Generalized connection graph method for synchronization in asymmetrical networks

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1 Physic D 4 006) Generlized connection grph method for synchroniztion in symmetricl networks Igor Belykh,, Vldimir Belykh b, Mrtin Hsler c Deprtment of Mthemtics nd Sttistics, Georgi Stte University, 30 Pryor Street, Atlnt, GA 30303, USA b Mthemtics Deprtment, Volg Stte Acdemy, 5, Nesterov st., Nizhny Novgorod, , Russi c School of Computer nd Communiction Sciences, Ecole Polytechnique Fédérle de Lusnne EPFL), Sttion 14, 1015 Lusnne, Switzerlnd Avilble online 5 October 006 Abstrct We present generl frmework for studying globl complete synchroniztion in networks of dynmicl systems with symmetricl connections. We extend the connection grph stbility method, originlly developed for symmetriclly coupled networks, to the generl symmetricl cse. The principl new component of the method is the trnsformtion of the directed connection grph into n undirected grph. In our method for symmetriclly coupled networks we hve to choose pth between ech pir of nodes. The extension of the method to symmetricl coupling consists in symmetrizing the grph nd ssociting weight to ech pth. This weight involves the node unblnce of the two nodes. This quntity is defined to be the difference between the sum of connection coefficients of the outgoing edges nd the sum of the connection coefficients of the incoming edges to the node. The synchroniztion condition for this symmetrized-nd-weighted network then lso gurntees synchroniztion in the originl symmetricl network. c 006 Elsevier B.V. All rights reserved. Keywords: Synchroniztion; Connection grph; Stbility; Pth length 1. Introduction The incresing interest in synchroniztion in limit-cycle nd chotic dynmicl systems [1 3] hs led mny reserchers to consider the phenomenon of synchroniztion in lrge complex networks of coupled oscilltors see, e.g. [4,5] for smpling of this lrge field). Much of this reserch hs been inspired by technologicl nd biologicl exmples, including coupled synchronized lsers [6, 7], networks of computer clocks [8], nd synchronized neuronl firing [9,10]. Networks of identicl or slightly non-identicl oscilltors often synchronize, or else form synchronous ptterns tht depend on the symmetry of the underlying network [11,1]. The strongest form of synchrony in oscilltor networks is complete synchroniztion when ll oscilltors do the sme Corresponding uthor. Tel.: ; fx: E-mil ddress: ibelykh@gsu.edu I. Belykh). thing t the sme time. An importnt problem in the study of complete synchrony is how the stbility of synchronized behvior, where the behvior could be fixed point, limit cycle or chotic ttrctor, is influenced by the network topology nd kind of interction. This problem ws intensively studied for networks of biologicl oscilltors [13 16], nd more generlly of limit-cycle nd chotic oscilltors [17 19,1 44]. Most methods for determining stbility of synchroniztion in linerly coupled networks of chotic systems re bsed on the clcultion of the eigenvlues of the connection mtrix nd term depending minly on the dynmics of the individul oscilltors see, e.g., [17 7]). Pecor nd Crroll [] developed generl pproch to the locl stbility of complete synchroniztion for ny liner coupling network rchitecture. This pproch, clled the Mster Stbility function, is bsed on the clcultion of the mximum Lypunov exponent for the lest stble trnsversl mode of the synchronous mnifold nd the eigenvlues of the connection mtrix. This powerful method is widely used in locl stbility studies of synchroniztion /$ - see front mtter c 006 Elsevier B.V. All rights reserved. doi: /j.physd

2 I. Belykh et l. / Physic D 4 006) in complex oscilltor networks [8 33]. Globl stbility results bsed on the clcultion of the connection mtrix eigenvlues were lso derived for oscilltor networks coupled vi undirected [34,35] nd directed grphs [36]. These studies show tht both locl nd globl stbilities of complete synchroniztion depend on the eigenvlues of the Lplcin connection mtrix. We hve previously developed n lternte wy to estblish synchrony which does not depend on explicit knowledge of the spectrum of the connection mtrix [40]. This connection grph method combines the Lypunov function pproch with grph theoreticl resoning. It gurntees complete synchroniztion from rbitrry initil conditions nd not just locl stbility of the synchroniztion mnifold. It is lso pplicble to timedependent networks. This pproch ws originlly developed for undirected grphs nd pplied to globl synchroniztion in complex networks [41,4]. More recently, we showed tht the method cn be directly pplied to symmetriclly coupled networks with node blnce [43,44]. Node blnce mens tht the sum of the coupling coefficients of ll edges directed to node equls the sum of the coupling coefficients of ll the edges directed outwrd from the node. We proved tht for node blnced networks it is sufficient to symmetrize ll connections by replcing unidirectionl coupling with bidirectionl coupling of hlf the coupling strength. The bound for globl synchroniztion in this undirected network then holds lso for the originl directed network. In this pper we extend our pproch to networks with rbitrry symmetricl connections. The connection grph of such network is directed nd the coupling coefficient from node i to node j is in generl different from the coupling coefficient for the reverse direction. The new ingredient of the method is the trnsformtion of the directed connection grph into n undirected weighted grph. This is done by symmetrizing the grph nd ssociting