Lag synchronization of hyperchaotic complex nonlinear systems via passive control

Transcription

1 Appl. Mah. Inf. Sci. 7, No. 4, (213) 1429 Applied Mahemaics & Infomaion Sciences An Inenaional Jounal hp://dx.doi.og/ /amis/7422 Lag synchonizaion of hypechaoic complex nonlinea sysems via passive conol Emad E. Mahmoud Depamen of Mahemaics, Faculy of Science, Sohag Univesiy, Sohag 82524, Egyp Received: 22 Dec. 212, Revised: 12 Feb. 212, Acceped: 4 Ma. 212 Published online: 1 Jul. 213 Absac: In his pape we sudy and invesigae he Lag synchonizaion (LS) of hypechaoic complex nonlinea sysems by using passive conol heoy. LS of hese hypechaoic complex sysems via passive conol has no eceived enough aenions, as fa as we know, in he lieaue. Based on he popey of he passive sysem he passive conolle is saed o achieve LS of wo idenical hypechaoic complex Loenz sysems. These complex sysems appea in many impoan fields of physics and engineeing. The analyical esuls of he conolles, which have been calculaed by using ou scheme, ae esed numeically and good ageemen is obained. Keywods: Lag synchonizaion, Passive conol, Minimum phase, Hypechaoic, Complex 1 Inoducion Thee ae seveal kinds of synchonizaion beween wo chaoic o hypechaoic sysems have been idenified as complee (o full) synchonizaion [1, 2], genealized synchonizaion [3], phase synchonizaion [4] and lag synchonizaion (LS) [5,6,7,8,9,1,11,12,13,14,15]. They epesen he diffeence in he degee of coelaion beween ineacing sysems [6]. Among hese synchonizaions, complee synchonizaion is he songes in he degee of coelaion and descibes he ineacion of wo idenical sysems, leading o hei ajecoies emaining idenical in he couse of empoal evoluion, i.e., x d ()=x (), whee x d () and x () ae he saes of he dive and esponse sysems, especively. Genealized synchonizaion, as inoduced fo dive-esponse sysems, is defined as he pesence of a funcional elaionship beween he saes of he esponse and dive sysems, i.e., x d () = f(x ()). Phase synchonizaion is he siuaion whee wo coupled hypechaoic sysems keep hei phases in sep wih each ohe while hei ampliudes emain uncoelaed. In he ypical synchonizaion egimes, lag synchonizaion has been poposed as he coincidence of he saes of chaoic sysems in which one of he sysems is delayed by a finie ime. Many expeimenal invesigaions and compue simulaions of chaos synchonizaion in unidiecional coupled exenal caviy semiconduco lases [16, 17, 18] have demonsaed he pesence of lag ime beween he dive and esponse lases inensiies. The simila expeimens fo chaoic cicuis [19] have also demonsaed he complee synchonizaion, i.e., he saes of wo chaoic sysems emain idenical in he couse of empoal evoluion, is pacically impossible fo he pesence of he signal ansmission ime and evoluion ime of esponse sysem iself. Thus, knowledge of he lag synchonizaion is of consideable pacical impoance. In LS he sae of he esponse sysem a ime is asympoically synchonous wih he dive sysem a ime τ, namely, lim x () x d ( τ) =, whee x d () and x () ae he saes of he dive and esponse eal sysems, especively. Complee synchonizaion is special case of LS when τ =. Recenly, some conol mehods, such as obseve-based scheme, [2] impulsive conol, [21] pojecive appoach, [22, 23] and adapive conol, [24, 25] have been applied o he lag synchonizaion fo chaoic and hypechaoic sysems. Passiviy is pa of a boade and geneal heoy of dissipaiviy, which can be found in Refs. [26, 27] and vaious ohes efeences heein. The main idea of passiviy heoy is ha he passive popeies of sysem can keep he sysem inenally sable. So, o make he Coesponding auho emad eluan@yahoo.com

