Global Synchronization of Directed Networks with Fast Switching Topologies

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1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Sociey and IOP Publishing Ld Vol. 52, No. 6, December 15, 2009 Global Synchronizaion of Direced Neworks wih Fas Swiching Topologies LU Xin-Biao, 1 QIN Bu-Zhi, 2, and LU Xin-Yu 3 1 Deparmen of Auomaion, Hohai Universiy, Nanjing , China 2 Deparmen of Auomaion, Nanjing College of Chemical Technology, Nanjing , China 3 Shandong Meallurgical Research Insiue, Jinan , China (Received December 30, 2008; Revised March 30, 2009) Absrac Global synchronizaion of a class of direced dynamical neworks wih swiching opologies is invesigaed. I is found ha if here is a direced spanning ree in he fixed ime-average of nework opology and he ime-average is achieved sufficienly fas, hen he nework will reach global synchronizaion for sufficienly large coupling srengh. PACS numbers: X Key words: global synchronizaion, direced nework, fas swich 1 Inroducion Synchronizaion of complex dynamical neworks has received a lo of ineress in recen years. [1 21] Alhough many researches have been focused on he local sabiliy of synchronizaion sae when he iniial saes of nodes are near o each oher, [1 3] global synchronizaionmeans of neworks despie of he differences beween nodes iniial saes has also been invesigaed [4,10 13] For example, i is found ha a direced nework globally synchronizes for sufficienly large coupling srengh beween nodes if here is a spanning direced ree in he nework. [4] I is known ha opologies of many real world neworks evolve wih ime. [12 18] Recenly, synchronizaion in neworks wih ime-varying coupling srenghs has been considered. [12 14] A connecion graph sabiliy mehod is proposed, and a bound is esablished based on considering he oal lengh of all pahs hrough edges on neworks wih ime-varying connecion. [15 16] I is found ha a ime-varying nework could propagae sufficien informaion o allow synchronizaion of coupled oscillaors, despie of an insananeously disconneced opology. [17] Furhermore, i is proved ha if he nework locally synchronizes for he fixed ime-average of he opology and he ime-average is achieved sufficienly fas, he nework locally synchronizes for swiching opologies. [18] In his paper, global synchronizaion of direced neworks wih swiching opologies is invesigaed. I is proved ha if here exiss a direced spanning ree in he fixed ime-average of nework opology and he imeaverage is achieved sufficienly fas, he nework wih swiching opology will reach global synchronizaion for sufficienly large coupling srengh beween nodes. The res of he paper is organized as follows. In he preliminary Sec. 2, some definiions and a lemma are inroduced. Main resul of his paper and simulaions are invesigaed in Secs. 3 and 4, respecively. In he las Secion, conclusions are presened. 2 Synchronizaion of Direced Neworks Consider a nework of N linearly coupled idenical oscillaors, wih each oscillaor being an n-dimensional dynamical sysem. The sae equaions of he nework are ẋ() = [f(x 1 (), ),..., f(x N (), )] T + κ(c() D())x(), (1) where x() = (x 1 (),..., x N ()) T, x i () = [x i1 (), x i2 (),..., x in ()] T R n is he sae variable of node i. f describes he dynamics of each isolaed node, κ is coupling srengh beween nodes, D() = (d ij ()) n n is an inner coupling marix. C() = (c ij ()) N N is he coupling marix which is defined as following: If here is a connecion beween node i and node j, hen c ij () < 0 (i j); oherwise, c ij () = 0 (i j). c ii () = N,j i c ij() (i = 1, 2,..., N). 