Adaptive complex modified hybrid function projective synchronization of different dimensional complex chaos with uncertain complex parameters

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1 Nonlnea Dyn (26) 83:9 2 DOI.7/s ORIGINAL PAPER Adapve complex modfed hybd funcon pojecve synchonzaon of dffeen dmensonal complex chaos wh uncean complex paamees Jan Lu Shuang Lu Julen Clnon Spo Receved: 4 Febuay 2 / Acceped: Sepembe 2 / Publshed onlne: Sepembe 2 Spnge Scence+Busness Meda Dodech 2 Absac Ths pape focuses on he adapve modfed hybd funcon pojecve synchonzaon wh complex funcon ansfomaon max (CMHFPS) fo dffeen dmensonal chaoc (hypechaoc) sysems wh complex vaables and unknown complex paamees. The chaoc sysems ae consdeably dffeen fom hose n he exsng closely elaed leaue. Moeove, he ansfomaon max n hs ype of chaos synchonzaon s no a squae max, and s elemens ae complex funcons. In pacula, by consucng appopae Lyapunov funcons dependen on complex vaables, he adapve conolles ae desgned o synchonze dffeen dmensonal complex chaos (hypechaos) wh complex paamees n he sense of CMHFPS, and he complex updae laws fo esmang unknown complex paamees of complex chaoc sysems ae also gven. Fnally, wo examples ae pe- J. Lu (B) School of Mahemacal Scences, Unvesy of J nan, Jnan 222, Shandong, People s Rupublc of Chna e-mal: lujan99@63.com S. Lu School of Conol Scence and Engneeng, Shandong Unvesy, Jnan 26, Shandong, People s Rupublc of Chna e-mal: slu@sdu.edu.cn J. C. Spo Depamen of Physcs, Unvesy of Wsconsn-Madson, Madson, WI 376, USA e-mal: cspo@wsc.edu sened o llusae he effecveness and febly of he heoecal esuls. Keywods Complex chaos (hypechaos) Complex paamee Complex funcon ansfomaon max Dffeen dmensons Modfed hybd funcon pojecve synchonzaon Inoducon In 982, Fowle e al. [] deved ognally he Loenz equaons wh complex vaables and complex paamees o descbe a wo-laye model of he baoclnc nsably follows: ẋ = σ(y x), ẏ = x xz ay, () ż = 2 ( xy+ x ȳ) bz, whee he Raylegh numbe and paamee a ae complex numbes defned by = + j 2, a = jδ, and σ, b,, 2,δae eal and posve. The complex vaables x, y and eal vaable z of Eq. () ae elaed, especvely, o elecc feld and he aomc polazaon ampludes and he populaon nveson n a ng le sysem of wo-level aoms [2,3], an oveba denoes complex conjugae vaable and a do epesens devave wh espec o me, chaoc moon of Eq. () s shown n Fg.. Afewad, some eseach woks n complex felds have been acheved on dynamcs and conol of

2 J. Lu e al. Fg. Phe po of he aaco fo complex chaoc Loenz sysem () wh complex paamees σ = 2, = 6+.2 j, a =.6 j, b =.8 and nal values x() = j, y() = +.2 j, z() = z z x (a) on (x, z) plane y (b) on (y, z) plane chaos [4]. In 27, Mahmoud e al. [] suded bc popees and chaoc synchonzaon of he complex Loenz model follows: ẋ = σ(y x), ẏ = x xz y, (2) ż = 2 ( xy+ x ȳ) bz, whee σ >, >, b > ae eal paamees. The Loenz model (2) s embedded n () and can be ecoveed when 2 = δ =. In ecen ye, seveal ohe such examples have been poposed, noably he so-called complex Chen, Lü sysems [6] and complex hypechaoc Loenz, Lü sysem [7,8] and so on. Acually, many sysems whch nvolve complex vaables have played an mpoan ole n many ae, ncludng loadng of beams and plaes [9], opcal sysems [], plma physcs [], oo dynamcs [2] and hghenegy acceleaos [3] and secue communcaons [4]. Due o s mpoance and boad applcaons, he synchonzaon of complex chaoc sysems h aaced gea aenon n he l few decades well, such global synchonzaon (GS) [6,], complee synchonzaon (CS) [6,7], an-synchonzaon (AS) [8], lag synchonzaon (LS) [9,2], pojecve synchonzaon (PS) and modfed pojecve synchonzaon (MPS) [2], modfed funcon pojecve synchonzaon (MFPS) [22]. I s clea ha he scalng max s always chosen eal max o eal-valued funcon max n he above synchonzaon. In ode o ensue ha he ansmed sgnals have songe an-aack ably and ananslaed capably han ha ansmed by he usual ansmsson model, scalng