Inner-Outer Synchronization Analysis of Two Complex Networks with Delayed and Non-Delayed Coupling

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1 ISS , Eglad, UK Joural of Iformao ad Compug Scece Vol. 7, o., 0, pp Ier-Ouer Sycrozao Aalyss of wo Complex eworks w Delayed ad o-delayed Couplg Sog Zeg + Isue of Appled Maemacs, Zeag Uversy of Face ad Ecoomcs, Hagzou Zeag, 3008, P.R. Ca (Receved ovember 8, 0, acceped December, 0 Absrac. I s paper, wo kds of sycrozao bewee wo complex eworks w o-delayed ad delayed couplg are dscussed by e pg corol meod, a s, er sycrozao ad ouer sycrozao. Based o e Lyapuov sably eory ad lear marx equaly (LMI, some suffce codos for e sycrozao are derved by addg lear feedback corollers o a par of odes, ad e lear feedback corollers are desged. Uder suable codos, o oly er sycrozao bu also ouer sycrozao ca be asympocally aceved. umercal smulaos are preseed o sow e effecveess of e proposed sycrozao sceme. Keywords: Complex ework; Ier-ouer sycrozao; Pg corol; o-delayed ad delayed couplg;. Iroduco I e pas few years, e corol ad sycrozao problem of complex eworks as bee exesvely vesgaed varous felds due o s may poeal applcaos [-3]. Ad varous corol scems cludg pg corol [4-5], adapve corol [6], mpulsve corol [7-9], ec., ave bee used o sudy e above problem. Geerally speakg, ework sycrozao ca be classfed o er sycrozao ad ouer sycrozao [0]. I bref, e sycrozao a ework s called er sycrozao,.e., e sycrozao of all e odes w a ework, as bee vesgaed recely [3-9]. O e oer ad, ouer sycrozao [0-4] occurs bewee wo or more complex eworks regardless of sycrozao of e er ework. Oe mpora example s e fecous dsease a spreads bewee dffere commues. erefore, ow o realze e sycrozao bewee dffere eworks s very eresg ad callegg work. L e al. [0] poeered sudyg e ouer sycrozao bewee wo udrecoally coupled complex eworks ad derved aalycally a crero for em avg e decal opologcal srucures. e adapve-mpulsve sycrozao bewee wo complex eworks w o-delayed ad delayed couplg was dscussed Ref. [4]. Aoer eresg work s ow o realze e er sycrozao sde eac ework ad e ouer sycrozao bewee wo dffere eworks smulaeously. Recely, Su e al. [5] vesgaed e ybrd sycrozao problem of wo coupled complex eworks by usg e lear feedback ad e adapve feedback corol meods, bu e me delay was gored. o smulae more realsc eworks, me delay sould be ake o accou. Su e al. [6] suded wo kds of sycrozao bewee wo dscree-me eworks w me delays, cludg er sycrozao w eac ework ad ouer sycrozao bewee wo eworks. I e above leraures [0-5], e eworks are coupled by full saes of odes e eworks, wc meas all e saes e drve ework mus be rasmed o e respose ework. However, for e complexy of e ework, s dffcul o realze e sycrozao by addg corollers o all odes. o reduce e umber of e corollers, a aural approac s o corol a complex ework by pg par of e odes e eworks. As far as e auors kow, ere s few work o pg sycrozao bewee wo coupled dyamcal eworks, aloug some pg corol scemes ave bee proposed for er sycrozao. Movaed by e above dscussos, s paper wll focus o e sycrozao problem of wo coupled + Correspodg auor. E-mal address: sogzeg0738@yaoo.com.c Publsed by World Academc Press, World Academc Uo