2 143 Emad E. Mahmoud: Lag synchonizaion of hypechaoic complex... sysem sable, one can design a conolle which endes he closed loop sysem passive using passiviy heoy. Duing he las decade he passiviy heoy has played an impoan ole in designing an asympoically sabilizing conolle fo nonlinea sysems [28, 29, 3]. These efeences show ha he main feaue of he passive conol appoach lies in he fac ha he conolle design includes, a a fundamenal level, he sysem sucual popeies ha can be exploied o solve a given conol poblem. I is also shown ha he passive conol has many advanages: e.g. clea physical inepeaion, less conol effo equied o ease in implemenaion. Theefoe, passive conol has been widely invesigaed and applied [27, 28, 29, 3]. LS via passiviy is no deeced ye in he lieaue. So, he moivaion of his pape is o invesigae he LS of n-dimensional hypechaoic complex nonlinea sysems by using passive conol heoy and compae ou esuls wih hose in [31]. In [31] LS of n-dimensional hypechaoic complex nonlinea sysems is invesigaed by using nonlinea conol scheme. The oganizaion of his pape is as follows. Secion 2 pesens he descipion of n-dimensional hypechaoic complex nonlinea sysems. Basic concepions of passiviy heoy ae saed in Secion 3. In Secion 4 we achieve LS beween wo idenical hypechaoic complex Loenz sysems. The analyical foms of he conolles of his example ae deived based on passiviy heoy. In Secion 5, he numeical esuls ae illusaed o emphasize he validiy of he heoeical invesigaions. Finally, conclusions ae dawn in Secion 6. 2 Descipion of hypechaoic complex nonlinea sysems A complex dynamical sysem is called hypechaoic if i is deeminisic, has long-em apeiodic behavio, and exhibis sensiive dependence on he iniial condiions. A hypechaoic complex aaco is defined as a complex chaoic aaco wih a leas wo posiive Lyapunov exponens. The sum of Lyapunov exponens mus be negaive o ensue ha sysem is dissipaive. I is even moe complicaed han chaoic complex sysems and has moe unsable manifolds. Due o hypechaoic complex sysems wih chaaceisics of high capaciy, high secuiy and high efficiency, i has a boadly applied poenial in nonlinea cicuis, secue communicaions, lases, neual newoks, biological sysems and so on. Theefoe, eseach on hypechaoic complex nonlinea sysems is exemely impoan nowadays [31, 32, 33, 34, 35,36,37,38,39,4,41,42]. Conside he n dimensional hypechaoic complex nonlinea sysem as follows: ż= ψ(z, z), (1) whee he sae complex veco z=(z 1, z 2,..., z n ) T C n, z j = η j1 + iη j2 ( j = 1,2,...,n), i = 1, ψ = (ψ 1,ψ 2,...,ψ n ) T is a smooh nonlinea veco field, Do epesens deivaive wih espec o ime and an oveba denoes complex conjugae vaiables. In his pape we sudy he phenomenon of he LS of hypechaoic