weight to ech edge of the undirected grph nd to ech pth between ny two nodes. This weight involves the node unblnce of the two nodes. This quntity is defined to be the difference between the sum of connection coefficients of the outgoing edges nd the sum of the connection coefficients of the incoming edges to the node. As in the cse of node-blnced networks, the synchroniztion criterion derived for this symmetricl network then gurntees synchroniztion in the symmetricl directed network. The lyout of this pper is s follows. First, in Section, we stte the problem under considertion. Then, in Section 3, we derive grph-bsed criterion for globl synchroniztion in symmetriclly coupled networks nd formulte the min theorem of our method. In Section 4, we show how to pply the generlized connection grph method to severl exmples of concrete networks. We strt with the simplest network of two unidirectionlly coupled oscilltors, then we continue with str-configurtion nd directed network with n irregulr topology. In Section 5, brief discussion of the obtined results is given.. Problem sttement.1. Systems under study We consider network of n intercting nonliner l-dimensionl dynmicl systems oscilltors). We ssume tht the individul oscilltors re ll identicl, even though our results cn be generlized to slightly non-identicl systems. The composed dynmicl system is described by the n l ordinry differentil equtions ẋ i = Fx i ) + d ik t)px k, i = 1,..., n, 1) k=1 where x i = xi 1,..., xl i ) is the l-vector contining the coordintes of the ith oscilltor, the function F : R l R l is nonliner nd cpble of exhibiting periodic or chotic solutions, nd P is projection opertor tht selects the components of x i tht re involved in the interction between the individul oscilltors. Without loss of generlity, we consider vector version of the coupling with the digonl mtrix P = digp 1, p,..., p l ), where p ν = 1, ν = 1,,..., s nd p ν = 0 for ν = s + 1,..., l. Note tht ll the results tht re obtined in this pper re directly pplicble to other possible cses where the projection mtrix P is nondigonl see [44] for the proof of synchroniztion mong oscilltors with non-digonl coupling). The connection mtrix D with entries d ik is n n n mtrix with zero row-sums nd nonnegtive off-digonl elements such tht d ik = 0 nd d ii = d ik, i = 1,..., n. k=1 k=1;k i This ensures tht the coupling is of diffusive nture on n rbitrry coupling grph) nd ny solution xt) for single oscilltor is lso solution of the coupled system 1). The connection mtrix D is ssumed to be symmetricl without ny further constrints. This is in contrst to our previous ppers, where we required the symmetry of the connection mtrix [40] or the zero column-sums property of n symmetricl connection mtrix [43]. The coupling mtrix D is ssocited with the edge-weighted directed connection grph D, where to ech individul system corresponds node nd for ech pir of nodes i, j with i j nd such tht d i j > 0, there is n edge directed from from j to i. The weight ssigned to this edge is d i j. The connection grph is ssumed to be connected. We dmit n rbitrry time dependence in the coupling mtrix even if t is not explicitly stted everywhere. All constrints nd criteri for the coupling mtrix re understood to hold for ll times t... Type of synchroniztion considered In this pper, we concentrte on the strongest form of synchrony, nmely, globl complete synchroniztion. Complete synchroniztion is defined by the invrint hyperplne M = {x 1 t) = x t) = = x n t)}. The mnifold M hs

3 44 I. Belykh et l. / Physic D 4 006) 4 51 the dimension of single oscilltor, nd is often clled the synchroniztion mnifold. Completely synchronous solutions of ll types multi-stble, periodic, nd chotic solutions) re constrined to this mnifold. Definition.1. Network 1) synchronizes completely, if lim x it) x j t) = 0 for i, j. ) t The system 1) exhibits globl complete synchroniztion, if the condition ) holds for ny solution. Locl complete synchroniztion rises in the system 1), if ny solution of the system 1) tht strts sufficiently close to the synchroniztion mnifold synchronizes completely. A centrl gol of our study is the bility to predict when the completely synchronous stte is globlly stble. In prticulr, we wnt to derive upper bounds for globl complete synchroniztion in the network 1). In wht follows we present the stbility nlysis of synchroniztion in oscilltor networks with n rbitrry directed connection grph, under the constrint tht the grph llows synchroniztion of ll the nodes. Indeed, synchrony in directly coupled networks is only possible if there is t lest one node which directly or indirectly influences ll the others [36]. In terms of the connection grph, this mounts to the existence of uniformly directed tree involving ll the vertices. A str-coupled network where secondry nodes drive the hub is counter exmple, where such tree does not exist nd synchroniztion is impossible..3. Hypothesis mde One cn formlly divide oscilltor networks 1) into two clsses of coupled systems with different synchroniztion behvior. Networks from the first clss globlly synchronize when the coupling is mde sufficiently lrge nd remin synchronized up to n infinitely lrge vlue of coupling. Of course, globl synchroniztion in such networks is lwys preceded by locl synchroniztion, typiclly rising t lower coupling strength. There my be multiple locl synchroniztion thresholds, ech ssocited with n unstble periodic orbit [3]. This lrge clss of networks contins mjority of known limit-cycle nd chotic systems. Exmples include coupled Lorenz systems [40], Chu circuits [0], Hindmrsh