2 Sog Zeg: Ier-Ouer Sycrozao Aalyss of wo Complex eworks w Delayed ad o-delayed Couplg dyamcal eworks w bo delayed ad o-delayed couplg va pg corol meod, cludg er sycrozao w eac ework ad ouer sycrozao bewee wo eworks. Some crera for e sycrozao are derved. Aalycal resuls sow a wo eworks ca realze e sycrozao: e ouer sycrozao bewee e drve-respose eworks, ad e er sycrozao e drve ework ad e respose ewwork, respecvely.. Problem descrpo ad prelmares I s paper, we cosder e wo coupled complex dyamcal ework cossg of learly coupled decal dyamcal odes, w eac ode beg a -dmesoal dyamc sysem respecvely. e drve coupled complex ework s caracerzed by x & ( = f( x ( + c a Γ x ( + c b Γ x ( τ, =,,...,. ( = = Cosder e respose coupled complex dyamcal ework as follows: = = y & ( = f( y ( + c a Γ y ( + c b Γ y ( τ + u, =,,...,. ( were x ( ( (, (,..., ( = x x x R s e drve sae vecor of e ode, ( ( (, (,..., ( y = y y y R s e respose sae vecor of e ode, f : R R s a smoo fuco, e cosa c > 0 ad c > 0 deoe e odelayed ad delayed couplg sreg respecvely, τ > 0 s e me delay. u are lear corollers o be desged. Γ = dag( γ, γ,, γ ad Γ = dag( γ, γ,, γ are posve defe dagoal er couplg marces of e eworks. A ( a R =? ad B= ( b R? are e odelayed ad delayed weg cofgurao marces respecvely, were a ad b are defed as follows: If ere s a coeco from ode o ode (, e e couplg a 0; oerwse, a = 0 (, ad e dagoal elemes of marx A are defed as =, a = a, =,,.... B as e same meag as a of marx A. Suppose C([ 0 τ, 0], R be e Baac space of couous vecor-valued fucos mappg e erval [ 0 τ, 0] o R w e orm φ = sup ( 0 τ s φ s. For e fucoal dffereal equao 0 (, s al codos are gve by x ( = φ ( C( [ 0 τ, 0], R. I s assumed a ( as a uque soluo w respec o ese al codos. For e fucoal dffereal equao (, s al codos are gve by y ( ( ([ 0, 0], = ϕ C τ R. Ad, a leas, ere exss a cosa ( =,, L, suc a ϕ( φ( for [ 0 τ, 0]. I order o derve our ma resuls, some ecessary defos ad lemmas are eeded. Defo e coupled ework ( ad e coupled ework ( are sad o aa er ad ouer sycrozao smulaeously f lm( x ( x ( = 0, lm( y ( y ( = 0, lm( y ( x ( = 0,, =,,.... Assumpo (A (see[7] Assumg a ere s a posve-defe dagoal marx P = dag( p, p,..., p ad a dagoal marx D = dag( d, d,..., d suc a f sasfes e followg equaly: ( x y P( f( x, f( y, Δ( x y η ( x y ( x y, for some η > 0, all x, yî R ad > 0. Lemma (see [7] Assumg A R sasfes e followg codos: JIC emal for corbuo: edor@c.org.uk