max h been exended o he complex doman o ake no accoun of pojec synchonzaon vey ecenly. Zhang e al. [23] dscussed MPS wh complex scalng facos (CMPS) of uncean eal chaos and complex chaos. Mahmoud e al. [24] acheved CMPS of wo cean chaoc complex sysems. Sun e al. [2] noduced combnaon synchonzaon wh complex scalng max. Lu and Zhang [26] dscussed FPS wh complex funcon max (CFPS) of coupled chaoc complex sysem wh known eal paamees and s applcaons n secue communcaon. I should be noed ha he afoemenoned papes only consde chaoc synchonzaon of he same dmensonal complex-vaable sysems, and he saes of he dve and esponse sysems synchonze by a dagonal max, so each sae vaable of esponse sysem synchonzes one of dve sysem by a specal scalng faco. As a mae of fac, he synchonzaon can be caed ou hough dffeen dmensonal oscllaos, especally he sysems n communcaon [27], bologcal scence and socal scence [28], whee he dve and esponse sysems could synchonze by a desed ansfomaon max, no a squae max. By means of sae ansfomaon, mulple sae vaables n dve sysem wll be nvolved fo a coespondng sae vaable of esponse sysem by especve scalng facos (o funcons). I s obvous ha ansfomaon max s abay and moe unpedcable han dagonal scalng max. Theefoe, Luo and Wang [29] noduced hybd modfed funcon pojecve syn-

3 Adapve complex modfed hybd funcon pojecve chonzaon (MHFPS) of wo dffeen dmensonal complex chaoc sysems. Moeove, he complex funcon ansfomaon max s moe unpedcable han eal funcon ansfomaon max n [29], wll gealy ncee he complexy and dvesy of he synchonzaon. As a genealzaon of synchonzaon, dependng on he fom of he ansfomaon max, MHFPS wh complex funcon ansfomaon max (CMHFPS) conans MHFPS, MFPS, CFPS. Theefoe, s meanngful and valuable o sudy CMHFPS fo complex chaos. To he bes of ou knowledge, he CMHFPS of dffeen dmensonal complex sysems h aely been exploed. Fuhemoe, he above dscussed chaoc sysems have mosly been lmed o complex chaoc sysems wh eal paamees such sysem (2). Thee ae small numbe of papes dscussed chaos synchonzaon wh complex paamees. Fo nsance, Mahmoud e al. [] dscussed GS of complex nonlnea equaons fo deuned les wh cean complex paamees by sepaang eal and magnay p of complex vaables. Recenly, Lu e al. [3] dscussed modfed hybd pojec synchonzaon wh complex ansfomaon max (CMHPS) fo dffeen dmensonal hypechaoc and chaoc complex sysems wh complex paamees whou sepaang eal and magnay p of complex paamees. Howeve, fa he auhos know, CMHFPS of dffeen dmensonal chaoc complex sysems wh complex paamees s seldom epoed n he leaues. In addon, n eal physcal sysems, paamees of sysems ae pobably unknown o may change fom me o me. In hs ce, s well known ha he adapve conol s an effecve mehod o ealze he synchonzaon of chaoc sysems wh unknown paamees. I s woh nong ha adapve synchonzaon fo eal sysems o complex ones wh eal paamees have been developed [8]. Howeve, adapve synchonzaon of complex chaoc sysems wh uncean complex paamees s moe less. Only n [3], Lu e al. noduced adapve complex modfed pojecve synchonzaon (CMPS) of wo same dmensonal complex chaoc (hypechaoc) sysems wh uncean complex paamees. How o acheve CMHFPS, whch coves CMPS and CMHPS, beween wo dffeen dmensonal complex chaoc sysems wh uncean paamees va he adapve conol? Insped by he above dscusson, n hs pape, we focus on adapve CMHFPS of dffeen dmensonal chaoc sysems wh complex vaables and unknown complex paamees. In moe deal, he dsngushng feaue of hs pape ae efned follows. Fs, he sysems unde nvesgaon ae emakably moe geneal han hose n he closely elaed leaue [ 8,4,6,8,9,2 29]. Ths can be seen fom a compon beween sysems () and (2). A chaoc sysem wh complex vaables and unknown complex paamees poduces moe complex and unpedcable sgnals. I s well known ha he adopon of chaoc sysems wh complex vaables h been noduced fo secue communcaon, whee doublng he numbe of vaables by noducng complex sysem o enhance he conens and