complex nonlinea sysems of he fom (1) via passive conol heoy which is no deeced ye in he lieaue. Remak 1: The n dimensional hypechaoic complex nonlinea sysem (1) can be ewien in he eal fom as: η = Ψ(η), (2) whee η =(η 11, η 12, η 21, η 22,..., η n1, η n2 ) T R 2n is a sae eal veco, ( j= 1,2,...,n), η j1, η j2 ae eal funcions and Ψ is a smooh nonlinea veco field. Remak 2: Mos of hypechaoic complex sysems can be descibed by (1), such as complex Loenz, Chen and Lü sysems [31, 32]. Fo example, he hypechaoic complex Loenz sysem [33,34] is: ż 1 = α(z 2 z 1 )+iz 4, ż 2 = γz 1 z 2 z 1 z 3 + iz 4, ż 3 = 1/2( z 1 z 2 + z 2 z 1 ) βz 3, ż 4 = 1/2( z 1 z 2 + z 2 z 1 ) σz 4, whee z = (z 1, z 2, z 3, z 4 ) T, ψ = (ψ 1, ψ 2, ψ 3, ψ 4 ) T = (α(z 2 z 1 ) + iz 4, γz 1 z 2 z 1 z 3 + +iz 4, 1/2( z 1 z 2 + z 2 z 1 ) βz 3, 1/2( z 1 z 2 + z 2 z 1 ) σz 4 ) T, α, γ and β ae posiive paamees, z 1 = η 11 + iη 12, z 2 = η 21 + iη 22 ae complex funcions and η jl ( j = 1,2, l = 1,2), z 3 = η 31, z 4 = η 41 ae eal funcions and σ is a conol paamee. This hypechaoic example has been inoduced ecenly in ou wok [33]. Fo he case α = 2, β = 5, γ = 4 and σ = 13 we calculaed he Lyapunov exponens as: ζ 1 = , ζ 2 =.243, ζ 3 =, ζ 4 = , ζ 5 = , ζ 6 = and is Lyapunov dimension is D = Theefoe sysem (3) has a hypechaoic behavio since ζ 1 and ζ 2 ae posiive, see Fig. 1, fo moe dynamical popeies, see Ref. [33]. 3 Basic concepions of passiviy heoy in hypechaoic complex nonlinea sysems We conside he hypechaoic complex nonlinea sysems wih he conolle in he geneal fom as follows: { ż= ψ(z, z)+φ(z, z)ξ, (4) y=h(z, z), whee Ξ = (v 1,v 2,...,v m ) T is he inpu (o conolle ), v s = v s1 + iv s2 (s=1,2,...,m), n>m, y=(y 1,y 2,...,y m ) T is he oupu, ψ and he m columns of φ ae smooh veco fields and h is a smooh mapping. The eal fom of sysem (4) can be wien as: { η = Ψ(η)+ Φ(η)v, (5) λ = H(η), (3)

3 Appl. Mah. Inf. Sci. 7, No. 4, (213) / a 3 2 b c b Fig. 1: Hypehaoic aacos of sysem: (a) In(η 1, η 3, η 6 ) space. (b) In(η 1, η 3, η 6 ) space. (c) In(η 2, η 3, η 5 ) space. (d) In(η 1, η 4, η 6 ) space. whee v = (v 11, v 12, v 21, v 22,..., v m1, v m2 ) T and λ = (λ 1, λ 2, λ 3, λ 4,..., λ 2m ) T. Fo evey iniial condiion η() and evey inpu v( ), hee is a maximally defined soluion η( ) of he sysem (5), and coesponding oupu λ( ) [26,27]. We suppose ha he veco field ψ has a leas one equilibium poin. Wihou loss of genealiy, we can assume ha he equilibium poin is η =. If he equilibium poin is no a η =, we can shif he equilibium poin o η = by coodinae ansfom [43]. Definiion 1 [43]: The sysem (5) is a minimum phase sysem if L Φ H() = H η Φ(η) is nonsingula and η = is one of asympoically sabilized equilibium poins of Ψ(η). Definiion 2 [44]: The sysem of he fom (5) is said o have elaive degee [1, 1,..., 1] a η = if he maix L Φ H() is nonsingula. Definiion 3 [45]: The sysem (5) is passive if he following wo condiions ae saisfied: (1) Ψ(η) and Φ(η) exis, Ψ and he 2m columns of Φ ae smooh veco fields and H(η) is a smooh mapping. (2), hee is a eal value ν saisfying he inequaliy: v T (ς)λ(ς)dς ν, (6)