3 Joural of Iformao ad Compug Scece, Vol. 7 (0 o., pp = ( a 0(, a = a, =,,... ; =, ( A s rreducble, e, oe as ( Real pars of e egevalues of A are all egave excep a egevalue 0 w e mulplcy. ( A as e rg egevecor (,, L, correspodg o e egevalue 0. ( Le ξ = ( ξ, ξ, L, ξ be e lef egevecor of A correspodg o e egevalue 0 sasfyg ξ =, e, we ca le ξ > 0 old for all =,,.... Lemma (see [6] If G = ( g R s a rreducble marx ad sasfes g = g 0(, ad =, g = g,, =,,..., e, all egevalues of e marx G = G dag( k, L, k l,0, L,0 are egave, were k > 0, =,, L, l are posve cosas. Lemma 3 (see[8] If e marx A= ( a sasfes a = a 0( ad a = =,, =,,.... e for ay wo vecors x = ( x, x,..., x ad y = ( y, y,..., y, we ave x Ay = a ( x x ( y y. > Lemma 4. Le Q ad R be wo symmerc marces, ad marx S as suable dmeso. e f ad oly f bo R < 0 ad 3. Sycrozao aalyss Q S 0 < S R Q SR S < 0. I s seco, we wll make e drve-respose coupled complex dyamcal eworks aceve e sycrozao, a s, we realze e er sycrozao sde eac ework ad ouer sycrozao bewee em smulaeously. We assume e couplg marx A s rreducble, ad le ξ = ( ξ, ξ, L, ξ be e lef egevecor of A correspodg o e egevalue 0 w ξ > 0. e we defe U =Ξ ξξ ad Ξ= dag( ξ, ξ,..., ξ. s Ξ A+ A Ξ We also le A = A dag( k, L, k l,0, L,0 ad ( Ξ A =. Sce e couplg marx A s rreducble, e ΞA s rreducble, a s, Ξ A+ A Ξ s also rreducble. From Lemma, we kow ( ΞA s s egave defe. For coveece of wrg, we deoe x% ( = ( x% (, x% (,, x% (, e% ( = ( e% (, e% (,, e% (. =,, L,. e drve ework ca be wre compac form as: X& = F( X + ( ca Γ X( + ( cb Γ X( τ (3 were X ( = ( x(, x(,..., x (, F( X = ( f( x, f( x,..., f( x, s e Kroecer produc of wo marces. Wou loss of e geeraly, assume a e frs l odes l are seleced ad ped w e lear feedback corollers, wc are descrbed by a, JIC emal for subscrpo: publsg@wau.org.uk

4 4 Sog Zeg: Ier-Ouer Sycrozao Aalyss of wo Complex eworks w Delayed ad o-delayed Couplg ck (,, Γ y x l u = 0, + l. were k ( l are ay posve cosas. Le e( = y ( x (, e e followg error dyamcal ework ca be obaed: e& ( = f( y( f( x( + caγ e( + cbγe( τ ck Γe(, l, = = (5 e& ( = f( y( f( x( + c aγ e( + c bγe( τ, + l. = = e we ave e followg resul: eorem. Suppose A olds. If ere exss a sem-posve defe marx M ad a posve defe marx Q ad cosas c, c, δ, γ, γ ( =,, L, suc a s cγ UB δu + ( cγξ A + M Z = 0 (6 cγ ( UB M ad s ( cγ ΞBQ B Ξ δξ+ ( cγ Ξ A + Q + < 0 (7 4 e, we ca aceve e sycrozao, a s, e er sycrozao sde eac ework ad ouer sycrozao bewee em smulaeously. Proof. Accordg o Ref. [9], we defe e sycrozao sae of e drvg ework as s( = ξ x (, e, we ave e er sycrozao error δ x ( = x ( s(. = We cosruc a Lyapuov fuco caddae as V = V+ V (8 were V = ξ( δx( Pδx( + p x ( θ Mx( θ dθ X ( U P X % % = τ = = + p x% ( θ Mx% ( θ dθ ad V τ = ξe( Pe( + p e( Qe( d. = % θ % θ θ τ = = By Assumpo ad Lemma 3, e dervave of V alog e raecores of (3 s τ = = V & = X ( U P X & + p x % ( Mx % ( p x % ( Mx % ( τ = X ( U P[ F( X + ( c A Γ X( + ( c B Γ X( τ ] px% ( Mx% ( px% ( τ Mx% ( τ = = ( [ ( ( ( ] ( [( ( c ] X ( c % % % x% = = + = X U P F x I Δ X + X U P I Δ + A Γ + X ( U P( B Γ X ( τ + px ( Mx ( px ( τ M ( τ = U ( x x P( f( x f( x Δ( x x + X ( U P[( I Δ + ( c A Γ] X( > (4 JIC emal for corbuo: edor@c.org.uk