secuy of he ansmed nfomaon. Wha s moe, he sysem paamees ae unknown complex numbes, and have moe choce han eal paamees, whch make he sysems poduce moe unpedcable sgnals. Second, dffeen fom he eal-valued funcon max n pevous synchonzaon [22,29], he ansfomaon max s composed of complex-valued funcons n CMHFPS of dffeen dmensonal complex chaos. As a genealzaon of synchonzaon, dependng on he fom of he ansfomaon max, CMHFPS conans CFPS, CMHPS, CMPS, MHFPS, MFPS, and exend ecenly pevous woks [6,8,2 24,26,29 3]. Thd, unlke he schemes poposed n he leaue [ 8,4 26,29], we consuc appopae Lyapunov funcons dependen on complex vaables and do no sepaae eal and magnay p of he uncean complex paamees o complex vaables. The adapve conolle s desgned o synchonze dffeen dmensonal complex chaos wh complex paamees n he sense of CMHFPS, and he complex updae laws fo esmang unknown complex paamees ae also gven whou sepaang eal and magnay p of he complex paamees. The emande of hs pape s sucued follows. The defnon of CMHFPS s noduced fo dffeen dmensonal complex chaoc sysems wh uncean complex paamees n Sec. 2, followed by, he geneal schemes of adapve CMHFPS ae desgned n Sec. 3. Secon 4 s devoed o smulaon. The adapve CMHFPS beween complex chaoc Loenz dve sysem wh uncean complex paamees and complex hypechaoc Loenz esponse sysem wh uncean eal paamees well complex hypechaoc

4 2 J. Lu e al. Table Types of funcon synchonzaon Sengs he max D() D() = D () + jd () C m n, non-squae D() R m n, non-squae D() = D () + jd () C n n, non-dagonal D() R n n, non-dagonal D() = dag{d (), d 2 (),...,d n ()} R n n D() = dag{d(), d(),...,d()} C n n D() = dag{d(), d(),...,d()} R n n Type of synchonzaon CMHFPS MHFPS CMGFPS MGFPS MFPS CFPS FPS Lü dve sysem wh uncean eal paamees and complex Loenz esponse sysem wh uncean complex paamees ae aken wo examples o demonsae he effecveness and febly of he poposed scheme. Fnally, Sec. daws some conclusons. Noaon C n sands fo n dmensonal complex veco space. If z C n s a complex veco, hen z = z + jz, j = s he magnay un, supescps and sand fo he eal and magnay p of z, especvely, z H, z T ae he conjugae anspose and anspose of z, especvely and z mples he 2-nom of z.ifz s a complex scala, z ndcaes he modulus of z and z s he conjugae of z, whle M H () s he conjugae anspose of M(), povded ha M() s a complex max. 2 The defnon of CMHFPS of complex chaos (hypechaos) wh complex paamees and poblem descpons Fs, we consde he followng geneal m-dmensonal complex chaoc (hypechaoc) dve sysem ż() = H(z,), (3) and n-dmensonal complex chaoc (hypechaoc) esponse sysem ẇ() = R(w,) + v(z, w, ), (4) whee z = (z, z 2,..., z m ) T C m, w = (w, w 2,..., w n ) T C n ae complex sae veco, and v = v + jv C n s he conol npu. Nex, we noduce he defnon of CMHFPS wh complex funcon ansfomaon max of complex chaoc sysems wh complex paamees follows. Defnon Fo he dve sysem (3) and esponse sysem (4), s sad o be CMHFPS wh D() beween w() and z(), f hee exss a conolle v(z, w, ) such ha lm w() D()z() =, whee D() = + D () + jd (), he elemens of D() should be connuously dffeenal funcons wh bounded. Remak The n m max D() s called a complex funcon ansfomaon max. Seveal knds of funcon synchonzaon ae specal ces of CMHFPS, such complex modfed genealzed funcon pojecve synchonzaon (CMGFPS), complex funcon pojecve synchonzaon (CFPS), modfed hybd funcon pojecve synchonzaon (MHFPS), modfed genealzed funcon pojecve synchonzaon (MGFPS), modfed funcon pojecve synchonzaon (MFPS), funcon pojecve synchonzaon (FPS), see Table. Remak 2 If D() s a complex consan max, he poblem becomes CMHPS, CMPS and CPS fo complex dynamcal sysems [23,24,3,3]. The geneal m-dmensonal complex dve chaoc sysem wh unknown complex paamees s consdeed ż() = H(z,) = F(z)A + f(z), () whee z = z + jz C m and z = (z, z 2,...,z m )T, z = (z, z 2,...,z m )T. A = (a, a 2,..., a s ) T C s s a s complex veco of unknown paamees, F(z) s a m s complex max and s elemens ae funcons of complex sae vaables, and f = ( f, f 2,..., f m ) T s a m veco of complex nonlnea funcon. On he ohe hand, he n- dmensonal complex esponse chaoc sysem wh he conolle s depced ẇ() = R(w,) + v = G(w)B + g(w) + v, (6) whee w = w + jw C n and w = (w,w 2,..., wn )T, w = (w,w 2,...,w n )T. B = (b, b 2,...,