2.1 Fixed Topology A firs, he nework wih fixed opology is considered in [4]. This means ha in (1) C() = C for all, he following lemma is obained. Lemma [4] Assume ha: (i) C is a zero row sums marix wih nonposiive offdiagonal elemens. (ii) f(x(), ) + κd()x() is V -uniformly decreasing for some symmeric posiive definie V, ha is (x y) T V (f(x, ) + κd()x f(y, ) κd()y) µ x y 2 for some µ > 0 and all x, y R n and all. (iii) V D() 0 and is symmeric for all. (iv) The underlying weighed direced graph conains a spanning direced ree. Then he nework (1) globally synchronizes for sufficienly large coupling srengh κ. Suppored by he Naural Science Foundaion of Hohai Universiy under Gran No Corresponding auhor, buzhiqin@126.com

2 1020 LU Xin-Biao, QIN Bu-Zhi, and LU Xin-Yu Vol Swiching Topology When he differences of he nodes saes are small, synchronizaion of neworks wih swiching opology is invesigaed in [18]. I was found ha if he nework wih fixed ime-average opology can reach synchronizaion and he ime-average is achieved sufficienly fas, he nework wih swiching opologies will reach synchronizaion. However, he invesigaed nework in [18] is undireced. For he direced nework wih swiching opologies, a sufficien condiion of he nework reaching synchronizaion is obained, especially despie of he differences of he nodes iniial saes. Theorem Suppose here exiss a consan T for which he marix-valued funcion C() is such ha C = 1 T for all and he sysem +T C(τ)dτ, (2) ẋ() = [f(x 1 (), ),..., f(x N (), )] T +( C D())x() (3) saisfies he condiions (i) (iv) in he Lemma. This means he sysem (3) globally synchronizes. Furhermore, suppose ha he oscillaor has bounded slope such ha (f(v, ) f(v, ))/(v v ) l, where l > 0 for all v, v R n and v v. Then here exiss ε > 0 such ha for all fixed ε (0, ε ), he sysem globally synchronizes. ẋ() = [f(x 1 (), ),..., f(x N (), )] T + (C(/ε) D())x() (4) 3 Simulaion As a ypical example, he Chua s circui is considered as each isolaed node s dynamics in he complex dynamical nework: ẏ 1 = 9y y [ y y 1 1 ], ẏ 2 = y 1 y 2 + y 3, ẏ 3 = y 2. (5) Then he Chua s circui is chaoic, as shown in Fig. 1. Fig. 1 Chaoic behavior of Chua s circui. A se of hree nework opologies in Figs. 2(a) 2(c) are seleced, and each of hem is disconneced. However, he union of hem, as shown in Fig. 2(d), has a direced spanning ree wih he roo node being Node 9. This implies ha he condiion (iv) of he lemma is saisfied and here is a leas one pah beween he roo node and each oher node. Fig. 2 (a) (c) are nework opologies G 1 G 3, respecively, while (d) is he union of hem.

3 No. 6 Global Synchronizaion of Direced Neworks wih Fas Swiching Topologies 1021 The coupling marix of G 1, G 2, and G 3 are denoed by C 1, C 2, and C 3, respecively. The weigh of each edge in Figs. 2(a) 2(c) is assigned o 1. Le C = (1/εT) εt 0 C(/ε)d =(C 1 + C 2 + C 3 )/3 denoes he coupling marix of he union of C 1, C 2, and C 3. Since he marix C i (i = 1, 2, 3) is a zero row sums marix wih nonposiive off-diagonal elemens, he marix C is also a zero row sums marix wih nonposiive off-diagonal elemens. This means he condiion (i) in he lemma is saisfied. For Chua circui, when he coupling srengh is κ = 50, he marix V = I, and he marix D() = I, where I is an