Synchronization of Complex Network System with Time-Varying Delay Via Periodically Intermittent Control

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1 Sychrozao of Complex ework Sysem wh me-varyg Delay Va Perodcally Ierme Corol JIAG Ya Deparme of Elecrcal ad Iformao Egeerg Hua Elecrcal College of echology Xaga 4, Cha Absrac he sychrozao corol problem of some geeral complex dyamcal eworks wh me me-varyg delay s vesgaed A perodcally erme corol s desged for he sysem sychrozao By Lyapuov sably heory wh he Zero heorem, obaed suffce codo ha sychrozao expoeally sable coroller exs bewee he complex ework sysems whch he characerscs are varable delay Fally, he feasbly of he proposed mehod s demosraed by umercal examples Keywords - complex ework, erme corol, me-varyg delay I IRODUCIO Complex eworks wdely exs varous pheomea of he aure ad huma socey I he rece e years, he heory ad applcao of complex eworks has become a ho opc a home ad abroad for he breakhrough of he research mehods of complex eworks Oe of he mos eresg ad sgfca pheomea complex dyamcal eworks s sychrozao amog all ework s dyamcal odes I fac, sychrozao s oe of he mos ypcal collecve behavor ad basc moos aure [-] I order o realze he sychrozao, adapve rules [3-6], mpulsve mehods [7- ], or a combao of hem [-3] have bee used chaoc sysems I rece years, Ierme corol has bee cocered by people for s covee applcao, savg eergy, easy realzao ad so o egeerg pracce [4-6] Ierme corol was used as a effecve corol mehod o corol a class of lear ecoomc sysems frsly, ad he was wdely used maufacurg, raspor, secure commucaos ad chaoc sychrozao ad oher felds I paper [4], Ca suded he problem of sychrozao for a class of complex delayed dyamcal eworks va pg perodcally erme corol I paper [5], Pa cosder he sochasc quas-sychrozao for he delayed eworks wh parameer msmaches ad sochasc perurbao msmach by usg erme corol We kow ha he me-delay exss formao rasformao of complex ework sysems, hs me-delay have wo possbles: couplg delay ad ode delay, secodly, he delay s ofe me-varyg praccal problems, so s of grea mporace for he sudy of dual me-varyg delay complex ework sysems hs paper begs wh he complex eworks sysem wh me-varyg couplg delays ad me-varyg odes delays, he sudy o he sychrozao of complex eworks s very meagful Based o he above descrpo, he sychrozao corol problem of some geeral complex dyamcal eworks wh me me-varyg delay s vesgaed I order o make he sysem sychrozed, A perodcally erme corol s desged for he sysem sychrozao By Lyapuov sably heory wh he Zero heorem, obaed suffce codo ha sychrozao expoeally sable coroller exs bewee he complex ework sysems whch he characerscs are varable delay Fally, he feasbly of he proposed mehod s demosraed by umercal examples II PROBLEM DESCRIPIO Wh each ode as he sae varable, cosderg he followg me-delay complex dyamc eworks: x& f(, x, x ( ())) G x ( ()),,,, () j j x ( x, x,, x ) R are sae varables Where, of he h ode; he me delay (), () may be ukow bu s bouded by a kow cosa, e,, G ( Gj ) R s a cosa couplg marx of he ework bewee odes, whch Gj s defe as follow: f here s a coeco bewee he ode ad he ode j( j ), he G j G j ; oherwse, Gj G j ( j ), ad he dagoal elemes of marx G are defe by G Gj G j,,,,, j, j Hypohess For he vecor-valued fuco f (, x (), x ( ())), suppose he uform sem-lpschz codo wh respec o he me holds, e, for ay DOI 53/IJSSSa6A8 8 ISS: x ole, pr