5 Adapve complex modfed hybd funcon pojecve 3 b q ) T C q s a q complex veco of unknown paamees, G(w) s a n q complex max and s elemens ae funcons of complex sae vaables, and g = (g, g 2,...,g n ) T s a n veco of complex nonlnea funcon. The conolle v = v + jv C n needs o be desgned. Remak 3 Mos of he well-suded complex chaoc and hypechaoc sysems can be wen he fom of sysem (), such complex Loenz sysem, complex Chen sysem, complex Lü sysem, complex Van de Pol oscllao, complex Duffng sysem, and complex hypechaoc Loenz sysem, complex hypechaoc Lü sysem. Remak 4 In many pevous woks, he numbes of vaables and paamees ae equal n chaoc sysems. In ou pape, he condon s no necessay. Tha s, m = s n sysem () and n = q n sysem (6). In hs pape, we dscuss he CMHFPS beween wo dffeen dmensonal complex chaoc sysems,.e. m > n o m < n. And CMHFPS eo s defned e() = w() D()z(), whee e() = e () + je () C n and e = (e, e 2,...,e n )T, e = (e, e 2,...,e n )T. The essenal conol goal s o desgn an adapve conolle v such ha synchonzaon eo ends o zeo,.e. lm e() = lm w() D()z() =. (7) Adapve CMHFPS schemes of dffeen dmensonal complex chaoc sysems wh uncean complex paamees Accodng o he defnon of CMHFPS, we have ė() = ẇ() d(d()z()) d = ẇ() (Ḋ()z() + D()ż()). (8) Se J(z, ) = d(d()z()) = (J d, J 2,...,J n ) T s a complex veco. Theoem Fo gven complex funcon ansfomaon max D() and nal condons w(),z(), f he adapve conolle s desgned v = R(w, ˆB, ) + J(z,Â, ) Ke = G(w) ˆB g(w) + Ḋ()z() + D()(F(z) + f(z)) Ke, (9) and he complex updae laws of complex paamees ae seleced {  = (D()F(z)) H e γ a Ã, () ˆB = G H (w) e γ b B, whee γ a = dag{γ a,γ a2,...,γ }, γ b = dag{γ b, γ b2,...,γ bq } and K = dag{k, k 2,...,k n } ae eal posve defne maces, k,( =, 2,...,n) ae couplng senghs, Â, ˆB ae paamee esmaons of he unknown vecos A and B, à =  A and B = ˆB B ae he paamee eo vecos, especvely, hen adapve CMHFPS beween he esponse sysem (6) and dve sysem () s acheved ympocally, and Â, ˆB convege o he ue values of he complex consan vecos A and B, especvely, () s called complex updae laws of unknown complex paamee vecos A and B. Poof Inseon of (), (6) and (9) no(8)gves ė() = ẇ() d(d()z()) d = ẇ() J(z, A, ) = R(w, B, ) +[ R(w, ˆB, ) + J(z,Â, ) Ke] J(z, A, ) = G(w)B + g(w) G(w) ˆB g(w) + Ḋ()z() +D()(F(z) + f(z)) Ke Ḋ()z() D()(F(z)A + f(z)) = D()F(z)( A) G(w)( ˆB B) Ke = D()F(z)à G(w) B Ke, () whee à =  A and B = ˆB B ae he paamee eo vecos, especvely. Inoducng he followng Lyapunov funcon canddae V (e, Ã, B, ) = ( ) e H e + à H à + B H B, (2) 2 fom complex updae laws () and à = Â, B = ˆB, he me devave of he V () along he ajecoes of he eos sysem () eads V () = [ (ė) H e+e H (ė)+ à H Ã+à H à + B H B + B H B ] 2 = 2 [((D()F(z))à G(w) B Ke) H e + e H (D()F(z)à G(w) B Ke) + (( D()F(z)) H e γ a Ã) H à + à H [( D()F(z)) H e γ a Ã]

6 4 J. Lu e al. + (G H (w) e γ b B) H B + B H (G H (w)e γ b B)] = e H Ke à H γ a à B H γ b B <. (3) Snce V () <, bed on he Lyapunov sably heoy, he eo veco e() +. So, one aves a CMHFPS wh desed complex funcon ansfomaon max D() beween wo dffeen dmensonal complex chaoc sysems () and (6) by usng he conolle (9) and complex updae laws (). Accodng o (), when +, he paamee eos à and B also convege o zeo, whch shows he esmaon of unknown complex paamee vecos  and ˆB n boh he dve and esponse sysems also convege o he seleced ue values. The poof s compleed. Theoem 2 Suppose he paamee B of he esponse sysem (6) s known a po. Then, fo gven complex funcon ansfomaon max D() and nal condons w(), z(), f he adapve conolle s desgned v = R(w, B, ) + J(z,Â, ) Ke = F(w)B g(w) + Ḋ()z() + D()(F(z) + f(z)) Ke, (4) and he complex updae law of complex paamee s seleced  = (D()F(z)) H e γ a Ã, () whee γ a = dag{γ a,γ a2,...,γ }, and K = dag{k, k 2,..., k n } ae eal posve defne maces, k,( =, 2,...,n) ae couplng senghs,  s paamee esmaon of he unknown veco A, à =  A s he paamee eo veco, especvely, hen adapve CMHFPS beween he esponse sysem (6) and dve sysem () s acheved ympocally, and esmaon  conveges o he ue value of he complex consan veco A. Poof Inoduce he followng Lyapunov funcon canddae V (e, Ã, ) = 2 (eh e + à H Ã). (6) Then s smla o he poof n Theoem and hus s omed. Theoem 3 Suppose he paamee A of he dve sysem () s known a po. Then, fo gven complex funcon ansfomaon max D() and nal condons w(), z(), f he adapve conolle s desgned v = R(w, ˆB, ) + J(z, A, ) Ke = G(w) ˆB g(w) + Ḋ()z() + D()(F(z)A + f(z)) Ke, (7) and he complex updae law of complex paamee s seleced ˆB = G H (w) e γ b B, (8) whee γ b = dag{γ b,γ b2,..., γ bq }, and K = dag{k, k 2,..., k n } ae eal posve defne maces, k,( =, 2,...,n) ae couplng senghs, ˆB s paamee esmaon of he unknown veco B, B = ˆB B s he paamee eo veco, especvely, hen adapve CMHFPS beween he esponse sysem (6) and dve sysem () s acheved ympocally, and ˆB conveges o he ue value of he complex consan veco B. Poof Inoduce he followng Lyapunov funcon canddae V (e, B, ) = 2 (eh e + B H B). (9) Then s smla o he poof n Theoem and hus s omed. Theoem 4 Suppose boh paamee vecos A and B ae known a po. Then, fo gven complex funcon ansfomaon max D() and nal condons w(), z(), f he adapve conolle s desgned v = R(w, B, ) + J(z, A, ) Ke = G(w)B g(w) + Ḋ()z() + D()(F(z)A + f(z)) Ke, (2) whee K = dag{k, k 2,..., k n } s eal posve defne max, k,( =, 2,...,n) ae couplng senghs, hen CMHFPS beween he esponse sysem (6) and dve sysem () s acheved ympocally. Poof Inoduce he followng Lyapunov funcon canddae V (e, ) = 2 eh e. Then s smla o he poof n Theoem and hus s omed. Remak Compaed wh po wok [ 8,6,8,2 24,26,29], we am a CMHFPS scheme of complex chaoc sysems wh uncean complex paamees,

7 Adapve complex modfed hybd funcon pojecve and desgn complex updae laws of uncean paamees and adapve conolle whou sepaang he eal and magnay p of he complex sae vaables o complex paamees; hus, he concluson s vey concse and ee o be appled. Remak 6 Noe ha he complex funcon ansfomaon max D() h no effec on he me devave V ; hus, one can adjus complex funcon ansfomaon max abaly whou aleng he conol obusness. Hence, he complex funcon ansfomaon max can be used nfomaon sgnal fo he communcaon scheme. In pacula, when D() s eal, Theoem 4 ae also appled o acheve MHFPS wh eal funcon ansfomaon max of complex chaoc sysems wh uncean paamees [29]. Remak 7 If ehe of he wo maces A, B s complex paamee veco, hen Theoem 4 ae also be appled o ealze CMHFPS of chaoc complex sysems wh paly complex paamees. In pacula, when paamee maces A and B ae eal paamee vecos, Theoem 4 ae also appled o acheve CMHFPS of chaoc complex sysems wh eal paamees. 4 Numecal examples Thoughou hs secon, n ode o demonsae he effecveness and febly of he poposed synchonzaon scheme n Sec. 3, examples ae shown fo wo knds of ces: CMHFPS beween wo dffeen dmensonal chaoc complex sysems bed on he nceed ode fo m < n and educed ode fo m > n, especvely. (23) 4. Adapve CMHFPS of complex chaoc Loenz whee ρ exp( jθ) = ρ(cos θ + j sn θ). Noe ha dve sysem wh complex paamees and z complex hypechaoc Loenz esponse sysem 3, w 3 R n sysems (2) and (22), we choose he scalng funcon d wh eal paamees 33 () R n (23) o make e 3 = w 3. cos(π/)z 3 R fo he convenence of eal dscusson. In ode o llusae adapve nceed ode CMHFPS, The conolle s consuced accodng o (9) n Theoem s sumed ha 3-dmensonal complex Loenz sys- ˆb (w 2 w )+(.π jz +.â (z 2 z )) exp( jπ/) k e v = ˆb 2 w + w 2 + w w 3 w 4 + (.π jz 2 + (â 2 z â 3 z 2 z z 3 )) exp( jπ/) k 2 e 2 ˆb 3 w 3 2 ( w w 2 + w w 2 ).