Rose nd Hodgkin Huxley-type neuron models [4], coupled driven chotic pendul [44], Duffing oscilltors, etc. The second, nrrower clss corresponds to the rrys in which incresing the coupling between systems destbilizes the loclly stble synchronous stte. This phenomenon is chrcterized by the bsence of globl synchroniztion nd is clled short-wvelength bifurction [18]. The stndrd exmples used in pplictions of the mster stbility function re x-coupled Rössler oscilltors nd electronic circuits modeled fter the Rössler equtions, where the qudrtic nonlinerity is replced by piecewise liner function [4]. These unexpected desynchroniztion trnsitions in x-coupled Rössler systems cn be explined by singulrity of the function defining the coordintes of the coupled system s equilibri. This leds to the ppernce of equilibri, outside the synchroniztion mnifold, tht re ssocited with desynchronous sttes [45]. Evidently, the connection grph method to prove globl synchroniztion, tht we extend in this pper, is only pplicble to the first clss of networks dmitting globl complete synchroniztion. It is difficult to determine priori wht clss the network 1) with chosen type of oscilltor belongs to. However, if we prove globl complete synchroniztion in the simplest network 1) of two oscilltors of the given type, then we will be ble to conclude tht lrger network 1) contining tht oscilltor is lso globlly synchronizble nd belongs to the first clss of networks. Therefore, to solve the problem of globl synchroniztion in network of oscilltors, given their individul dynmics nd coupling structure, one should lwys strt from the question of whether or not two oscilltors of the given type re cpble of globl synchroniztion. This mounts to imposing the following constrint; Hypothesis 1 Sufficient Condition). Globl synchroniztion in the network 1) of two unidirectionlly coupled oscilltors with coupling strength d 1 is globlly stble, provided tht d 1 exceeds the threshold. We will reformulte this hypothesis in more technicl form in Section 3. The existence of the threshold is the principl requirement of our method such tht Hypothesis 1 hs to be proven for ech prticulr sitution for the concrete individul node s dynmics nd the projection mtrix P). The proof involves the construction of Lypunov function long with the ssumption of the eventul dissiptiveness of the coupled system 1) [40]. For the chotic systems from the first clss, listed bove [0, 40,4,44], the criticl vlue of coupling cn be expressed explicitly through the prmeters of the individul oscilltor. 3. Grph-bsed criterion for network synchroniztion 3.1. Stbility system for the difference vribles Since we re interested in complete synchroniztion, we introduce the difference vribles X i j = x j x i for ny i nd j. Similrly to our previous works [40,43], we cn write the stbility system for the difference vribles Ẋ i j = Fx j ) Fx i ) + {d jk P X jk d ik P X ik }, i, j = 1,..., n. 3) k=1 The function difference Fx j ) Fx i ) cn be rewritten in compct vector form [ ] 1 Fx j ) Fx i ) = DFβx j + 1 β)x i )dβ X i j, 0 where D F is n l l Jcobi mtrix of F. Hence, we obtin [ ] 1 Ẋ i j = DFβx j + 1 β)x i )dβ + 0 X i j {d jk P X jk d ik P X ik }, i, j = 1,..., n. 4) k=1

4 I. Belykh et l. / Physic D 4 006) We will prove globl complete synchroniztion in the system 1) by showing tht the equilibrium O = {X i j = 0, i, j = 1,..., n} cn be mde globlly symptoticlly stble by incresing coupling. The first term in Eq. 4) is unstble nd defines the divergence of trjectories within the individul dynmicl systems. The second term n k=1 {d jk P X jk d ik P X ik } representing the coupling structure fvors the stbility nd cn overcome the unstble term, provided tht the coupling is strong enough. We will derive the stbility conditions in two steps. I. We first study globl stbility of synchroniztion in the simplest network 1) composed of two unidirectionlly coupled systems cf. Hypothesis 1). In this cse, the stbility system for the difference vribles 4) is reduced to the system [ ] 1 Ẋ 1 = DFβx + 1 β)x 1 )dβ X 1 d 1 P X 1. 5) 0 We need to show tht the origin of the difference eqution system 5) is globlly symptoticlly stble. This cn be done by pplying the Lypunov function method. We choose the Lypunov function of the form W 1 = 1 X T 1 H X 1, 6) where H = digh 1, h,..., h s, H 1 ), h 1 = 1,..., h s = 1, nd the l s) l s) mtrix H 1 is positive definite. To ensure the globl stbility of the origin, the derivtive Ẇ 1 with respect to the system 5) [ ] 1 Ẇ 1 = X1 T H DFβx + 1 β)x 1 )dβ d 1 P X 1, 0 X i j 0 7) hs to be negtive. This is true under the condition of Hypothesis 1. II. To study globl stbility of synchroniztion in the network 1) with rbitrry network topologies, we construct the Lypunov function for the system of the difference vribles 4) V = 1 4 i=1 j=1 Xi T j H X i j, 8) where H is the mtrix in 6). The corresponding time derivtive hs the form V = 1 1 i=1 i=1 Ẇ i j + 1 j=1 j=1 k=1 i=1 Xi T j P X i j j=1 {d jk X T ji H P X jk + d ik Xik T H P X i j}, 9) where W i j = 1 X T i j H X i j nd is the synchroniztion threshold in the two-oscilltor network with the stbility system 5). Fig. 1. Node unblnce : The grph representtion of the ith-column sum of the connection mtrix D. Here, the node unblnce D c i = d 5 + d 4 ) d 3 + d + d 1 ). After some lgebric mnipultions see [43] for the detils of this pssge), we obtin the following inequlity s V h n d ) i j + d ji Xi ν j + s h Dc i + D c j X ν i j, 10) where D c i = n k=1 d ki nd D c j = n k=1 d kj re the ith nd jth column sums of the connection mtrix D, respectively. To fcilitte cross-pper