, at c/τ slightly below0 .
5. The smooth transition seen in the plot of σ ′ (rightcolumn) reveals that initially the network does not syn-chronize in the fixed point but in a time-dependent state,and as c increases and the delay distribution enlarges, itgradually approaches the fixed point. We assess the effect of self-feedback links by setting η ii = 1 ∀ i , in the previously studied network topologies.In a large enough network (as the one studied so far,with N = 200 maps), the simulations show that feedbacklinks have a small influence on the network synchroniz-ability when the connectivity is low (when h b i . . N ),and their effect is negligible when h b i is larger. With 2Dcolor-coded plots is difficult to appreciate the influence ofthe feedback links; therefore, we plot in Fig. 6 the globalsynchronization indicators vs. the coupling strength, for afixed value of h b i , that is as low as possible.We consider three regular topologies: (i) a ring withtwo neighbors per node and a self-feedback link in eachnode (black solid line); (ii) the same ring without the self-feedback links (dot-dashed line, blue online); and (iii) aST network composed by the ring with two neighbors pernode (without self-feedback links) and a centrally con-nected node (dashed line, red online). It can be noticedthat with no delays and with uniform delays (left and τ < b > N=8000 0.5 1010203040 N=8000 0.5 1010203040 −80−60−40−200 N=200
Fig. 5. (Color online) Plot of σ (left column) and σ ′ (rightcolumn) in logarithmic scale, in the parameter space (normal-ized width of the delay distribution, c/τ , average connectivity, h b i ) for three system sizes: N = 200 (top row), N = 400 (mid-dle row), and N = 800 (bottom row). Parameters are τ = 5and ǫ = 1, SW topology. central columns in Fig. 6), the ST network synchronizesfor ǫ large enough, while the ring, with and without self-feedback links, does not. With randomly distributed de-lays (right column in Fig. 6), the three networks do notsynchronize for any value of ǫ .Let us now consider the influence of feedback links ina smaller network of N = 20 maps. Figure 6 displays theglobal synchronization indicator while Figs. 7-9 presentsome examples of the space and time evolution of the net-work. The figures are done by representing in a color scalethe variable x i ( t ), with the space index i on the horizontalaxis and the time index t on the vertical axis. The left col-umn of Figs. 6-9 shows results for networks without feed-back loops, and the right column, for the same networkswith feedback loops. A relevant effect of the feedback canbe observed on both, the global macroscopic indicator andon the microscopic network configuration. There are situ-ations in which feedback loops enhance coherence, givingrise to spatially more ordered patterns (e.g., in Fig. 7,top and bottom row), while in others, on the contrary,feedback loops destroy the spatial coherence of the pat-tern (e.g., in Fig. 9). Interestingly, it can be observed thatwith random delays, Fig. 9, the synchronization is not al-ways on a homogeneous state, but there are also staticpatterns with spatial ”antiphase” arragement. The char-acterization of these patterns is in progress and will bereported elsewhere. arcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... 7 σ ε σ ’ σ ’ Fig. 6. (Color online) Plot of σ (top row) and σ ′ (bottomrow) vs. the coupling strength for a NN network with twoneighbors per node and a feedback link (solid line), for the samering without feedback links (dot-dashed line, blue online) andfor the ST network composed by the ring plus a central node(dashed line, red online). The delays and other parameters areas in the previous figures: no-delays (left column), homoge-neous delays (central column) and heterogeneous delays (rightcolumn). ε σ ε σ σ Fig. 7. (Color online) Synchronization indicator σ for a smallnetwork of N = 21 nodes without feedback loops (left column),and with feedback loops (right column). The delay distribu-tions are: no delays (top row), fixed delays (middle row), anddistributed delays (bottom row). The topologies are: a nearest-neighbors ring (black), a ring with the addition of a centralnode (red) and a pure star network composed by a central hubconnected to N = 20 nodes (blue). i t i Fig. 8. (Color online) Synchronization patterns with instan-taneous interactions. The network topologies are: a nearest-neighbors ring of N = 20 nodes (top row); a N = 20 nearest-neighbors ring with the addition of a central node (middle row)and a star network composed by a central hub connected to N = 20 nodes (bottom row). Without feedback loops (leftcolumn), with feedback loops (right column). The couplingstrength is ǫ = 1 i t i Fig. 9. (Color online) Synchronization patterns with homoge-neous delays ( τ = 5, c = 0). The network topologies are as inFig. 7. (a),(b) ǫ = 1; (c)-(f) ǫ = 0 .