2 x(), y() R, here exs posve cosas L, L, such hasem-lpschza: [ x ( ) y ( )] [ f( x, ( ), x ( )) f(, y ( ), y ( ))] L[ x y] [ x y] L [ x( ) y( )] [ x( ) y( )] () Lemma For ay m -dmesoal real varable vecor X, Y, ad K s posve defe symmerc m m marces, ad P R, here exss he followg marx equaly: X PY X PK P X Y KY (3) Defo A complex dyamcal ework s sad o acheve he asympocal er sychrozao, f x x x s( ), (4) Where s R s a soluo of arge ode sasfyg s f ( s( )), for our sychrozao scheme, le us defe he error vecor as follows: e x s( ) (5) he am of hs paper s make he ode x, (,,, ) acheve sychrozao uder he corol aco ha meas he error sae for odes sasfed: lm e,,,, (6) III MAI RESULS I hs par, for sysem (), desged Ierme corol o make he sysem sychrozed ad sable x& f(, x, x( )) Gjxj( ) k( x s),,,, (7) Where k () s he erme feedback corol ga defed as follows: k m, m k () (8) m,( m) s he corol perod, s called he corol wdh, m,, L Accordg o Eq(5) ad he sysem (), /, he error dyamcs s gve o ˆ ( ) () (,, ) j () j( ()) () () j j e e f e e G e e e u L e () e () L e ( ()) e ( ()) G e () e ( ()) ke () e() ˆ(,, e & f e e ) Gjej ( ()) ke, m,( m ),,,, (9) e fˆ(, e, & e ) Gjej ( ( )), ( m ),( m),,,, Where ˆ(,, f e ) (,, ( ( ))) (, ( ), ( ( ))) e f x x f s s, Whe he error sysem(9) s asympocally sable, he sysem() acheve sychrozao heorem Suppose ha Hypohess holds s a posve cosa sasfg he codos, If here exs posve cosas a, a, ad dagoal marx K mee he followg codos ( L a) I K () L ( a a) () a L () a( ) (3) Where max (( GI) ( G I)), K dag( k, k,, k ), s a uque posve soluo of he equaly a( L )exp( ),he he corolled dyamcal ework () s global sychrozao expoeally Proof: le: e ( ) ( e,, e ),ad cosruc a Lyapuov fuco: V( e) e ()() e e () e() (4) Usg Hypohess, codos () ad (), he dervave of V () wh respec o me alog he soluos of Eq (9) ca be he calculaed as follows, whe m,( m ), for m,, L le K dag( k, k,, k ) (e) Le ()() e Le ( ())( e()) e ()( GI)( e()) KIe ()() e Accordg o Eq(3),we ca ge e ( GI)e( ()) e e e ( ())( GI) ( GI) e( ()) DOI 53/IJSSSa6A8 8 ISS: x ole, pr

3 le max (( GI ) ( G I )) (e ) Le e Le ( ())e( ()) e( e ) ( ) e( ()) e ( ()) KIe ( e ) ( ) e (( L a) I K) Ie Le ( ()) e( ()) e ( ()) e( ()) ae e av () LV ( ()) V ( ()) whe ( m ),( m ), for m,, L hs coradcs he secod equaly (7), ad so (6) holds ow, we prove ha for [, ) ˆ ( e) e f(, e, e ) Gje ej( ()) R () W () hm exp{ a}, h (9) e ( L ( a a)) Ie Le ( ()) e( ()) Oherwse, here exss a [, ), such ha e ( ()) e( ()) ( a a) e e R ( ), R &( ) () ( a av ) () LV ( ()) V ( ()) R (), () amely, we have For,, usg Eqs(5)(6)(),we (e ) av () LV ( ()) V ( ()) m,( m) oba Ve &( ) ( a av ) () LV ( ()) V ( ()) ( m),( m) W ( ) hmexp{ a( )} I he followg, we wll prove ha codos () ad W ( ) hmexp{ a( )} (3) mples Usg Eqs(8), ()ad (), we oba V () sup s V(s)exp, R &( ) W exp ahm exp{ a} W ( a a)exp V Lexp V( ) Deoe g( ) a( L )exp( ), exp V( ) ahmexp{ a} Sce a L, we have g(), ( ( a a)) W Lexp( ) W( ) g( ),ad g& ( ),Usg he couy ad he exp( ) W ( ()) ahmexp{ a} mooocy of g( ), he equaly ( a( L )exp( )) hmexp{ a} a( L )exp( ) has a uque posve hs coradcs he secod equaly (), ad so (7) holds Cosequely, for [, ) soluo, ake M sup sv (s) ad W ( ) hmexp{ a} hmexp{ a( ) } W() exp V,, Le Q () W () hm, where O he oher had, follows from Eqs(5), (6)ha for h s a cosa I s easy o see ha Q (), (,) (5) ex, we wll prove ha Q (), (, ) (6) Oherwse, here exss a [, ), such ha Q ( ), Q &( ) (7) Q (), (8) Usg Eqs (5), (7) ad (8), we oba Q& ( ) W( ) exp( ) ( ) W( ) aexp( ) V( ) Lexp( ) V ( ) exp( ) V ( ) ( a) W( ) Lexp( ) W( ) exp( ) W( ) ( a( L )exp( )) hm [, ) W () hm hmexp{ a( ) } so W ( ) hmexp{ a( ) }, [, ) Smlarly, we ca prove ha for [,( ) ) W ( ) hmexp{ a( ) } ad for [( ), ) W ( ) hmexp{ a( ) } By duco, for [ m,( m ) ) W ( ) hmexp{ am ( ) } hmexp{ a( ) } () For [( m ),( m ) ) W ( ) hm exp{ a( ( m) ) } hm exp{ a( ) } (3) DOI 53/IJSSSa6A8 83 ISS: x ole, pr