ideniy marix. Obviously, he fixed imeaverage nework saisfies he condiions (ii) (iii) in he lemma. Therefore, from he lemma, he nework wih he coupling marix C may reach global synchronizaion in heory. Figure 3 shows he saes of nodes in he nework wih he coupling marix C. As can be seen from Fig. 3, he saes of dimension x i1 reach synchronizaion afer 0.7 s; he saes of dimension x i2 reach synchronizaion afer 0.9 s; he saes of dimension x i3 reach synchronizaion afer 1 s. This means ha for > 1 s, he nework reaches synchronizaion. Fig. 3 Dynamics of he fixed nework in Fig. 2(d) and each node being an isolaed Chua s circui and coupling srengh κ = 50. (a) x i1(); (b) x i2(); (c) x i3(). Fig. 4 Dynamics of he nework wih swiching opologies in Figs. 2(a) 2(c) and each node being an isolaed Chua s circui. Coupling srengh κ = 50 and swiching ime ε = s. (a) x i1(); (b) x i2(); (c) x i3(). When he node is Chua circui, he inequaion (f(v, ) f(v, ))/(v v ) l saisfies wih he larges slope bound being l = Combined wih he nework wih he coupling marix C saisfies he condiions (i) (iv) of he lemma,

4 1022 LU Xin-Biao, QIN Bu-Zhi, and LU Xin-Yu Vol. 52 i is found ha he nework wih swiching opologies G 1, G 2, and G 3 may globally reach synchronizaion when he swich ime is sufficienly small according o he heorem. Figure 4 shows ha he dynamics of nework wih swiching opologies G 1, G 2, and G 3, where he coupling srengh is κ = 50 and he swiching ime is ε = 0.003s. I is found ha he saes of dimension x i1 reach synchronizaion afer 0.6 s; he saes of dimension x i2 reach synchronizaion afer 1 s; he saes of dimension x i3 reach synchronizaion afer 1 s. This means ha for > 1 s, he nework reaches synchronizaion. he whole en nodes can reach global synchronizaion afer 1s. Figure 5 shows ha he dynamics of he same nework as used in Fig. 4. Here, he swiching ime increases o ε = 0.03 s. I is found ha he saes of all hree dimensions can no reach synchronizaion. This implies ha differen opologies mus be swiched fas o make he nework achieve synchronizaion. Fig. 5 Dynamics of he nework wih swiching opologies in Figs. 2(a) 2(c) and each node being an isolaed Chua s circui. Coupling srengh κ = 50 and swiching ime ε = 0.03 s. (a) x i1(); (b) x i2(); (c) x i3(). 4 Conclusion Global synchronizaion of direced neworks wih fas swiching opologies is invesigaed. If here is a direced spanning ree in he fixed ime-average of nework opology and he ime-average is achieved sufficienly fas, he nework wih swiching opologies globally synchronizes for sufficienly large coupling srengh beween nodes. This resul can be easily exended o undireced neworks which for every edge in he undireced neworks which can be considered as wo direced edges. Synchronizaion of direced neworks wih swiching opologies and imedelay needs o be furher invesigaed. Acknowledgemens The auhors hank he anonymous referees for heir useful suggesions o revise his paper. Appendix Proof Since he sysem (3) saisfies he condiions (i) (iv) in he Lemma, here exiss a symmeric irreducible zero row sums marix U N N wih nonposiive off-diagonal elemens. Consruc a Lyapunov funcion Then, i is found ha g(x()) = 1 2 x()t (U V )x(). (A1) ġ(x()) = x() T (U V )ẋ() f(x 1 (), ) + D()x 1 () = x() T (U V ) f(x N (), ) + D()x N () + x() T (U V )( C D() I D())x() i<j U ij (x i () x j ()) T V (f(x i (), ) + D()x i () f(x j (), ) D()x j ()) i<j U ij ( µ x i () x j () 2 ) T.