4 1432 Emad E. Mahmoud: Lag synchonizaion of hypechaoic complex... o, hee exis a consan ε > and a eal-valued consan ν which saisfy he inequaliy: v T (ς)λ(ς)dς+ ν ε λ T (ς)λ(ς)dς. (7) Definiion 4 [46]: The sysem (5) is said o be passive if hee is a nonnegaive funcion V : η R, called soage funcion, which saisfies V() =, such ha: V(η) V(η()) λ T (ς)v(ς)dς. (8) The physical meaning of passive sysem is ha he enegy of he nonlinea sysem can be inceased only hough he supply fom an exenal souce. In ohe wods, a passive sysem canno soe moe enegy han ha supplied exenally. Remak 3: The paameic vesion of he nomal fom fo he sysem (5) which saisfying definiion 1 is: { µ = ϕ (µ)+ χ(µ, λ) λ, (9) λ = Ω(µ, λ)+ κ(µ, λ)v, whee a new coodinae of he sysem (5) is (µ, λ), locally defined in he neighbohood of he oigin, µ R 2n 2m and κ(µ, λ) is nonsingula fo all (µ, λ) in he neighbohood of he oigin. By designing a suiable conolle v, he sysem (9) may be passive. Thus, he equilibium poin of he sysem (9) can be asympoically sabilized by applying he nonlinea conolle v [47]. If he sysem (9) descibes an eo dynamical sysem wih ime lag, hen he synchonizaion beween he dive and esponse sysems is achieved. Remak 4: Seing λ = in sysem (9), yields he zeo dynamic sysem: µ = ϕ (µ). (1) The sysem (9) is called minimum phase, if is zeo dynamics is asympoically sable. Theoem 1 [47]: If he sysem (5) has a elaive degee [1, 1,..., 1] a η = and sysem (5) is a minimum phase sysem, he sysem (9) will be equivalen o a passive sysem and asympoically sabilized a an equilibium poin if we le he local feedback conol as follows: ( ) W(µ) T v = κ 1 (µ, λ)[ Ω(µ,λ) χ(µ, λ) (11) µ ε λ+ξ], whee he Lyapunov funcion of ϕ (µ) is W(µ), ε is a posiive eal value and ξ is an exenal signal veco ha is conneced wih he efeence inpu. 4 LS via passive conol Le us now invesigae he LS of sysem (3), which has been inoduced in [31], based on he passiviy heoy. The dive and he esponse sysems ae hus defined, especively, as follows: and ż d 1 = α(zd 2 zd 1 )+izd 4, ż d 2 = γzd 1 zd 2 zd 1 zd 3 + izd 4, ż d 3 = 1/2( z d 1 zd 2 + zd 1 zd 2) βz d 3, ż d 4 = 1/2( z d 1 zd 2 + zd 1 zd 2) σz d 4, ż 1 = α(z 2 z 1 )+iz 4, ż 2 = γz 1 z 2 z 1 z 3 + iz 4 + v 1, ż 3 = 1/2( z 1 z 2 + z 1 z 2 ) βz 3 + v 2, ż 4 = 1/2( z 1 z 2 + z 1 z 2 ) σz 4 + v 3, (12) (13) whee z d 1 = ηd 11 +iηd 12, zd 2 = ηd 21 +iηd 22, zd 3 = ηd 31, zd 4 = ηd 41, z 1 = η 11 + iη 12, z 2 = η 21 + iη 22, z 3 = η 31, z 4 = η 41, v 1 = v 11 + iv 12, v 2 = v 21, v 3 = v 31 ae complex and eal conol funcions, especively, which we need o deemine. In ode o obain he complex and eal conol signals v 1, v 2, v 3, he complex eo dynamical sysem akes he fom: whee 1 = α( 2 1 )+i 4, 2 = γ i 4 z 1 ()z 3 () +z d 1 ( τ)zd 3 ( τ)+v 1, 3 = β 3 + 1/2( z 1 ()z 2 ()+z 1 () z 2 () z d 1 ( τ)zd 2 ( τ) zd 1 ( τ) zd 2 ( τ))+v 2, 4 = σ 4 + 1/2( z 1 ()z 2 ()+z 1 () z 2 () z d 1 ( τ)zd 2 ( τ) zd 1 ( τ) zd 2 ( τ))+v 3, (14) 1 = z 1 () zd 1 ( τ), 2 = z 2 () zd 2 ( τ), 3 = z () z d ( τ), 4 = z 4 () zd 4 ( τ), (15) and 1 = 11 + i 12, 2 = 21 + i 22 and 3 = 31, 4 = 41 ae complex and eal eos funcions, especively. Sysem (14) in he eal fom: 11 = α( ), 12 = α( )+ 41, 21 = γ η31 d ( τ) 31 η11 ()+v 11, 22 = γ η31 d ( τ) 31 η12 ()+v 12, 31 = β η21 d ( τ)+ 12η22 d ( τ) + 21 η11 ()+ 22η12 ()+v 21, 41 = σ η21 d ( τ)+ 12η22 d ( τ) + 21 η11 ()+ 22η12 ()+v 31, (16) whee jl = η jl () ηd jl ( τ), j= 1, 2, 3, 4, l = 1, 2. Theoem 2: The eo dynamical sysem (16) is minimum phase sysem. Poof: Fis we can compue he maix L Φ H() as [46,47]: 1 1 L Φ H()= 1, (17) 1