4 Le h, from he defo of W (), we oba V Mexp{ [ a( )] } Mexp{ } (4) hs mples he cocluso ad he proof s complee IV UMERICAL EXAMPLE I hs seco, a umercal example s gve o show he effecveess of he proposed ework sychrozao crera For smplcy, we cosder a e-ode ework, whch each ode s a smple 3-dmesoal sable lear sysem x& f(, x, x( )) Gjxj( ),,,, Where he me delay s bouded s, he coupled cofgurao marx G ( Gj ) s mee he codos: G G G,,,, j j, j, j For he vecor-valued fuco; x& f(, x, x( )) x() ( x()) ( x( ())) 3 Where, x (,, 3, ) x x x R, 3 ( x) ( ( )( x x ),,) R, 3 ( x ( )) (,, s( x ( ))) R ( ) Le 3, 9, 5, 57, 95, 5,,, I s easy o verfy ha ( xj sj )( f(, xj, xj ( )) f(,s j, sj ( ))) max ( % )( xj sj ) ( xj sj ) / ( )( xj ( ) sj ( )) ( xj ( ) sj ( )) L ( xj sj ) ( xj sj ) L ( xj ( ()) s ( ())) j ( xj( ()) sj( ())) Where % ( dag( ( ),, )), L ( % ) max, L /( ),ca be deermed by choosg approprae parameer,we ca ge L, L Choose from he heorem, we ca ge he corol pu parameers k ad he corol wdh, f k <-5 ad 8, he sysem ca ge sable he sae error of ode s gve by Fgure 5 5 e() Fgure he sae error of odes From he Fg, we kow he sae of he sysem s sable he characerscs are varable delay Fally, he feasbly of he proposed mehod s demosraed by umercal V COCLUSIOS examples I hs paper he sychrozao corol problem of REFERECES some geeral complex dyamcal eworks wh me mevaryg delay s vesgaed A perodcally erme [] DJWas, SHSrogaz Collecve dyamcs of smallworld eworks aure, vol 393, o 6684, pp44 44, 998 corol s desged for he sysem sychrozao By Lyapuov sably heory wh he Zero heorem, obaed [] L M Pecora, L Carroll Sychrozao chaoc sysems PhysRevLe, vol 64, o 8, pp 8-84, 99 suffce codo ha sychrozao expoeally sable [3] LWag, HPDa, HDog, YHShe, YXSu Adapve coroller exs bewee he complex ework sysems whch sychrozao of weghed complex dyamcal eworks wh DOI 53/IJSSSa6A8 84 ISS: x ole, pr

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