reding, it is worth noticing tht ε i j nd µ i j in [43] stnd for d i j +d ji nd Dc i +Dc j, respectively. In terms of grphs, the column sum Di c = n k=1 d ki = k i d ki + d ii = k i d ki k i d ik mounts to the difference between the sum of the coupling coefficients of ll edges directed outwrd from node i nd the sum of the coupling coefficients of ll the edges directed to node i Fig. 1). We cll this quntity the node unblnce. In order to estblish synchroniztion, we hve to prove tht the right hnd side RHS) of the inequlity 10) is negtive qudrtic form. This is equivlent to the following inequlity between qudrtic forms s ) di j + d ji Xi ν j > s n ) 1 + Dc i + D c j X ν i j. 11) Here, the coupling coefficients d i j +d ji define the edges on the symmetrized connection grph obtined by replcing n edge directed from node i to node j nd nother edge in the reverse direction by n undirected edge with this men coupling coefficient. Consequently, the difference vribles X i j on the left hnd side LHS) of the inequlity 11) correspond to pirs of nodes directly connected by n edge on the symmetrized grph. At the sme time, the right hnd side RHS) contins the difference vribles between ny pir of nodes. The pivotl ingredient of the connection grph method for symmetriclly coupled networks [40] is to express ll difference vribles X i j, i, j = 1,..., n on the RHS through the connection grph vribles X i j on the LHS of the inequlity 11). This is done by estblishing bound on the totl length of ll chosen pths

5 46 I. Belykh et l. / Physic D 4 006) 4 51 pssing through n edge on the connection grph. In contrst to the symmetricl cse, these pth lengths will be ) weighted due to the presence of nonunit fctors 1 + Dc i +Dc j on the RHS of the inequlity 11). The terms n on the RHS of the inequlity 11) re ssocited with the sum node unblnce of nodes i nd j. If the term n is negtive for given i nd j, then it is fvorble for lowering the inequlity 11). Therefore, we could hve ssigned ll negtive terms Di c+dc j n to the LHS we discuss this possibility in Remrk fter formulting the min theorem). However, it turns out tht it is more dvntgeous, in generl, to incorporte the term n into the qudrtic form on the LHS only if it is negtive nd if i nd j re linked directly by n edge k of the symmetrized grph. We then denote n 1 Di c+dc j by D k. Tht is, we preserve the structure of the symmetrized grph, but mke the grph weighted: some its edges hve stronger coupling d i j +d ji + D k. If the term n 1 Di c+dc j is negtive, but there is no edge linking node i nd node j we leve it on the RHS of the inequlity 11) long the terms n 1 tht re positive. Note tht 1 + Dc i +Dc j my become negtive. In this cse we simply set it to 0 so tht the RHS remins positive qudrtic form. The redistribution of the terms Dc i +Dc j following: Xi ν j mounts to the To ech edge of the symmetrized connection grph, we ssocite the quntity D k defined by Di c + D c j D k = n, if Dc i + D c j < 0; 1) nd k links i nd j 0, otherwise. For ny pir of nodes i, j), we choose pth P i j nd ssocite to ech pth P i j its length LP i j ) defined by P i j, if Di c + D c j < 0; nd there LP i j ) = is link k between i nd j P i j χ 1 + D ) 13) i j, otherwise where D i j = Dc i +Dc j nd P i j is the number of edges in P i j. The function { x, if x 0 χx) = 0, if x < 0. s Thus, the inequlity 11) becomes d k + D k ) X ν i j where d k = d i j +d ji. > n s 1 + D ) i j Xi ν j, 14) Exctly s in [40], we now replce the connection grph vribles Xi ν j, j = 1, i > i on the LHS of the inequlity 14) by Yk ν, k = 1,..., m, where m is the number of edges of the symmetrized-nd-weighted grph nd pply the Cuchy Schwrz inequlity. This leds to the min theorem of this pper. 3.. Connection grph method for rbitrry symmetricl coupling Theorem 1 Sufficient Conditions). Under Hypothesis 1, globl complete synchroniztion is chieved in the network 1) with n rbitrry coupling grph D if for ll k d k + D k > n b k, where b k = LP i j ) 15) j>i; k P i j is the sum of the lengths LP i j ) of ll chosen pths P i j which pss through given edge k tht belongs to the symmetrized undirected grph. This weighted pth length LP i j ) is defined in Eq. 13) s follows P i j, if Di c + D c j < 0; nd there is link k between i nd j D LP i j ) = Pi j i c + D c j χ1 + ) = P i j χ 1 + D ) i j ; otherwise, where the function χ is the identity for positive nd 0 for negtive rguments. The men coupling coefficient d k = d i j +d ji defines n edge k on the undirected symmetrized grph. An extr coupling strength D k = Dc i +Dc j n is dded to the edges of the symmetrized connection grph for which the men node unblnce Di c + D c j is negtive. Remrk 1. In the cse where the directed connection grph is not uniformly directed tree involving ll nodes nd complete synchroniztion of ll the nodes is impossible, the condition for synchroniztion is simply impossible to stisfy. Remrk. The ssignment of the different terms Dc i +Dc j to the qudrtic forms on the left nd on the right sides of the inequlity 14) is somewht rbitrry. Another possibility is to ssign ll negtive terms to the LHS. This implies tht for those terms tht do not hve direct link between i n nd j n dditionl edge with connection coefficient hs to be dded between nodes i nd j to the symmetrized connection grph. The finl criterion 15) in Theorem 1 hs to be evluted with respect to this ugmented symmetrized connection grph. This leds to different set of inequlities tht my be dvntgeous in certin exmples. Theorem 1 directly leds to the following method to estblish our sufficient condition for globl complete synchroniztion.