5 No. 6 Global Synchronizaion of Direced Neworks wih Fas Swiching Topologies 1023 Noe ha U ij 0 for i < j. For each U ij 0 and δ > 0, and sufficienly large such ha if x i () x j () δ, hen ġ (µ/2) x i () x j () 2. This implies ha for large enough, x i () x j () δ. Therefore, lim x i () x j () = 0. For sysem (3), suppose ē() = (ē 1 (),..., ē N ()) T, ē i () = x i () s(), where s() is he synchronizaion sae s() = x i (), i = 1,...,N, which saisfies ṡ() = f(s()). Then from (5), consruc he following Lyapunov funcion as follows g(ē()) = 1 2ē()T (U V )ē(). (A2) According o he above analysis of (A1), i also can be obained ha lim x i () x j () = 0. Then here exis posiive scalars η, ρ, and ϕ η ē() 2 g(ē(), ) ρ ē() 2, (A3) ġ(ē(), ) ϕ ē() 2. For, hen (3) can be rewrien as ē i () = f(x i ()) f(s()) + κ c ij D()ē j (). For e i () = x i () s(), (4) can be rewrien as ė i () = f(x i ()) f(s()) + κ (A4) (A5) c ij (/ε)d()e j (). (A6) Because f(v, ) f(v, )/v v l, wihou loss of generaliy, assume v v > 0, hen l(v v ) f(v, ) f(v, ) l(v v ). Then (A6) can be rewrien as (A7) ė i () = f(x i ()) f(s()) + κ c ij (/ε)d()e j () l(x i () s()) + κ c ij (/ε)d()e j (), l(x i () s()) + κ c ij (/ε)d()e j () ė i () = f(x i ()) f(s()) + κ c ij (/ε)d()e j (). Combined (A5) and he righ of (A7), i is obained ē() = (li + κ C D())ē(). The righ of (A8) can be wrien as ė() = (li + κc(/ε) D())e(), (A8) (A9) (A10) (A11) where I is an ideniy marix. To esablish global sabiliy of (4), g(e(), ) is needed o be also a Lyapunov funcion for (4) if ε is sufficienly small. This claim is achieved by showing ha for sufficienly small value of ε, g(e(), +εt, ) = g(e(+εt), +εt) g(e(), ) is negaive definie for all g(e(), + εt, ) = 1 2 e( + εt)t (U V )e( + εt) 1 2 e()t (U V )e(). (A12) Suppose φ C (, 0 ) is he ransiion marix corresponding o li + κc(/ε) D(), i.e. e() = φ C (, 0 )e( 0 ). Then (A12) can be rewrien as g(e(), + εt, ) = 1 2 e()t (φ T c ( + εt, )(U V )φ c ( + εt, ) (U V ))e(). (A13) Similarly, le φ C(, 0 ) be he ransiion marix corresponding o li + κ C D(), he Peano Baker series represenaion of he ransiion marix is used o define φ C ( + εt, ) = I + By hypohesis, +εt + i=2 +εt +εt (li + κc(σ 1 /ε) D())dσ 1 σ1 σi 1 (li + κc(σ 1 /ε) D()) (li + κc(σ 1 /ε) D())dσ 1 dσ i. C(σ/ε)dσ = εt C. Then i is obained H( + εt, ) = φ C ( + εt, ) φ C( + εt, ) +εt = (li + κc(σ 1 /ε) D()) i=2 i=2 +εt σ1 (li + κ C σ1 D()) σi 1 σi 1 (li + κc(σ 1 /ε) D())dσ 1 dσ i (li + κ C D())dσ 1 dσ i.