πz 3 sn(π/) +[.( 2 ( z z 2 + z z 2 ) â 4 z 3 )] cos(π/) k 3 e 3, (24) em () wh unknown complex paamees dves 4- dmensonal complex hypechaoc Loenz sysem wh unknown eal paamees [7]. Theefoe, he dve chaoc Loenz sysem s gven ż = a (z 2 z ), ż 2 = a 2 z a 3 z 2 z z 3, (2) ż 3 = 2 ( z z 2 + z z 2 ) a 4 z 3, whee z = z + jz, z 2 = z 2 + jz 2 ae complex sae vaables and z 3 s a eal sae vaable, A = (a, a 2, a 3, a 4 ) T s unknown complex paamee veco. Equaon (2) s chaoc when a = 2, a 2 = j, a 3 =.6 j, a 4 =.8, see [] fo moe deals. The complex esponse hypechaoc Loenz sysem wh he conolle s wen ẇ = b (w 2 w ) + v, ẇ 2 = b 2 w w 2 w w 3 + w 4 + v 2, (22) ẇ 3 = b 3 w 3 + (/2)( w w 2 + w w 2 ) + v 3, ẇ 4 = b 4 w + b w 2 + v 4, whee w = w + jw, w 2 = w2 + jw 2 and w 4 = w4 + jw 4 ae complex sae vaables, and w 3 s a eal sae vaable, B = (b, b 2, b 3, b 4, b ) T s unknown eal paamee veco. Equaon (22) s chaoc when b = 4, b 2 = 3, b 3 = 3, b 4 =, b = 4and n he absence of he conolle v = (v,v 2,v 3,v 4 ) T, see [6] fo moe deals. The complex funcon ansfomaon max s aken D() =.exp( jπ/) exp( jπ/).cos(π/) 2exp( jπ/2) ˆb 4 w ˆb w 2 + (.π jz + 2â (z 2 z )) exp( jπ/2) k 4 e 4,

8 6 J. Lu e al. he complex updae laws of paamees ae gven accodng o ()  = (.exp( jπ/)e + 2exp( jπ/2)e 4 )( z 2 z ) γ a (â a ) exp( jπ/) z e 2 γ a2 (â 2 a 2 ) exp( jπ/) z 2 e 2 γ a3 (â 3 a 3 ). cos(π/)z 3 e 3 γ a4 (â 4 a 4 ), (2) and ( w 2 w )e γ b ( ˆb b ) w e 2 γ b2 ( ˆb 2 b 2 ) ˆB = w 3 e 3 γ b3 ( ˆb 3 b 3 ). (26) w e 4 γ b4 ( ˆb 4 b 4 ) w 2 e 4 γ b ( ˆb b ) In he numecal smulaons, he fouh-ode Runge Kua mehod s appled, he ue values of unknown paamees ae chosen A = (a, a 2, a 3, a 4 ) T = (2, j,.6 j,.8) T and B = (b, b 2, b 3, b 4, b ) T = (4, 3, 3,, 4) T, especvely. The nal condons of dve sysem (2) and esponse sysem (22) ae andomly aken z() = (2 +.2 j, +.2 j, ) T and w() = ( 2 j, 3 4 j,, 6 7 j) T. The nal values of esmaed paamees and conol sengh ae Â() = (â (), â 2 (), â 3 (), â 4 ()) T = (,,, ) T, ˆB() = ( ˆb (), ˆb 2 (), ˆb 3 (), ˆb 4 (), ˆb ()) T = (,,,, ) T, γ a = dag{γ a,γ a2,γ a3,γ a4 } = dag{6, 6,, 8}, γ b = dag{γ b,γ b2,γ b3,γ b4,γ b } = dag{, 2, 6,, 8} and K = dag{k, k 2, k 3, k 4 } = dag{, 2, 3, 4}. The adapve CMHFPS eos of sysems (2) and (22) convege ympocally o zeo demonsaednfg.2, whee he sold lne shows he eal pa of he eo and he doed lne pesens he magnay pa of he eo. The pocesses of paamees denfcaon of  and ˆB ae shown n Fgs. 3 and 4, especvely, whee he ed lne shows he eal pa of he paamee and he blue lne he magnay pa of he paamee. The esmaed values of he unknown paamees gadually convege o he seleced values A = (a, a 2, a 3, a 4 ) T = (2, j,.6 j,.8) T and B = (b, b 2, b 3, b 4, b ) T = (4, 3, 3,, 4) T, especvely. As expeced, he above esuls demonsae ha adapve CMHFPS h been acheved beween complex chaoc Loenz dve sysem (2) and complex hypechaoc Loenz esponse sysem (22), and ha all of unknown paamees n boh dve and esponse sysems ae denfed successfully wh he desgned conolle (24) and he complex updae laws (2) and (26) of paamees. 4.2 Adapve CMHFPS of complex hypechaoc Lü dve sysem wh eal paamees and complex Loenz esponse sysem wh complex paamees In ode o llusae adapve educed ode CMHFPS, s sumed ha 4-dmensonal complex hypechaoc Lü sysem wh unknown eal paamees [8]dves3- dmensonal complex Loenz sysem () wh unknown complex paamees. Theefoe, he dve sysem s wen ż = a (z 2 z ) + z 4, ż 2 = a 2 z 2 z z 3 + z 4, (27) ż 3 = a 3 z 3 + (/2)( z z 2 + z z 2 ), ż 4 = a 4 z 4 + (/2)( z z 2 + z z 2 ), whee z = z + jz, z 2 = z 2 + jz 2 ae complex sae vaables and z 3, z 4 ae eal sae vaable, e() e () e () e 2 () e 2 () e 3 () e 4 () e 4 () Fg. 2 The CMHFPS eo dynamc of sysems (2) and(22) wh he conolle (24), complex paamee updae laws (2), (26). Hee e = w.exp( jπ/)z, e 2 = w 2 exp( jπ/)z 2, e 3 = w 3.cos(π/)z 3, e 4 = w 4 2exp( jπ/2)z