5 Joural of Iformao ad Compug Scece, Vol. 7 (0 o., pp c % % % x% = = ( ( ( ( + X ( U P( B Γ X ( τ + px ( Mx ( px ( τ M ( τ αx U I X + X UI PΔ X + X c UA PΓ X % % % % = = U( x x ( x x X ( UI P X X ( cua P X ( > + X ( c UB PΓ X ( τ + px ( Mx ( px ( τ Mx( τ α + Δ + Γ % % % % = = + X ( c UB PΓ X ( τ + px ( Mx ( px ( τ Mx( τ α X ( U P X + p x (( U x ( + p x (( c ΞA x ( max( p % δ % % γ % = = px% c UB x% px% Mx% px% M% = = = + (( γ ( τ + ( ( ( τ x ( τ α ( x% = X ( U P X + p[ x (, x( ] Z max( p % % τ ( = x % τ From e codo (6 of eorem, we oba V & α X ( U P X max( p (9 Evaluag e me dervave of V alog e raecory of (5, oe obas = ξ ( [ ( ( ( ( Δ ( + ξ ( P[ Δ + Γ = = = V& e P f y f x e ] e e( c a e ( l ξe ( P e τ pe% ( Qe% = = = c k Γ e ( + c b Γ ( ] + ( pe % ( τ Qe % ( τ = s % % % % = = = α ξe ( e( + p e (( δ Ξ e ( + p e (( cγ ΞA e ( pe% c Be% pe% Qe% pe% Qe% = = = (( γ Ξ ( τ + ( ( ( τ ( τ ( e% = αξe( e( + p[ e% (, e% ( τ] Z = = e % ( τ s cγ ΞB δξ+ ( cγ Ξ A + Q were Z =. cγ B Ξ Q s ( cγ ΞBQ B Ξ From Lemma 4, we kow f δξ+ ( cγ Ξ A + Q+ < 0, e Z < 0. 4 us, we oba e, we ave V& αξe ( e( (0 = JIC emal for subscrpo: publsg@wau.org.uk

6 6 Sog Zeg: Ier-Ouer Sycrozao Aalyss of wo Complex eworks w Delayed ad o-delayed Couplg V& V& V& α X ( U P X ξe ( e( ( = + α max( p = erefore, e er sycrozao of e drve ework ( ad e respose ework (, respecvely, ad e ouer sycrozao of e wo eworks are aceved smulaeously. 4. umercal smulaos I s seco, o verfy ad demosrae e effecveess of e proposed meod, we cosder umercal examples, a s, e Lü caoc sysem as e ode dyamc sysem of e eworks. e Lü caoc sysem s descrbed by a( x x x& = bx + xx3, ( xx cx 3 we e parameers a= 36, b= 0, c= 3, e Lü sysem s caoc. I e umercal smulaos, for smplcy, we assume c = c =, τ = 3, Γ = dag(5,5,5, Γ = dag(0.,0.,0., l =, k = 5, k = 0, =,...,. Cosder a complex ework w omedelay ad me delayed couplg cossg of e Lü caoc sysem 6 odes o verfy e correcess of eorem. Coosg e couplg cofgurao marces A = ad B = ( e lef egevecor of A assocaed w egevalue 0 s ξ = (0.785,0.0847,0.476,0.547,0.06, 0.39, Ξ= dag(0.785, , 0.476, 0.547,0.06,0.39, U =Ξ ξξ. Usg e LMI oolbox MALAB, we oba M = ad Q = (4 e, e codos (6 ad (7 of eorem are sasfed. erefore, accordg o eorem, wo JIC emal for corbuo: edor@c.org.uk

7 Joural of Iformao ad Compug Scece, Vol. 7 (0 o., pp coupled eworks ca aceve e er ad ouer sycrozao, a s, e er sycrozao sde eac ework ad ouer sycrozao bewee em smulaeously. Ad e smulao resuls are sow Fgs. ad. Fg. (a-(c sows e sae varables of e drve-respose eworks. Fg. (d exbs e er sycrozao error of e drve ework ad a of e respose ework, respecvely. e ouer sycrozao errors are sow respecvely Fg.. e umercal resuls sow a e sceme for e drve-respose complex ework s effecve e eorem. Fg.. e sae of e varables (a x (red le ad y (blue do le (b x (red le ad y (blue do le (c x 3 (red le ad y 3 (blue do le; (d e er sycrozao error of e drve ework E ad a of e x respose ework E y, respecvely. Fg.. Ouer sycrozao errors of e drve- respose coupled eworks: (a e (b e (c e 3 (d 5. Cocluso E ouer I s paper, e pg corollers ave bee proposed o sudy e er sycrozao w eac JIC emal for subscrpo: publsg@wau.org.uk