6 Step 1. Determine the node unblnce for ech node D c i = nj=1 d ji. I. Belykh et l. / Physic D 4 006) Step. Symmetrize the connection grph by replcing the edge directed from node i to node j by n undirected edge with hlf the coupling coefficient d i j /. In the cse where there is n edge directed from node i to node j nd nother edge in the reverse direction, the pir of directed edges is replced by n undirected edge with men coupling coefficient d k = d i j +d ji. Step 3. Choose pth P i j between ech pir of nodes. Usully, the shortest pth is chosen. Sometimes, however, different choice of pths cn led to lower bounds [4]. Step 4. For ech pth P i j determine the men node unblnce of the endnodes i nd j. Identify pths of length 1, i.e. edges of the symmetrized grph, with negtive men node unblnce Di c + D c j. For these edges, clculte nd dd extr strength D k = Dc i +Dc j n to the symmetrized coupling d k. For ll other pths P i j, nmely, pths of length 1 with nonnegtive men node unblnce nd ny pths composed of t lest two edges, clculte the quntities D i j = Dc i +Dc j nd 1 + D i j. Associte weight 1 + D i j to the pth length of P i j if 1 + D i j > 0, nd zero weight, otherwise. Step 5. For ech edge k of the symmetrized-nd-weighted connection grph determine the inequlity d k + D k > n b k, where b k = LP i j ). j>i; k P i j Step 6. Combine the inequlities either to describe the set of common vlues for ll connection coefficients tht gurntee globl complete synchroniztion or to describe in generl the set of connection coefficient vectors tht gurntee synchroniztion if we llow for coefficients tht vry from link to link. Finlly, the bound for globl synchroniztion in the symmetrized-nd-weighted network holds lso for the originl symmetricl network. Let us show how to pply the generl method to three exmples of concrete symmetricl networks. 4. Exmples: Appliction of the method To find n upper bound for the synchroniztion threshold in concrete networks, we should follow the steps of the bove study Two unidirectionlly coupled oscilltors Consider the simplest directed network with n = nd coupling strength d Fig. )). Step 1. Determine the node unblnce for node 1 nd : D c 1 = d nd D c = d. Step. Symmetrize the grph s shown in Fig. b): d 1 = d+0 = d. Fig.. Simplest directed network nd its symmetrized nlog. The directed link is replced by the undirected edge with hlf the coupling strength. D Here, the men node unblnce, 1 c+dc = 0, so tht the symmetrize-ndweight opertion mounts to symmetriztion. The pth length P 1 lso remins unweighted. Step 3. Choose pth between ech pir of nodes. Here, the grph hs only one brnch. Steps 4. For ech pth determine the men node unblnce of the endnodes. Here, this quntity is equl to 0 : D1 c + Dc = d d = 0. Therefore, D k D 1 = 0 = 0 nd D i j D 1 = 0. Steps 5 6. For the edge 1 determine the inequlity: d + 0 > P 1. The pth length P 1 = 1 such tht the finl inequlity becomes d >. Recll tht by Hypothesis 1, is n upper bound for synchroniztion in this network such tht our method gives the correct synchroniztion bound. 4.. Str-coupling This is well-known coupling scheme where the network hs centrl hub this node is mrked s the first one) nd ll other nodes re linked to this node. The coupling coefficient d out of the edges directed outwrd from the hub differs from the coupling coefficient d in of the edges directed to the hub see Fig. 3). The criterion of Theorem 1 is pplied to this network s follows. Step 1. Clculte the node unblnce for ech node: D c 1 = n 1)d out d in ) nd D c i = d in d out ), i =,..., n. Step. Symmetrize the grph: d k = d out+d in, k = 1,..., n 1. Step 3. Choose pth between ech pir of nodes: P 1 j : 1 j, j =,..., n P i j : i 1 j, i, j =,..., n. Step 4. For ech pth P i j determine Dc i +Dc j for nodes i nd j nd its plce in the inequlities 15) P 1 j : Dc 1 + Dc j P i j : Dc i + D c j = n d out d in ), j =,..., n = d in d out, i, j =,..., n. For d in > d out, the term ssocited with P 1 j is negtive nd there is n edge linking nodes 1 nd j. Therefore, this term trnsforms into D k = n n d out d in ). For ll other pths P i j, the terms Dc i +Dc j = d in d out re positive nd become D i j = d in d out. For d in < d out, the term ssocited with P 1 j is positive nd therefore becomes D 1 j = n d out d in ). In ll other cses,

7 48 I. Belykh et l. / Physic D 4 006) 4 51 Fig. 3. ) Str-network with symmetricl connections: d out nd d in re different. b) Symmetrized-nd-weighted nlog of ) with bidirectionl connections with strength d out+d in + D k, k = 1,..., n 1. For d in > d out, D k = n d in d out ), nd for d out > d in, D k = 0. In generl, for both cses, the length of the chosen pth P i j is weighted. Fig. 4. ) Unidirectionlly coupled network with uniform coupling d. b) Symmetrized nlog of ) with weighted bidirectionl connections. Arrows indicte the direction of coupling long n edge; edges without rrows re coupled bidirectionlly. The width of the links my be thought of s the coupling strength. the terms Dc i +Dc j = d in d out re negtive but there is no direct link between i nd j for i, j =,..., n. Hence, these terms lso become D i j = d in d out. Step 5. For ech edge of the grph determine the inequlity 15). Here, ll edges re equivlent. Cse d in > d out : d out + d in > n + n [ 1 + n )χ Cse d out > d in : d out + d in > n d in d out ) [ 1 χ + n )χ 1 + d in d out 1 + n )]. ) d out d in ) 1 + d in d out )]. Both cses led to the sme sufficient condition for globl complete synchroniztion: d out > n 3 n 3 d in +. 16) Let us now check if the cse d out = 0 is comptible with our criterion for synchroniztion: 0 > n 3 n 3 d in +. 17) This condition cn only be fulfilled for n =. In this cse we re bck to the previous exmple. On the other hnd, for n > it is obvious tht synchroniztion is impossible. Indeed, secondry nodes of the network hve no interction t ll nd therefore they do not synchronize Irregulr network Consider the symmetricl seven-node network of Fig. 4). For simplicity, we chose equl coupling coefficients d for ll directed edges.