9 Adapve complex modfed hybd funcon pojecve 7 Fg. 3 The denfcaon pocess of unknown paamee veco A of dve sysem (2) a ( ) a 3 ( ) 2 ( ) a ( ) a a ( ) 4 a ( ) a a 3 2 ( ) ( ) Fg. 4 The denfcaon pocess of unknown paamee veco B of esponse sysem (22) ( ) b ( ) b ( ) b 3 2 ( ) b 4 ( ) b 4 2 ( ) b ( ) b 2 ( ) b 3 ( ) b 4 ( ) b A = (a, a 2, a 3, a 4 ) T s unknown eal paamee veco. Equaon (27) s chaoc when a =, a 2 = 36, a 3 = 4., a 4 = 2, see [8] fo moe deals. The complex esponse sysem wh he conolle s gven ẇ = b (w 2 w ) + v, ẇ 2 = b 2 w b 3 w 2 w w 3 + v 2, (28) ẇ 3 = b 4 w 3 + (/2)( w w 2 + w w 2 ) + v 3, whee w = w + jw,w 2 = w2 + jw 2 ae complex sae vaables, and w 3 s a eal sae vaable,

10 8 J. Lu e al. B = (b, b 2, b 3, b 4 ) T s unknown complex paamee veco, and he conolle v = (v,v 2,v 3 ) T s o be desgned. The complex funcon ansfomaon max s seleced.exp( jπ/) D() = exp( jπ/),.2 + sn.2 + cos (29) Noe ha z 3, z 4,w 3 R n sysems (27) and (28), we choose he scalng funcons d 33 (), d 34 R n (29)o make e 3 = w 3 (.2 + sn )z 3 (.2 + cos )z 4 R fo convenence of eal dscusson. The conolle s consuced accodng o (9) n Theoem The adapve CMHFPS eos of sysems (27) and (28) convege ympocally o zeo demonsaed n Fg., whee he doed lne shows he eal pa of he eo and he sold lne pesens he magnay pa of he eo. The pocesses of paamees denfcaon of  and ˆB ae shown n Fgs. 6 and 7, especvely, whee he ed lne shows he eal pa of he paamee and he blue lne pesens he magnay pa of he paamee. The esmaed values of he unknown paamees gadually convege o he seleced values A = (a, a 2, a 3, a 4 ) T = (42, 2, 6, ) T and B = (b, b 2, b 3, b 4 ) T = (2, j,.6 j,.8) T, especvely. v = ˆb (w 2 w ) + (.π jz +.(â (z 2 z ) + z 4 )) exp( jπ/) k e, v 2 = ˆb 2 w + ˆb 3 w 2 + w w 3 + (.π jz 2 +â 2 z 2 + z 4 z z 3 ) exp( jπ/) k 2 e 2, v 3 = ˆb 4 w 3 2 ( w w 2 + w w 2 ) + (.2 + sn )( â 3 z 3 + (/2)( z z 2 + z z 2 )) +(.2 + cos )( â 4 z 4 + (/2)( z z 2 + z z 2 )) + z 3 cos z 4 sn k 3 e 3, (3) he complex updae laws of paamees ae gven accodng o ().exp( jπ/)( z 2 z )e γ a (â a )  = exp( jπ/) z 2 e 2 γ a2 (â 2 a 2 ) (.2 + sn )z 3 e 3 γ a3 (â 3 a 3 ) (.2 + cos )z 4 e 3 γ a4 (â 4 a 4 ) (3) and ( w 2 w )e γ b ( ˆb b ) w ˆB = e 2 γ b2 ( ˆb 2 b 2 ) w 2 e 2 γ b3 ( ˆb 3 b 3 ). (32) w 3 e 3 γ b4 ( ˆb 4 b 4 ) In he numecal smulaons, he ue values of unknown paamees ae chosen A = (a, a 2, a 3, a 4 ) T = (42, 2, 6, ) T and B = (b, b 2, b 3, b 4 ) T = (2, j,.6 j,.8) T, especvely. The nal condons of dve sysem (27) and esponse sysem (28) ae andomly chosen z() = (+ j, + 6 j, 2, 2) T and w() = (2 +.2 j, +.2 j, ) T. The nal values of esmaed paamees and conol sengh ae Â() = (â (), â 2 (), â 3 (), â 4 ()) T = (, 6, 7+ j, 8+ j) T, ˆB() = ( ˆb (), ˆb 2 (), ˆb 3 (), ˆb 4 ()) T = (,,, + j) T, γ a = dag{γ a,γ a2,γ a3, γ a4 } = dag{6, 8,, 8}, γ b = dag{γ b,γ b2, γ b3,γ b4 }=dag{, 2, 6, 8} and K = dag{k, k 2, k 3 }=dag{,, 2}. As expeced, he above esuls demonsae ha adapve CMHFPS h been acheved beween complex hypechaoc Lü dve sysem (27) wh unknown eal paamees and complex chaoc Loenz esponse sysem (28) wh unknown complex paamees, and ha all of unknown paamees n boh dve and esponse sysems ae denfed successfully wh he desgned conolle (3) and he complex updae laws (3) and (32) of paamees. e() Fg. The CMHFPS eo dynamc of sysems (27) and(28) wh he conolle (3) and complex paamee updae laws (3), (32). Hee e = w.exp( jπ/)z, e 2 = w 2 exp( jπ/)z 2, e 3 = w 3 (.2 + sn )z 3 (.2 + cos )z 4 e () e () e 2 () e 2 () e 3 ()

11 Adapve complex modfed hybd funcon pojecve 9 Fg. 6 The denfcaon pocess of unknown paamee veco A of dve sysem (27) a ( ) ( ) a.. 3 a ( ) 4 a ( ) a 2 ( ) a ( ) 4 a 3 ( ) a 2 ( ) 2 Fg. 7 The denfcaon pocess of unknown paamee veco B of esponse sysem (28) ( ) b ( ) b 2 ( ) b ( ) b b ( ) 4 b ( ) 3 ( ) b ( ) 3 b 4 Dscusson and conclusons In hs pape, CMHFPS s noduced fo wo dffeen dmensonal chaoc sysems wh complex vaables and complex paamees. Wh he pesen mehod, n he complex space, he esponse sysem s ympocally synchonzed dffeen dmensonal dve sysem by a desed complex funcon ansfomaon max,