5 Appl. Mah. Inf. Sci. 7, No. 4, (213) / since he maix L Φ H() is nonsingula and accoding o he definiion 1 sysem (16) has elaive degee [1, 1,..., 1] a =, we le µ= [ µ 1 µ 2 ] T = [ ] T, λ= [ λ 1 λ 2 λ 3 λ 4 ] T = [ ] T, hus sysem (16) has he fom: µ 1 = α(λ 1 µ 1 ), µ 2 = α(λ 2 µ 2 )+λ 4, λ 1 = γµ 1 λ 1 µ 1 η31 d ( τ) λ 3η11 ()+v 11, λ 2 = γµ 2 λ 2 + λ 4 µ 2 η31 d ( τ) λ 3 η12 ()+v 12, λ 3 = βλ 3 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ) +λ 1 η11 ()+λ 2η12 ()+v 21, λ 4 = σλ 4 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ) +λ 1 η11 ()+λ 2η12 ()+v 31. (18) Sysem (18) can be epesened in he following fom: { µ = ϕ (µ)+ χ(µ, λ) λ, (19) λ = Ω(µ, λ)+ κ(µ, λ)v, whee T [ ] α αµ1 α ϕ (µ) =, χ(µ, λ)= αµ 2, κ(µ, λ) = 1, 1 γµ 1 λ 1 µ 1 η31 d ( τ) λ 3η11 () Ω(µ, λ)= γµ 2 λ 2 + λ 4 µ 2 η31 d ( τ) λ 3η12 () βλ 3 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ)+ρ, σλ 4 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ)+ρ whee ρ = λ 1 η 11 ()+λ 2η 12 (). Choose a soage funcion as: V(µ, λ) = W(µ)+ 1 2 λ T λ, = W(µ)+ 1 ( λ λ2 2 + λ3 2 + λ4 2 ), (2) whee W(µ)= 1 ( 2 µ µ 2 2 ), W()=. The zeo dynamics of he sysem (19) descibes he inenal dynamics and occu when λ =, i.e. µ=ϕ (µ). (21) Diffeeniaing W(µ) espec o, we ge: Ẇ(µ) = W(µ) µ µ= W(µ) µ ϕ (µ), = [ µ 1 µ 2 ][ αµ1 αµ 2 ] T, (22) = (αµ αµ 2 2). Since W(µ) > and Ẇ(µ) <, i can be concluded ha W(µ) is he Lyapunov funcion of ϕ (µ) and ϕ (µ) is globally asympoically sable which means ha he eo sysem (16) is minimum phase sysem. So sysem (16) can be equivalen o a passive sysem, and heoem 2 is poved. Because ou sysem (16) is minimum phase sysem and has veco elaive degee [1,..., 1]. So sysem (16) is appopiae o apply heoem 1. We can deive he conolle as: v 11 = αµ 1 (γµ 1 λ 1 µ 1 η31 d ( τ) λ 3 η11 ()) ελ 1+ ξ 1, v 12 = αµ 2 (γµ 2 λ 2 + λ 4 µ 2 η31 d ( τ) λ 3 η12 ()) ελ 2+ ξ 2, v 21 = ( βλ 3 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ) +λ 1 η11 ()+λ 2η12 ()) ελ 3+ ξ 3, v 31 = µ 2 ( σλ 4 + µ 1 η21 d ( τ)+ µ 2η22 d ( τ) +λ 1 η11 ()+λ 2η12 ()) ελ 4+ ξ 4. (23) The conolle in he final fom is: v 11 = α 11 (γ η31 d ( τ) 31 η11 ()) ε 21+ ξ 1, v 12 = α 12 (γ η31 d ( τ) 31 η12 ()) ε 22+ ξ 2, v 21 = ( β η21 d ( τ)+ 12η22 d ( τ) + 21 η11 ()+ 22η12 ()) ε 31+ ξ 3, v 31 = 12 ( σ η21 d ( τ)+ 12η22 d ( τ) + 21 η11 ()+ 22η12 ()) ε 41+ ξ 4. (24) The analyical fomula of he conolle (24), which has been calculaed by using ou scheme, is used wih sysem (16) o achieve he LS of hypechaoic aacos of ou example. 