8 I. Belykh et l. / Physic D 4 006) As before, we use the six-step process to derive the synchroniztion condition of Theorem 1. Step 1. Clculte the difference between the sum of the coupling coefficients of ll edges directed outwrd from node i nd the sum of the coupling coefficients of ll the edges directed to node i. Thus, determine the node blnce for ech node of the grph: D c 1 = d d = 0 D c 3 = d d = 0 D c 4 = 3d d = d Dc Dc = d 3d = d 5 = d d = d D c 6 = d d = d Dc 7 = d d = 0. Step. Symmetrize the grph by replcing ech directed edge by n undirected edge with hlf the coupling strength: d k = d, k = 1,..., 10 see Fig. 4b)). Step 3. Choose pth P i j between ny pir of nodes i, j of the symmetrized grph. It turns out tht it is often dvntgeous to choose pths tht contin edges with negtive men node unblnce this quntity will be clculted in Step 4.) Our choice of pths is P 1 : edge 1 P 13 : edges 1, P 14 : edge 8 P 15 : edges 1, 10 P 16 : edges 1, 7, 6 P 17 : edges 1, 7 P 3 : edge P 4 : edges, 3 P 5 : edge 10 P 6 : edges 7, 6 P 7 : edge 7 P 34 : edge 3 P 35 : edges, 10 P 36 : edges, 7, 6 P 37 : edges, 7 P 45 : edge 4 P 46 : edge 9 P 47 : edges 9, 6 P 56 : edge 5 P 57 : edges 5, 6 P 67 : edge 6. 18) Step 4. For ech pth P i j determine the men node unblnce for endnodes i nd j: P 1 : Dc 1 + Dc P 14 : Dc 1 + Dc 4 P 16 : Dc 1 + Dc 6 P 3 : Dc + Dc 3 P 5 : Dc + Dc 5 P 7 : Dc + Dc 7 P 35 : Dc 3 + Dc 5 P 37 : Dc 3 + Dc 7 P 46 : Dc 4 + Dc 6 = d P 13 : Dc 1 + Dc 3 = d P 15 : Dc 1 + Dc 5 = d P 17 : Dc 1 + Dc 7 = d P 4 : Dc + Dc 4 = d P 6 : Dc + Dc 6 = d P 34 : Dc 3 + Dc 4 = d P 36 : Dc 3 + Dc 6 = 0 P 45 : Dc 4 + Dc 5 = d P 47 : Dc 4 + Dc 7 = 0 = d = 0 = 0 = 3d = d = d = 3d = d P 56 : Dc 5 + Dc 6 = 0 P 57 : Dc 5 + Dc 7 = d P 67 : Dc 6 + Dc 7 = d. We now ctegorize the men node unblnce terms s follows. If Dc i +Dc j < 0 nd there is n edge k of the symmetrized D grph linking directly i nd j, we set D k = i c+dc j 7 nd dd this dditionl coupling strength to d k. This reltes to edges 1,, 6, 7, 10 see Fig. 4b)): D 1 = D1 c + Dc 7 = d D D = c + Dc = d 7 D D 6 = 6 c + Dc 7 7 = d D 7 = D c + Dc = d 7 D D 10 = c + Dc 5 7 = d 14. In ll other cses, the terms Dc i +Dc j re either nonnegtive or negtive but there is no direct link between i nd j, so tht ll these terms become D i j. Step 5. For ech edge of the grph determine the inequlity 15). Edge 1 link between nodes 1 nd ): d 1 + D 1 = d + d 7 > 7 b k, where b k = LP i j ). j>i;k P i j The chosen pths tht pss through the edge 1 re P 1, P 13, P 15, P 16, P 17 cf. 15)). Their weighted lengths LP i j ) re clculted in ccordnce with Eq. 13): LP 1 ) = P 1 = 1 since D c 1 + Dc 1 < 0; nd there is n edge between 1 nd LP 13 ) = P 13 χ 1 + D ) 13 = P ) = LP 15 ) = P 15 χ 1 + D ) 15 = P d ) = 1 + d ) LP 16 ) = P 16 χ 1 + D ) 16 = P 16 ψ 1 d ) = P 16 0, by ssumption: d > LP 17 ) = P 17 χ 1 + D ) 17 = P 17 =. Summing up ll the lengths, we obtin [ d d + d 7 > 7 ) + ] = 7 + d. 7 Therefore, the synchroniztion condition for the edge 1 becomes d >. Exctly s for the edge 1, we cn clculte the synchroniztion bounds for other edges. These bounds cn be summrized s follows edge 1: d > edge : d > 18 edge 3: d > 6 7 5

9 50 I. Belykh et l. / Physic D 4 006) 4 51 edge 4: d > edge 7: d > 10 9 edge 10: d > 5. edge 5: d > 3 5 edge 8: d > 5 edge 6: d > 5 edge 9: d > 3 Step 6. Combining the synchroniztion criteri for ll the edges, we tke the mximum constrint to chieve globl synchroniztion. This constrint corresponds to the wekest link. Here, the wekest link is the edge 6. This edge is bottle neck for synchroniztion of the entire network nd requires the mximum coupling strength to synchronize ll oscilltors of the network. Therefore we conclude tht for d > d = 5 19) we cn gurntee globl synchroniztion of the network. It is customry to discuss network synchroniztion in terms of eigenvlues of the connection mtrix D. It llows one to give necessry nd sufficient conditions for locl synchroniztion depending on usully numericlly clculted) Lypunov exponents of the individul systems. We hve previously shown tht the second lrgest eigenvlue lso llows one to obtin bound for globl synchroniztion [40]. For symmetricl coupling, the eigenvlues re typiclly complex nd, therefore, difficult to derive. Usully, for irregulr directed networks