6 1024 LU Xin-Biao, QIN Bu-Zhi, and LU Xin-Yu Vol. 52 Suppose li + κc(/ε) D() α, li + κ C D() α, a bound for H( + εt, ) is compued H( + εt, ) e 2εTα 1 2εTα. Noe ha φ C ( + εt, ) = H( + εt, ) + φ C( + εt, ), hen (A13) can be rewrien as (A14) g(e(), + εt, ) = 1 2 e()t (φ T c ( + εt, )(U V )φ c( + εt, ) (U V ))e() I is known from (A3) and (A4) ha From (A2) and (A18), Thus e()t (φ T c ( + εt, )(U V )H( + εt, ) + HT ( + εt, )(U V )φ c ( + εt, ) + H T ( + εt, )(U V )H( + εt, ))e(). (A15) U V ρ, φ C(, 0 ) ρ/ηe (ϕ/2ρ)( 0 ), g(ē(), ) e (ϕ/ρ)( 0) g(ē( 0 ), 0 ). g(ē( + εt), + εt) g(ē(), ) (e ϕεt/ρ 1)g(ē(), ) ρ(e ϕεt/ρ 1) ē() 2. g(e( + εt), + εt) g(e(), ) (e ϕεt/ρ 1)g(e(), ) ρ(e ϕεt/ρ 1) e() 2, 1 2 e()t (φ T c ( + εt, )(U V )φ c( + εt, ) (U V ))e() ρ(e ϕεt/ρ 1) e() 2. (A19) From (A14) and (A16) (A19) g(e(), + εt, ) [ρ(e ϕεt/ρ 1) + ρ( ρ/ηe ϕεt/2ρ )(e 2εTα 1 2εTα) + ρ 2 (e2εtα 1 2εTα) 2 ] e() 2. (A20) Defining he coninuously differeniable funcion q(ε, e a ()) o be he righ-hand side of (A20), i can be shown ha q(0, e()) = 0 and ( / ε)q(0, e()) = ϕt e() 2 < 0. Thus since q(ε, e()) as ε, here exiss ε, q(ε, e()) = 0 and q(ε, e()) < 0 for all ε (0, ε ). Thus g(e(), + εt, ) 0 for all ε (0, ε ). Similarly, he sabiliy of he lef of (A9) is obained. Therefore, he sabiliy of (A6) is obained. This means he sysem (4) globally synchronizes. The proof is complee. (A16) (A17) (A18) References [1] L.M. Pecora and T.L. Carroll, Phys. Rev. Le. 80 (1998) [2] X.B. Lu, X.F. Wang, X. Li, and J.Q. Fang, Physica A 370 (2006) 381. [3] W.L. Lu and T.P. Chen, Physica D 213 (2006) 214. [4] C.W. Wu, Nonlineariy 18 (2005) [5] J.H. Li, Physica D 190 (2004) 129. [6] P. Zhou, X.F. Cheng, and N.Y. Zhang, Commun. Theor. Phys. 50 (2008) 931. [7] J.H. Li, Chin. Phys. Le. 25 (2008) 413. [8] Y. Chen and X. Li, Commun. Theor. Phys. 51 (2009) 470. [9] J.H. Li, Commun. Theor. Phys. 49 (2008) 665. [10] H.X. Wang, Q.S. Lu, and Y.Q. Wang, Commun. Theor. Phys. 51 (2009) 475. [11] J.B. Zhang, Z.R. Liu, and Y. Li, Commun. Theor. Phys. 50 (2008) 925. [12] V.N. Belykh, I.V. Belykh, and M. Hasler, Physica D 195 (2004) 159. [13] I.V. Belykh, V.N. Belykh, and M. Hasler, Physica D 195 (2004) 188. [14] J. Io and K. Kaneko, Phys. Rev. Le. 88 (2002) [15] J.H. Lv and G.R. Chen, IEEE Trans. On Auomaic Conrol 50 (2005) 841. [16] D.H. Zanee and A.S. Mikhailov, Physica D 194 (2004) 203. [17] J.D. Skufca and E.M. Bol, Mahemaical Bio-Sciences and Engineering (MBE) 1 (2004) 347. [18] D.J. Siwell, E.M. Boll, and D.G. Roberson, SIAM, J. Applied Dynamical Sysems 5 (2006) 140. [19] X.B. Lu and B.Z. Qin, Commun. Theor. Phys. 51 (2009) 485. [20] P.D. Lellis, M.D. Bernardo, F. Sorrenino, and A. Tierno, Inernaional Journal of Compuer Mahemaics 85 (2008) [21] V.N. Belykh, G.V. Osipov, V.S. Perov, J.A.K. Suykens, and J. Vandewalle, Chaos 10 (2008)