5 Numeical esuls Numeical simulaions ae conduced in his secion o illusae he effeciveness of he designed conolle (24). We solve sysems (12) and (13) wih (24) numeically fo α = 2, β = 5, γ = 4 and σ = 13 fo which hypechaoic aaco exiss [33] and wih diffeen iniial condiions =, η11 d () = 1, ηd 12 () = 2, ηd 21 () = 3, η22 d () = 4, ηd 31 () = 5, ηd 41 () = 6 and η 11 () = 6, η12 () = 8, η 21 () = 3, η 22 () = 4, η31 () = 8,η 41 () = 1. We choose τ =.2, ε = 15 and ξ 1 = ξ 2 = ξ 3 = ξ 4 =. The vaiables saes duing he LS pocess beween sysems 12 and 13 ae shown in Figue 2. Fom i, one can see ha each η jl () convege o η d jl (), j = 1, 2, 3, 4, l = 1, 2 bu wih posiive ime lagged τ =.2. Figue 2 shows LS is achieved afe small ime ineval. The LS eos ae ploed in Figue 3, and as expeced fom he above analyical consideaions he LS eos jl convege o zeo as afe small value of. Compaing he numeical esuls in his eseach, emeging fom he Figues 2 and 3, wih hose in [31]. In [31] LS was achieved afe lage ime and his unlike ou

6 1434 Emad E. Mahmoud: Lag synchonizaion of hypechaoic complex a d b d c d 5 25 d d e f 12 8 d d Fig. 2: LS of sysems (12) and (13) wih (24): (a) η d 11 () and η 11 () vesus, (b) ηd 12 () and η 12 () vesus, (c) ηd 21 () and η 21 () vesus, (d) ηd 22 () and η 22 () vesus, (e) ηd 31 () and η 31 () vesus, (f) ηd 41 () and η 41 () vesus. esuls. We solve sysems (12) and (13) wih he same paamees and iniial condiions in [31]. Bu he numbe of conol funcions in ou pape is less han hose used in [31]. This shows he effeciveness of conolle (24) which has been calculaed by using heoem 1 in ou scheme. 6 Conclusion In engineeing applicaions, ime lag always exiss. Fo example in he elephone communicaion sysem, he voice one heas on he eceive side a ime is he voice fom he ansmie side a ime τ. So, sicly speaking, i is no easonable o equie he dive sysem o synchonize he esponse sysem a exacly he same ime. Theefoe, ecenly, much aenion has been given o he LS, in which he sae of he esponse sysem a ime is asympoically synchonous wih he dive sysem a ime τ. Unique o his pape is o sudy LS of hypechaoic complex nonlinea sysems by using passive conol heoy. LS of wo idenical hypechaoic complex Loenz sysems is achieved by applying he passiviy heoy on he eo dynamical sysem wih ime lag. We have shown ha he eo dynamical sysem is passive sysem and he

7 Appl. Mah. Inf. Sci. 7, No. 4, (213) / a.2 b d d c.2 d d d e.2 f d d Fig. 3: LS eos: (a) ( 11,) diagam, (b) ( 12,) diagam, (c) ( 21,) diagam, (d) ( 22,) diagam, (e) ( 31,) diagam, (f) ( 41,) diagam. conolle is deived by applying heoem 1 [47]. All he heoeical esuls ae veified by numeical simulaion of ou example. An excellen ageemen is found as shown in Figues 2 and 3. LS occus afe easonable value of as shown in Figue 2. Figue 3 displays he eo dynamical sysems. These eos appoach zeo afe small values of which shows he effeciveness of he conolle. Ou esuls in his pape ae bee han hose published in he [31]. Alhough he numbe of conol funcions in ou pape is less han hose used in [31]. Refeences [1] S. Boccalei, J.Kuhs, G. Osipov, D.L.Valladaes, C.S. Zhou, Physics Repos 366, 1-11 (22). [2] Z. Yan, P. Yu, Chaos, Solions and Facals 35, (28). [3] G. Alvaez, S. Li, F. Monoya, G. Paso, M. Romea Chaos, Solions and Facals 24, (25). [4] M. Juan, W. Xing-yuan, Physics Lees A 369, (27). [5] W. Guo, Nonlinea Analysis: Real Wold Applicaions 12, (211). [6] M. Zhan, G.W. Wei, C.H. Lai, Phys Rev E 65, (22). [7] M.G. Rosenblum, A.S. Pikovsky, J. Kuhs, Physics Review Lees 78, (1997).