8 8 Sog Zeg: Ier-Ouer Sycrozao Aalyss of wo Complex eworks w Delayed ad o-delayed Couplg ework ad ouer sycrozao bewee wo coupled complex eworks w delayed ad o-delayed couplg. W e Lyapuov sably eory ad adapve corol eory, some crera are derved. Aalycal resuls sow a wo eworks ca realze e sycrozao: e ouer sycrozao bewee e drve-respose eworks, ad e er sycrozao e drve ework ad e respose ewwork, respecvely. A umercal example as bee gve o sow e usefuless of e eorecal resul. 6. Ackowledgemes s work was suppored by e aoal aural Scece Foudao of Ca (o Refereces [] D. J. Was, S. H. Srogaz. Collecve dyamcs of small-world eworks. aure. 998, 393: [] A. L. Barbaas, R. Alber. Emergece of scalg radom eworks. Scece. 999, 86: [3] C.W. Wu. Sycrozao Complex eworks of olear Dyamcal Sysems. World Scefc, Sgapore, 007. [4] X. L, X. Wag, G. Ce. Pg a complex dyamcal eworks o s equlbrum. IEEE ras. Cr. Sys. 004, I 5: [5]. Ce, X. Lu, W. Lu. Pg complex eworks by a sgle coroller. IEEE ras. Cr. Sys. 007, I 54: [6] S. We, S. Ce, W. Guo. Adapve global sycrozao of a geeral complex dyamcal ework w odelayed ad delayed couplg. Pys. Le. 008, A 37: [7] J. Zou, L. Xag, Z. Lu. Sycrozao complex delayed dyamcal eworks w mpulsve effecs. Pys. 007, A 384: [8] P. L, J. Cao, Z. Wag. Robus mpulsve sycrozao of coupled delayed eural eworks w uceraes. Pys. 007, A 373: 6-7. [9] H. Jag, Q. B. Impulsve sycrozao of eworked olear dyamcal sysems. Pys. 00, A 374: [0] C. L, W. Su, J. Kurs. Sycrozao bewee wo coupled complex eworks. Pys Rev E. 007, 76: [] H. W. ag, L. Ce, J. Lu, C K. se. Adapve sycrozao bewee wo complex eworks w odecal opologcal srucures. Pys. 008, A 387: [] Y. L, Z. Lu, J. Zag. Sycrozao bewee Dffere eworks. C. Pys. Le. 008, 5: [3] C. L, W. Su, J. Kurs. Ouer sycrozao of coupled dscree-me eworks. Caos. 009, 9: [4] S. Zeg, G. Dog, Q. B. Impulsve sycrozao of complex eworks w o-delayed ad delayed couplg. Pys. Le. 009, A 373: [5] W. Su, Z. Ce, Y. B. Lü, S. H. Ce. A rgug ybrd sycrozao peomeo of wo coupled complex eworks. Appl. Ma. Compu. 00, 6: [6] W. Su, R. Wag, W. Wag, J. Cao. Aalyzg er ad ouer sycrozao bewee wo coupled dscreeme eworks w me delays. Cog eurody. 00, 4: 5-3. [7] W. L. Lu,. P. Ce. ew approac o sycrozao aalyss of learly coupled ordary dffereal sysems. Pysca. 006, D 3: [8] C. W. Wu, L O Cua. Sycrozao a array of learly coupled dyamcal sysems. IEEE ras Crc Sys. 995, I 4: [9] X. W. L,. P. Ce. Expoeal sycrozao of e learly coupled dyamcal eworks w delays. Cese Aals of Maemacs. 007, 8B: JIC emal for corbuo: edor@c.org.uk