one cn only clculte these eigenvlues numericlly. Thus, by the eigenvlue pproches to globl synchroniztion, the synchroniztion bound ssocited with the second lrgest eigenvlue of the connection mtrix is d > d = / Reλ. 0) Note tht this is true only for networks llowing globl synchroniztion nd for which the threshold cn be rigorously derived. Actully in the context of our qudrtic Lypunov function the criterion 0) is the optiml bound for the synchroniztion threshold. The choice of pths P i j we mde for clculting the bound 19) is suboptiml so tht the condition d = 5 gives n overestimte: 5 versus / Reλ, where λ = 1 is the second lrgest eigenvlue of the connection mtrix ssocited with the network of Fig. 4). Here, λ is clculted numericlly. A different choice of pths P i j cn led to lower thresholds tht re closer to the optiml bounds chievble by the eigenvlue method. Typiclly, our grph method becomes more effective nd gives more correct informtion on the qulittive dependence of the synchroniztion limits on prmeters of the network, while the number of oscilltors composing the network increses. Clcultion of weighted pth lengths cn be quite lborious tsk for networks with complicted coupling schemes. However, once the clcultion scheme is constructed, bound giving n explicit dependence of the synchroniztion threshold on the network size nd topology cn be obtined for more complicted cses one cn use MAPLE). 5. Conclusions We hve given sufficient condition for globl complete synchroniztion in n rbitrry network of diffusively coupled identicl dynmicl systems. The condition is composed of set of inequlities which hve to be stisfied, one inequlity for ech edge of the connection grph. Ech inequlity involves term tht depends only on the individul dynmicl systems, nmely the coupling strength tht gurntees globl synchronizing of two systems. The other terms of the inequlity depend only on the grph structure nd on the coupling coefficients. The new component of the method for synchroniztion in symmetricl networks is the use of the symmetrize-nd-weight opertion. This mounts to replcing ech direct link between node i node j by n undirected edge with coupling strength tht depends on the node unblnce between the two nodes. Different weights re ssocited with ech pth between ny two nodes of the network. These weights lso depend on the node unblnce between the endnodes of the pth. The synchroniztion criterion for this symmetrized network lso gurntees globl stbility of synchroniztion in the originl directed network. In smll nd lso in sufficiently regulr networks, the condition cn be written down explicitly. In other networks, combintoril lgorithm of polynomil complexity cn estblish the inequlities on the coupling coefficients tht gurntee globl complete synchroniztion. The min computtionl tsk is to determine pth between ny two nodes of the grph, typiclly the shortest pth. We impose no restriction on the interction between the individul systems other thn diffusive coupling, i.e. coupling mtrix with non-negtive off-digonl elements nd zero row sums. In prticulr, we do not impose symmetry on the coupling mtrix. This mens tht the coupling between ny two systems my be either bsent, unidirectionl, or bidirectionl with not necessrily equl coupling coefficients for both directions. Of course, when complete synchroniztion is never possible, such s in networks without uniformly directed tree, the condition for synchroniztion is simply impossible to stisfy. Since our pproch is bsed on Lypunov functions, the inequlities we obtin re conservtive. However, compring with numericl simultions, we hve noticed tht often they give correct informtion on the dependence of the synchroniztion limits on prmeters of the network. Note tht one cn lso use the eigenvlues of the connection mtrix for the Lypunov function pproch. By the eigenvlue method, we my obtin, in the cse of symmetricl networks with fixed, time-independent connections, better bound for globl synchroniztion thn with the connection grph method. However, the eigenvlues of connection mtrices ssocited with irregulr grphs re difficult to clculte nlyticlly. The generlized connection grph method hs n dvntge over the eigenvlue method in studying networks with time-dependent coupling coefficients. Specificlly, within the frmework of our method, the time-dependent coupling coefficients cn be hndled without problems, wheres inequlities in coupling coefficients do not necessrily result in

