Multi-Objective Control and Clustering Synchronization in Chaotic Connected Complex Networks*

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1 Mul-Objecve Conrol and Cluserng Synchronzaon n Chaoc Conneced Complex eworks* JI-QIG FAG, Xn-Bao Lu :Deparmen of uclear Technology Applcaon Insue of Aomc Energy 043, Chna Fjq96@6.com : Deparmen of Auomaon, College of Elecrcal Engneerng, Hoha Unversy, anjng, 0098, Chna Absrac: - The convenonal chaos conrol s only o reach sngle conrol objecve a each me. Dfferen conrol objecve only can be realzed a anoher me. Obvously, ha s no enough for applcaons of chaos conrol, especally for chaoc conneced complex neworks. Therefore mul-objecve conrol (MOC and cluser synchronzaon are naurally rased and deeded n pracce, as well as some mehods are developed and nvesgaed n dfferen complex dynamcal neworks, such as n communcaon engneerng, bologcal sysems, and socal neworks. In hs arcle, we revew and summarze wo effecve conrol and synchronzaon mehods for mul-objecve and cluser synchronzaon respecvely for any chaoc conneced complex nework. To demonsrae he mehod effecveness, we ake wo ypcal examples beam ranspor nework (BT and Kuramoo chaoc oscllaor s conneced nework.. Key-Words: - Chaoc conneced complex nework, mul-objecve conrol, cluser synchronzaon, adapve conrol, Kuramoo oscllaor s conneced nework, beam ranspor nework. Inroducon As a par of complex sysem research, complex neworks are suded across many or cluser synchronzaon. Snce many echnologcal, socal and bologcal neworks may conss of or dvde no several sub- -neworks or so-called communes, each one felds of scence and echnology, from naural scence o socal scence and echnology. Especally he WWW and Inerne pervades nearly all of felds. Scenss have dscovered ha many complex neworks have an underlyng archecure governed by shared organzng prncples, such as Small World, Scale-free and Superfamly. These nsghs have mporan mplcaons for a hos of applcaons. Synchronzaon and Conrol of ofen has her own behavor or dfferen knd of funcon. Man objecve/goal of Chaos conrol and synchronzaon nclude[-5]: unsable equlbrum pons, unsable perodc orbs, unsable me or spaal-emporal chaos (paern and so on. I s noed ha usually Only one conrol objecve can be realzed a one same me, bu s dffcul o reach mul-objecve a same me n radonal Chaos s one of very mporan ssues n chaos conrol and synchronzaon. Complex eworks. An ncreasng neress Obvously, ha s no enough for have been focused on synchronzaon and conrol of chaos n nonlnear complex neworks, especally on mul-objecve conrol applcaons of chaos conrol n pracce. Mul-objecve conrol (MOC and cluser synchronzaon(cs,ps are developed snce ISS: ISB:

2 s needed n communcaon engneerng, bologcal sysems, socal neworks and so on. Paral and complee synchronzaon s a unversal phenomenon n real world neworks. As an mporan branch of synchronzaon, he coordnaed ensemble of coupled phase oscllaors s a paradgm for many naural processes n physcal, bologcal and chemcal sysems, such as laser arrays, Josephson juncons, neural neworks, and so on [6,7]. CS ofen exss n bologcal scence [8] and communcaon engneerng [9]. So-called CS ha nodes n he same group synchronze wh each oher, bu here s no synchronzaon beween nodes n dfferen sub-nework [0-4]. Usually by choosng dfferen couplng scheme, he sably of seleced CS n coupled Josephson equaons s nvesgaed []. The resuls mples ha he sably of CS depends srongly on he selecon of he couplng schemes. Dfferen couplng schemes may lead o dfferen CS. On he anoher hand, for a gven nearesneghborhood nework wh zero-flux, an effecve approach o deermne some possble saes of CS s supposed [3]. However, s dffcul o oban hese couplng scheme. To realze MOC and CS, we wll exend he pnnng approach n Ref.[5,6] o reach dfferen desred objecve above n dfferen sub-neworks. In addonal, he mehod of adapve conrol approach[7,8] s appled o adapvely oban he bes conrol law for he curren sae of he chaoc dynamcal nework wh Kuromoo phase oscllaors. Adapve sraegy based on local nformaon was proposed n [9-] o make he nework acheve complee synchronzaon. In hs paper we wll ake beam ranspor nework (BT wh small-world (SWor scale-free (SF[], and Kuromoo phase oscllaor s nework as wo ypcal examples o show he MOC and CS. Frs of all, we apply a global lnear couplng conrol and local lnear error feedback mehod o realze he MOC of halo-chaos (equlbrums and perodc sae n he BT-SW and BT-SF respecvely. ex, a novel dsrbued conrol sraegy s proposed o make he nework acheve CS, where boh he conrol sraegy of one node and he edge srengh are adjused only accordng o s local nformaon. The whole nework s randomly dvded no hree sub-neworks, and he desnaon of each sub-nework s he average phase of all he nodes n he sub-nework. All smulaon sudes show ha he MOC and adapve sraegy mehod are effecve for any chaoc conneced neworks separaed no several dfferen sub-neworks. The new effecve conrol and synchronzaon mehods above may be poenal prospecve applcaons for based-chaos secure communcaon.. Descrpon of Basc Idea, Model and Mehod Was and Srogaz (WS proposed he small-world model n he frs me 998 [3] and scale-free model[4]. Scenss have hen made he mprovemen o he WS and he SF models wh dfferen formaon mechansm. Le us consder lnearly and symmercally coupled dencal dynamcal sysems, every node express n-dmensonal dynamcs sysem [5,7-]: (, x = f x + c M H( x, =,,,, j j j= ( n where x = ( x, x,, xn R s he sae vecor of he h node, c s he couplng srengh, M,j s he coupled marx (M,j =-k for =j; M,j= for j ; M,j= 0 oherwse, ISS: ISB:

3 H(x j makes he oupu funcon. The mehod of he maser sably funcon s presened o be used as a measure for he sably of he synchronous sae. For he BT, we have node dynamcal equaons f(x of beam halo-chaos n he BT s gven by[] : dx d = x, dx K = ( a+ bcos( x x + +, ( d x x 3 3 dx 3 = ω. d where (a, b,ω,k are he parameers of he subsysems. f a=.65, b=.5, ω = π, K=5 and he nal condons are aken randomly, such as (x (0,x (0,x 3 (0=(.0,0.5,π, Lyapunov exponens ( ,0, are obaned n one posve value means s n he halo-chaos. When he lnear couplng conroller G s appled only on varable x, he second equaon above s gven by: dx K = ( a+ bcos( x3 x+ + + G, (3 3 d x x where G = c a ( x x, =,,,. The j= j j G s desgned as a lnear couplng funcon of x for each node. We consder he BT wh sze =00 and he local lnks K=6, he nal condons of every halo-chaoc oscllaor are que dfferen: x [0 ], x [0], x 3 =π, c= or 5, he rewrng probably p=0.04, 0.4, 0.9. We compue he BT wh he WS model and he oher SW model, respecvely. The maxmum synchronzaon conrol error s defned as: max, { j } D ( = max D ( n = max [ ( xk, ( xjk, ( ], k= j, =,,,. (4 Sablzed perod or one halo-chaos sae n he BT s synchronzed f lm D ( = 0. max For an exend Kuramoo model of coupled phased dencal nodes, he evoluon of he dynamcal varable funcon s wren as follows: θ( = ω + a (sn( θ ( θ( u(, j j j=, j =,,,, (5 where ω s naural frequences dsrbued wh a gven probably densy. A= ( a j s he couplng marx. u ( s a conrol npu added o each node. When he npu u ( equals zero, (5 reduces o he normal exend Kuramoo model nvesgaed n [5-8]. If here s a connecon beween node and node j, hen = > 0( j, oherwse aj aj a j = = 0( j. Le a = a, a j hen (5 can be rewren as follows: j=, j θ( = ω + a ( sn(( θ ( θ( u(, j j j= =,,,, (6 Assume nodes fnally acheve M cluser synchronzaon and he whole nodes spls no M clusers (objecves. Whou loss of generaly, he ses of subscrps of hese clusers are G {,,, } =, { +, + } G = +,,, GM, { +, } = + + M,, where M =. When he nodes acheve he followng saes: j ISS: ISB:

4 θ( = θ( = = θ ( = s ( nework be ncluded n a se V, randomly =,, dvde he se no q( q p dsjon θ + ( = θ + ( = = θ ( ( + = s subses (sub-neworks V,, Vq. The = +,, + (7 correspondng sze of hese subses θ + + ( θ ( ( ( M θ M + = = = = s are,, q, respecvely, wh = + + M +,, + + = q. he nework s consdered o reach cluser synchronzaon, where s ( s he desred sae of node a me. In order o make he nework acheve CS, he npu s desgned as follows: u ( = s ( s ( and he couplng srengh aj j ( beween nodes s adjused by he followng adapve sraegy. a ( = β sn(( θ ( s ( ( θ ( s ( j j j (8 where, β > 0 s he adapve gan. Obvously, f here s only one subnework n he nework, hen for random node and node j, here s s becomes = s j. Then equaon (8 a ( = β sn( θ ( θ ( (9 j j where, Eq.(9 s he adapve sraegy. 3. Mul- objecve conrol mehod n dfferen subneworks In hs secon our purpose s o conrol dfferen objecve, such as equlbrums and perods, n dfferen sub-neworks of he complex nework (so-called communes, respecvely. A frs, le all he nodes n he For smplcy, we reorder he nodes as follows: V = {,,, } V = { +, +,, + } Vq = { + + q +, + + q +,, + + q + q} (0 Whou loss of generaly, we assume ha he objec s o make he saes of he nodes n V o one equlbrum x ( q. Therefore, he desnaon s rewren as: x = = x = x e x+ = = x + = x e ( x = = x = x + + q q + q eq In order o acheve hs desnaon, add a conroller o he node as follows: x = f ( x ( + c aj x j x + u j = where he conroller e ( u = c a ( x x hcd( x x j e e j = ( V, q Here d > 0, (3 e ISS: ISB:

5 xe j =,, xe j = +,, + = j = + + q +,, xeq + + q + q me-nvaran opology srucure, whch s used o ge rd of he effec of dfferen desnaons beween node neghbors. For q = and s, all he nodes are xe needed o be conrolled o one equlbrum, hen he former par s zero. The laer par s a sae feedback n subneworks. For each sub-nework (subse, here s only one node (4 has he sae feedback conrol par. Denoe and δ x( = x( xe where V and =, +,, q + h = qhen he nework sysem ( can be 0 oherwse rewren as: (5 dδ x ( Obvously, he conroller has wo pars. = f ( x (, fx ( e + c ( a δx j j d j= The former par s a consan for a fxed global couplng srengh c and a dδ x ( = f ( x(, f( xe + c a δ x ( j (7 d (6 and he nework wh he conroller (3 s wren as: ~ j j= BT-SW where To reach wo equlbrums x eq and x eq ~ ~ ~ n he BT-SW: a = a d, a = a,,,, d a a = d + + q +, + + q q +, + + q + x ~ = = x = x eq, oherwse a = a. j j x eq I can be seen from Res.[9] ha four + = = x = x heorems are proved o realze sablzed 8. only one equlbrum by usng a sngle conroller n complex neworks.i s noed ha n Refs. [9] and [30]only one We selec I nodes n local sub-nework V and I nodes n sub-nework V hen wo conrollers are desgned by objecve00goal can be acheved f couplng u = cd( x xeq effcen c s large enough and he couplng marx s symmerc. We would lke o ( =,,, I, emphasze here ha s easy o smlarly u = cd( x xeq prove ha he heorems above n Ref.[9] ( = +, +,, + can be exended o realze MOC of chaos or halo- chaos n chaoc conneced complex nework. To demonsrae, we ake wo ypcal examples o show he effecveness 9 The conrollers u and u are appled no followng equaons respecvely, of MOC mehod above as follows. I, 3. Mul-equlbrums conrol n he ISS: ISB:

6 x = f ( x + c a h( x, x + u =,, I j j j = x = f ( x + c a h( x, x = I +,, j j j = x = f ( x + c a h( x, x + u = +,, + I + +, j j j = x = f ( x + c a h( x, x = + I +,, +, j j j = where couplng funcon h(x j j eq eq x j s hx (, x = ( x x ( x x Smulaon resuls are shown n Fg., where I =4 I =4 couplng srengh c=00d=7.5 wo equlbrums x eq = [.4,. 0 0, 0] n V and x eq =[-.4, 0, 0] n V can be sablzed respecvelywe also compued he Lyapunov exponens are :λ = andλ = , hese negave values show ha wo sablzed equlbrums afer he MOC conrollng. a b Fg. Two equlbrums (x eq = [.4, 0, 0] and x eq =[-.4, 0, 0] n BT-SW are reached. (alocal nework V c=00, d=7.5 (b Local nework V, c=00, d= Sablzng one equlbrum and one perod n BT-SW To sablze one equlbrum n subse V and one perod n subse V, x = = x = x eq x + = = x = S where x eq =[.4 0 0]conrol objecve S s perod- We desgn global error couplng funcon h(x x j hx (, x = ( x s ( x x j j eq and wo local conrollers u and u follows u = cd( x xeq ( =,,, I, u = d( x + ( = +, +,, + I are as The Conrolled resuls s gven n Fg., where I =4 and x eq = [.4, 0, 0] sablzed n V,, c=00, d =7.5d =5I =4 perod- sablzed n V.Afer conrol, we have he Lyapunov exponenλ = n V and Lyapunov exponenλ = -.6 n V whch mply he equlbrum and perod- are ISS: ISB:

7 sablzed respecvely. b (a Conrolled resuls for x eq = [.4, 0, 0] and perod-., (a Sablzed equlbrum n V and (b Sablzed perod-n V ; d =5c=00, d =7.5d =5I =4,I =4. 4. Adapve Cluser Synchronzaon In hs secon, we ake n coupled phase oscllaor s nework as he second example o show adapve CS conrol. Here, we ake he nvesgaed nework sze s 00, and he whole nodes are randomly dvded no hree subneworks, whch means M = 3. The desred sae s ( of node a me s chosen as follows: θ ( + + θ ( s ( = s ( = = s ( = θ + ( + + θ ( + s + ( = s + ( = = s (, + = θ + + ( + + θ( s + + ( = s + + ( = = s( == 3 3 where hree group sze, and (3 are 4, 38 and 38, respecvely. The nodes nal phases are randomly dsrbued n [0, π ] and he nal edge srenghs are se o zero. Whou loss of generaly, he values of ω are se o zero. 4. BA scale-free neworks A frs, a BA scale-free nework s consruced wh m= m 0 = 5 and = 00. For dealed generaon algorhm of BA scale-free nework, s nroduced n [0]. The dsrbued conroller (5 s added o each node wh he edge srengh s adapve accordng o (6. As can be seen from Fg.3(a, for > 0s, 4 red doed lnes reduces o one red doed lne, 38 magena doed lnes reduces o one magena doed lne, and he res 38 blue sold lnes reduces o one blue sold lne. Ths mples ha he nework reaches he desred CS. ISS: ISB:

8 θ(=,,..., Edge Srengh (a (b Fg. 3. Cluser synchronzaon n he exend Kuramoo model, wh he underlyng nework beng BA scale-free nework and he adapve gan β = 0.. (a θ ( ; (b edge srenghs aj (. In Fg. 3(b, for he edge srengh consruced. In deal, he rewrng aj beween node and node j, keep consan for > 0s. Combned wh Fg.3(a, s found ha when he nework reaches cluser synchronzaon, he edge srenghs are fxed, whch mples ha he lef sde of (6 wll be zero. For mos edge srenghs, her values are less han 0.5. If we do no adop he adapve couplng sraegy, he nework wll reach CS when all edge srenghs are bgger han Ths means ha many unnecessary edge srenghs are wased f he edge srenghs do no change wh he adapve sraegy. Defne he me of he nework achevng CS as s, able shows s for dfferen adapve gans. As can be seen ha, he ncrease of he adopve gan reduces he me s. Ths can be undersood by Eq.(6, he bgger adopve gan wll lead o he adapve sraegy adjus rapdly, whch reduces he me s. 4. WS small-world neworks In order o compare wh he BA scale-free nework, a WS small-world nework wh he same nework sze and average degree as hose of BA scale-free nework used n he above secon s Table. In BA nework he me of globally achevng CS for adapve gans and m= m 0 = 5. Adapve The me of reachng CS, s s gan For BA ework probably s p = 0., nework sze s = 00, and k = 5. As shown n Fg. 4(a, for > s, 4 red doed lnes reduces o one red doed lne, 38 magena doed lnes reduces o one magena doed lne, and he res 38 blue sold lnes reduces o one blue sold lne. Ths mples ha he nework reaches he desred cluser synchronzaon. Compared wh he resuls n Fg.3, he me ha he nework frs reaches cluser synchronzaon s obvously reduced. ISS: ISB:

9 In Fg. 4(b, for a j > s, he edge srengh keep consan. Combned wh Fg. 4(a, s found ha when he nework reaches cluser synchronzaon, he edge srenghs no longer change. For mos edges, he edge srenghs are all less han 0.8. Compared wh he edge srenghs n Fg. 3(b, when he nework reaches cluser synchronzaon, he values of edge srenghs are obvously reduced. We change he rewrng probably and ge smlar resuls. Here, he dealed fgures are omed. Therefore, for he same nework sze and average degree, BA scale-free nework s dffcul o become cluser synchronzaon han he WS small-world nework. For he WS small-world nework, he nework becomes easy o reach cluser synchronzaon wh he ncrease of rewrng probably θ(=,,..., Edge Srengh (a (b Fg. 4. Cluser synchronzaon n he exend Kuramoo model, wh he underlyng nework beng WS small-world nework and he adapve gan β = 0.. (a θ ( (b edge srenghs a (. j Table shows s for dfferen he rewrng probables and adapve gans. As can be seen ha, for he same value of adapve gan, he me s changes lle wh he ncrease of rewrng probably. On he oher sde, for he same value of rewrng probably, he ncrease of adopve gan reduces he me s. 5. Concluson In hs paper, he mos mporan resuls s ha mul-objecve conrol and cluser synchronzaon can be realzed n wo ypcal chaoc conneced neworks, he BT-SW and coupled phased oscllaor s nework wh SW/SF. We dvde he whole nework (se no several dsjon sub-neworks (subses as s needed n pracce. Then we apply he global couplng appled o he whole nework and combned lnear error feedback conroller no a few nodes n dfferen sub-neworks respecvely, fnally mul-objecve (dfferen equlbrums, perod and chaos s reached afer mul-objecve conrol. In addonal, a novel dsrbued conrol sraegy o make a nework wh coupled phased oscllaors acheve cluser synchronzaon was also proposed. In hs approach, he npu added o each node s based on self and s neghbors desnaon due o each solaed node knows local nformaon. Moreover, he edge srengh beween nodes s adapve accordng o s local nformaon. These make he approach easy o apply wh low Table. The me of he nework globally achevng cluser synchronzaon for dfferen he rewrng ISS: ISB:

10 probables and adapve gan. Rewrng Adapve The me of probably gan reachng CS (sec cos. By wo classcal nework opologes, he effecveness of he mehods above was confrmed by numercal smulaon. Therefore, he wo mehods above can be exended o oher complex dynamcal neworks wh nonlnear/chaoc dynamcs and wll be helpful o comprehend he underlyng mechansm of complex dynamcal nework. I has a poenal prospecve of applcaons n based-chaos/ halo-chaos secure communcaon n complex nework [3,3]. Auhor would lke o hank hs sudens X. B Lu and Q. Lu for helpng smulaon and draw fgures References: [] G. Chen and X. Dong,FROM CHAOS TO ORDER:Mehodologes, Perspecves and Applcaons 998USA, [] Eds. by G. Chen and X Yu, Chaos Conrol Theory and Applcaons, Sprnger,003. [3] G. Chen,J. Q. Fang, Y. Hong, H. S. Qn, Chaper 6: Inroducon o Chaos Conrol and An-Conrol, Advanced opcs n nonlnear conrol sysems, Eded by T. P. Leung & H. S. Qn, pp93-45, World Scenfc Publshng, 00. [4] J. Q. Fang, Tamng Chaos and Developng Hgh-Tech,(n Chnese Bejng, Aomc Energy Press, 00. [5] Q. Fang, Tamng Halo and Explorng ework Scence (n Chnese, Bejng, Aomc Energy Press, 008. [6] Y.Kuramoo, Chemcal oscllaons, waves,urbulence (Sprnger, ew York, 99. [7] S.H.Srogaz, Physca D 43, (000. [8] K. Kaneko, Physca D 75.55(994 [9].F.Rulkov, Chaos, 6,6(996. [0] M.I.Rabnovch, e al., Phys.Rev. E 60. R30(999. [] D.H.Zanee, A.S.Mkhalov, Phys.Rev. E 57,76(998. [] W. X. Qn, G. Chen, Physca D 97,375 (004. [3] V.. Belykh, I. V. Belykh, E. Mosekde, Phys. Rev. E,63, 0366 (00. [4] S. Jalan, R. E. Amrkar, C. K. Hu, Phys. Rev. E, 7,06(005. [5] Xang L, Xaofan Wang and Guanrong Chen, Pnnng a complex dynamcal neworks o s equlbrum, IEEE Transacons on Crcus and Sysems-I, Regular Papers, 5(0, (004, [6] Tanpng Chen, Xwe Lu and Wenlan Lu, Pnnng complex neworks by a sngle conroller, arxv:mah-ph/ v. [7] J.Zhou, J. A. Lu, J.Lu, IEEE ISS: ISB:

11 Transacons on Auomac Conrol,5 (4, (006. [8] P. D. Lells, M. D. Bernardo, F. Garofalo, Chaos, 8, 0370 (008. [9] P. D. Lells, M. D. Bernardo, F. Garofalo, Chaos, 8, 0370 (008. [0] W L Lu. Chaos(S ,7(, 3(007. [] Q. S. Ren,. J. Y Zhao.Physcal Revev E (007 [] J. Q. Fang, G. R. Chen, Prog. In. Phys.3(0033; J. Q. Fang, Z. S. Wang and G. R. Chen, Theor. Phys. 4( [3] D. J. Was, S. H. Srogaz, aure, 393( [4] A. L. Barabas, R. Alber,, Scence, 86 ( S. H. Srogaz, aure,40(0068. [5] H Hong, M Y Cho, B J Km Phys. Rev. E (00 [6] Y Moreno, A F Pacheco.Europhyscs Leers (004 [7] T Ichnomya Phys. Rev. E (004 [8] X L. Physca A. 360( 69(006 [9] Tanpng Chen, Xwe Lu and Wenlan Lu, Pnnng complex neworks by a sngle conroller, arxv:mah-ph/ v. [30] Xang L, Xaofan Wang and Guanrong Chen, Pnnng a complex dynamcal neworks o s equlbrum, IEEE Transacons on Crcus and Sysems-I, Regular Papers, 5(0, (004, [3] J. Q. Fang, X. H. YU, Chn. Phys. Le, ( [3] J. Q. Fang, X. F. Wang and Z.G. Zheng, Progress n Physcs, 9( ISS: ISB:

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

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