8 1436 Emad E. Mahmoud: Lag synchonizaion of hypechaoic complex... [8] C.K. Ahn, Inenaional Jounal of Moden Physics B 25, (211). [9] J. Feng, A. Dai, C. Xu, J. Wang, Compues & Mahemaics wih Applicaions 61, (211). [1] C. Li, X. Liao, K. Wong, Physica D 194, (24). [11] Y. Sun, J. Cao, Physics Lees A 364, (27). [12] T.H. Lee, Ju H. Pak, Chin. Phys. Le. 26, 957 (29) [13] M. Dehghani, M. Haei, SICE-ICASE Inenaional Join Confeence, (26). [14] J. Chen, H. Liu, J. Lu, Q. Zhang, Communicaions in Nonlinea Science and Numeical Simulaion 16, (211). [15] D. Wang, Y. Zhong, S. Chen, Communicaions in Nonlinea Science and Numeical Simulaion 13, (28). [16] E.M. Shahvediev, S. Sivapakasam, K.A. Shoe, Physics Lees A 292, (22). [17] S. Taheion, Y.C. Lai, Physics Review 59, (1999). [18] J. Yu, J. Liu, X. Chen, J. Liu, Applied Mahemaics & Infomaion Sciences 5, (211). [19] X. Huang, J. Xu, W. huang, F. Zhu, Aca Phys. Sin. 5, (21). [2] C. Li, X. Liao, K. Wong, Chaos, Solions and Facals 23, (25). [21] C. Li, X. Liao, R. Zhang, Chaos, Solions & Facals 23, (25). [22] Q. Zhang, J. Lu, Physics Lees A 37, (28). [23] G. H. Li, Chaos, Solions & Facals 41, (29). [24] Y. Sun, J. Cao, Physics Lees A 364, (27). [25] J. Ren, Applied Mahemaics & Infomaion Sciences 6, (212). [26] J.C. Willems, Ach. Raion. Mech. Anal. 45, (1972). [27] C.I. Bynes, A. Isidoi, J.C. Willems, IEEE Tans. Auom. Conol 36, (1991). [28] A.L. Fdakov, D.J. Hill, Auomaica 34, (1998). [29] Y. Wen, IEEE Tans. Cicuis Sys. I 46, (1999). [3] A.Y. Pogomsky, In. J. Bifu. Chaos 8, (1998). [31] G.M. Mahmoud, E.E. Mahmoud, Nonlinea Dyn. 67, (212). [32] G.M. Mahmoud, M.E. Ahmed, E.E. Mahmoud, In. J. Moden Physics C 19, (28). [33] E.E. Mahmoud, J Fank. Ins. 349, (212). [34] E.E. Mahmoud, G.M. Mahmoud, Chaoic and Hypechaoic Nonlinea Sysems, Lambe Academic Publishing, Gemany, ISBN , 211. [35] G.M. Mahmoud, T. Bounis, E.E. Mahmoud, In. J. Bifuca. Chaos 17, (27). [36] G.M. Mahmoud, E.E. Mahmoud, Nonlinea Dyn. 61, (21). [37] G.M. Mahmoud, E.E. Mahmoud, Mahemaics and Compues in Simulaion 8, (21). [38] G.M. Mahmoud, E.E. Mahmoud, M.E. Ahmed, Nonlinea Dyn. 58, (29). [39] G.M. Mahmoud, E.E. Mahmoud, M.E. Ahmed, In. J. Appl. Mah. Sa. 12, 9-1 (27). [4] G.M. Mahmoud, E.E. Mahmoud, In. J. of Bifucaion and Chaos 21, (211). [41] E.E. Mahmoud, Mahemaical and Compue Modelling 55, (212). [42] X. Chen, C. Liu, Nonlinea Analysis: Real Wold Applicaions 11, (21). [43] F. Wang, C. Liu, Physica D, 225, 55-6 (27). [44] D.Q. Wei, X.S. Luo, B. Zhang, Y.H. Qin, Nonlinea Analysis: Real Wold Applicaions 11, (21). [45] K. Kemih, Chaos, Solions Facals 41, (29). [46] W. Xiang-Jun, L. Jing-Sen, C. Guan-Rong, Nonlinea Dyn. 53, (28). [47] G.M. Mahmoud, E.E. Mahmoud, A.A Aafa, J. Vib. Conol: doi:1.1177/ D. Emad E. Mahmoud was bon in sohag, Egyp, in He eceived his B. Sc. Degee in mahemaics fom souh valley univesiy 24 wih vey good wih honos degee, and his M. Sc fom Sohag Univesiy, Sohag, egyp 27. He eceived his Ph.D. in dynamical sysems, Sohag Univesiy, Sohag, egyp 21. He has published ove 18 papes in inenaional efeeed jounals and wo books in Lambe Academic Publishing. The aeas of inees ae synchonizaion and conol of nonlinea dynamical sysems (complex sysems) and nonlinea diffeenail equaions.