10 I. Belykh et l. / Physic D 4 006) corresponding inequlities in eigenvlues. This implies tht, in generl, the eigenvlue method cnnot be pplied to networks with time-vrying coupling structure. We should remrk tht our generlized method is vlid for networks of slightly nonidenticl oscilltors. In this cse, perfect synchroniztion cnnot exist nymore, but pproximte synchroniztion is still possible. We hve previously shown tht in the cse of symmetriclly coupled networks, similr globl stbility conditions of pproximte synchroniztion cn be derived within the frmework of the connection grph method [40]. This crries over to symmetricl heterogenous networks. Finlly, let us remrk tht the results of this pper re generliztions of our previous ppers on symmetric, or symmetric node-blnced coupling to rbitrry symmetric coupling. Acknowledgments I.B. cknowledges the finncil support of the Georgi Stte University Reserch Initition Progrm Grnt FY07) nd Criplo Foundtion fellowship. V.B. cknowledges the support from the RFBR grnt No ), NWO-RFBR grnt No ), nd RFBR-MF grnt No ). M.H. cknowledges the support from the SNSF grnt No ) nd EU Commission FP6-NEST project N ). References [1] H. Fujisk, T. Ymd, Prog. Theor. Phys ) ) 885. [] V.S. Afrimovich, N.N. Verichev, M.I. Rbinovich, Rdiophys. Quntum Electron ) 795. [3] L.M. Pecor, T.L. Crroll, Phys. Rev. Lett ) 81. [4] J. Kurths, S. Boccletti, C. Grebogi, Y.-C. Li Eds.), Focus Issue: Control nd Synchroniztion in Chotic Dynmicl Systems, in: Chos, vol. 13, 003. [5] S. Boccletti, L.M. Pecor Eds.), Focus Issue: Stbility nd Pttern Formtion in Dynmics on Networks, in: Chos, vol. 16, 006. [6] L. Fbiny, P. Colet, R. Roy, D. Lenstr, Phys. Rev. A ) 487. [7] S.H. Strogtz, Nture ) 68. [8] D.L. Mills, IEEE Trns. Communictions ) 148. [9] C.M. Gry, W. Singer, Proc. Ntl. Acd. Sci. USA ) [10] R. Stoop, L.A. Bunimovich, K. Schindler, Nonlinerity ) [11] M. Golubitsky, I. Stewrt, A. Torok, SIAM J. Appl. Dyn. Syst. 4 1) 005) 78. [1] M. Golubitsky, I. Stewrt, Bull. Amer. Mth. Soc ) 305. [13] N. Kopell, G.B. Ermentrout, Mth. Biosci ) 87. [14] S.H. Strogtz, R.E. Mirollo, Physic D ). [15] D. Somers, N. Kopell, Physic D ) 169. [16] I. Belykh, E. de Lnge, M. Hsler, Phys. Rev. Lett ) [17] C.W. Wu, L.O. Chu, IEEE Trns. Circ. Syst. -I: Fundm. Theory Appl ) 161. [18] J.F. Hegy, L.M. Pecor, T.L. Crroll, Phys. Rev. Lett ) [19] L.M. Pecor, T.L. Crroll, G.A. Johnson, D.J. Mr, J.F. Hegy, Chos ) 50. [0] V.N. Belykh, N.N. Verichev, L.J. Kocrev, L.O. Chu, in: R.N. Mdn Ed.), Chu s Circuit: A Prdigm for Chos, World Scientific, Singpore, 1993, p. 35. [1] R. Brown, N.F. Rulkov, Chos ) 395. [] L.M. Pecor, T.L. Crroll, Phys. Rev. Lett ) 109. [3] L.M. Pecor, Phys. Rev. E ) 347. [4] K.S. Fink, G. Johnson, T. Crroll, D. Mr, L.M. Pecor, Phys. Rev. E ) [5] J. Jost, M.P. Joy, Phys. Rev. E ) [6] Y. Chen, G. Rngrjn, M. Ding, Phys. Rev. E ) [7] X.F. Wng, G. Chen, IEEE Trns. Circ. Syst. -I: Fundm. Theory Appl ) 54. [8] M. Brhon, L.M. Pecor, Phys. Rev. Lett ) [9] T. Nishikw, A.E. Motter, Y.-C. Li, F.C. Hoppenstedt, Phys. Rev. Lett ) [30] A.E. Motter, C. Zhou, J. Kurths, Phys. Rev. E ) [31] D.U. Hwng, M. Chvez, A. Amnn, S. Boccletti, Phys. Rev. Lett ) [3] M. Chvez, D.U. Hwng, A. Amnn, S. Boccletti, Phys. Rev. Lett ) [33] C. Zhou, A.E. Motter, J. Kurths, Phys. Rev. Lett ) [34] A.Yu. Pogromsky, H. Nijmeijer, IEEE Trns. Circ. Syst. -I: Fundm. Theory Appl ) 15. [35] C.W. Wu, Synchroniztion in Coupled Chotic Circuits nd Systems, in: World Scientific Series on Nonliner Science, Series A, vol. 41, World Scientific, Singpore, 00. [36] C.W. Wu, Nonlinerity ) [37] V.S. Afrimovich, S.N. Chow, J.K. Hle, Physic D ) 44. [38] V.S. Afrimovich, W.W. Lin, Dyn. Stbl. Syst ) 37. [39] K. Josić, Nonlinerity ) 131. [40] V.N. Belykh, I.V. Belykh, M. Hsler, Physic D ) 159. [41] I.V. Belykh, V.N. Belykh, M. Hsler, Physic D ) 188. [4] I. Belykh, M. Hsler, M. Luret, H. Nijmeijer, Int. J. Bifurct. Chos 15 11) 005) 343. [43] I. Belykh, V. Belykh, M. Hsler, Chos ) [44] I. Belykh, M. Hsler, V. Belykh, Int. J. Bifurct. Chos 006) in press). [45] V.N. Belykh, I.V. Belykh, M. Hsler, Phys. Rev. E 6 000) 633.