8. Without feedback loops(left column), with feedback loops (right column).
It is also interesting to notice that when the couplingstrength is small (roughly speaking, when ǫ < .
2) the de-pendence of the global synchronization indicators σ and σ ′ with ǫ , shown in Figs. 6 and 6, is very similar. Thissuggests that the global dynamics is almost independentof the topology, the connectivity, the network size and thedelay distribution. We refer to this region as the ”weakcoupling” region. It has been recently shown that in thisregion all the nodes exhibit a qualitatively similar sym- Marcelo Ponce C., C. Masoller, Arturo C. Mart´ı: Synchronizability of chaotic logistic maps... i t i Fig. 10. (Color online) Synchronization patterns with hetero-geneous delays ( τ = 5, c = 2). The network topologies are asin Figs. 7 and 8. (a),(b) ǫ = 0 .
9; (c),(d) ǫ = 1; (e),(f) ǫ = 0 . bolic dynamics, that, for instantaneous interactions, de-pends mainly on the network architecture and only to asmall extent, on the local dynamics [40]. We studied the synchronizability of a network focusing onthe roles of the connectivity, the topology, and the delaytimes that are associated with the links. The nodes weremodeled by chaotic logistic maps, and various topologiesand delay distributions were considered. For low connec-tivity (roughly speaking, when the mean number of linksper node is h b i < . N ), and for weak coupling ( ǫ < . σ and σ ′ [Eqs. (5) and (6)], the first one tends to zero whenthe network synchronizes isochronously [ x i ( t ) = x j ( t ) ∀ i , j ], while the second one tends to zero only when thenetwork synchronizes in the steady state [ x i ( t ) = x ∀ i ,with x = f ( x ) being the fixed point of the uncoupledmaps].When the coupling strength is weak the dependenceof the synchronization indicators with ǫ is very similarfor all topologies and delay distributions considered. Thissuggest that in this region of ”weak coupling” the net-work topology and the delay distribution play no relevant role in the dynamics. However, is important to remarkthat the global synchronization indicators employed, σ and σ ′ , have the limitation that they characterize onlyisochronal synchronization and steady-state synchroniza-tion respectively. In a complex network where the inter-actions among the units are not instantaneous, other syn-chronization patterns are also expected, such as stateswhere the nodes are synchronized but with lag-times be-tween them [38,39]. An interesting indicator to analyze infuture studies is the one that measures the average dis-tance between the present state of a map and the delayedstate of the maps interacting with it: σ = (1 /N ) X i (1 /b i ) X j η ij h ( x i ( t ) − x j ( t − τ ij )) i t . Also, it will be very interesting to analyze synchronizationpatterns using symbolic dynamics [40] and complexity in-dicators [41,42].
With heterogeneous delays we also found that there isa connectivity threshold below which the network does notsynchronize, regardless of the network size. This minimumaverage number of neighbors is also independent of the de-lay distribution. Above the minimum connectivity level, asthe width of the distribution increases, the plot of σ re-veals that there is a sharp transition to isochronous syn-chronization, while the plot of σ ′ reveals that the net-work does not synchronize in the fixed point but in a time-dependent state, that gradually approaches the fixed pointas the delays become more heterogeneous. Simulations werepreformed for SW topology; the study of how this con-nectivity threshold depends on the network topology is inprogress and will be reported elsewhere. We studied the influence of feedback loops in each nodeof the network; these feedback loops having the same de-lay distribution as the mutual interactions, and found thatwhen the network is large, feedback loops have very littleimpact on the global synchronization indicators. However,they affect synchronizability of small networks, enhancingor degrading the synchronization, depending on the net-work architecture and on the delay distribution. As a fu-ture study it will be interesting to analyze the interplay ofdelayed feedback with instantaneous mutual coupling andvice-versa, instantaneous feedback with delayed coupling.
ACM and MPC acknowledge financial support from CSICand PEDECIBA (Uruguay). CM acknowledges supportfrom the “Ramon y Cajal” Program (Spain) and the Eu-ropean Commission (GABA project, FP6-NEST 043309).
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