12 2 J. Lu e al. no a squae max. The adapve conolle and updae laws of unknown complex paamees ae desgned o make he esponse sysem become a complex funcon pojecon of he dve sysem. The pesened synchonzaon scheme s smple and heoecally goous. I s woh ponng ou ha suffcen cea on adapve CMHFPS ae deved by consucng appopae Lyapunov funcons dependen on complex vaables and employng adapve conol echnque. Que dffeen fom he schemes poposed n he leaue, we do no sepaae he eal and magnay p of complex vaables o complex paamees. Ths goes beyond he known esuls, and we hope he pefomed wok wll seve a gudelne fo fuhe sudes n chaoc synchonzaon of complex nonlnea sysems. Moeove, he CMHFPS beween complex chaoc Loenz dve sysem wh uncean complex paamees and complex hypechaoc Loenz esponse sysem wh uncean eal paamees s mplemened an example o dscuss nceed ode synchonzaon, and CMHFPS beween complex hypechaoc Lü dve sysem wh uncean eal paamees and complex Loenz esponse sysem wh uncean complex paamees s mplemened an example o dscuss educed ode synchonzaon, well. Numecal esuls ae ploed o show he apd convegence of eos o zeo and of he esmaons of unknown complex paamees o he seleced ue values. The CMHFPS bdges he gap beween dffeen dmensonal complex chaos wh complex paamees by a complex funcon ansfomaon max. The ansfomaon max s composed of complex funcons, whch ncees he complexy and scope of he synchonzaon and decs hgh secuy and lage vaey of secue communcaons. Wha s moe, moe choces of boh conol paamees and scalng funcons ae povded o ealze secue communcaons by chaos synchonzaon, songe an-aack ably and moe an-anslaed capacy ae senghened fo ou mehod. Ou fndngs ndcae ha he poposed scheme s paculaly effcen and of wde eal-wold applcably, a moe bgh fuue s wang fo secue communcaon and nfomaon pocessng. Acknowledgmens The auhos ae vey gaeful o he edos and he evewes fo he consucve commens and suggesons. Ths eseach w suppoed n pa by he Naonal Naue Scence Foundaon of Chna (Gan Nos , , 633), he Naue Scence Foundaon of Shandong Povnce, Chna (No. ZR24FL), Docoal Reseach Foundaon of Unvesy of Jnan (No. XBS3) and he Foundaon fo Unvesy Young Key Teache Pogam of Shandong Povncal Educaon Depamen, Chna. Refeences. Fowle, A.C., Gbbon, J.D.: The complex Loenz equaons. Phys. D 4, (982) 2. Gbbon, J.D., McGunnes, M.J.: The eal and complex Loenz equaons n oang fluds and le. Phys. D, 8 22 (982) 3. Fowle, A.C., Gbbon, J.D., McGunnes, M.J.: The eal and complex Loenz equaons and he elevance o physcal sysems. Phys. D 7, 3 (983) 4. Mahmoud, G.M., Bouns, T.: The dynamcs of sysems of complex nonlnea oscllaos: a evew. In. J. bfuc. Chaos 4, (24). Mahmoud, G.M., Alkhf, M.A.: Bc popees and chaoc synchonzaon of complex Loenz sysem. In. J. Mod. Phys. C 8, (27) 6. Mahmoud, G.M., Bouns, T., Mahmoud, E.E.: Acve conol and global synchonzaon of complex Chen and Lü sysems. In J. Bfuc. Chaos 7, (27) 7. Mahmoud, E.E.: Dynamcs and synchonzaon of new hypechaoc complex Loenz sysem. Mah. Compu. Model, (22) 8. Mahmoud, G.M., Mahmoud, E.E., Ahmed, M.E.: On he hypechaoc complex Lü sysem. Nonlnea Dyn. 8, (29) 9. Nayfeh, A.H., Mook, D.T.: Nonlnea Oscllaons. Wley, New Yok (979). Newell, A.C., Moloney, J.V.: Nonlnea Opcs. Addson Wesley, Readng (992). Rozhansk, V.A., Tsendn, L.D.: Tanspo Phenomena n Paally Ionzed Plma. Taylo Fancs, London (2) 2. Cvecann, L.: Resonan vbaons of nonlnea oos. Mech. Mach. Theoy 3, 8 88 (99) 3. Dlao, R., Alves-Pes, R.: Nonlnea Dynamcs n Pacle Acceleaos. Wold Scenfc, Sngapoe (996) 4. Wu, X.J., Zhu, C.J., Kan, H.B.: An mpoved secue communcaon scheme bed psve synchonzaon of hypechaoc complex nonlnea sysem. Appl. Mah. Compu. 22, 2 24 (2). Mahmoud, G.M., Bouns, T., Al-Khf, M.A., Aly, S.A.: Dynamcal popees and synchonzaon of complex nonlnea equaons fo deuned les. Dyn. Sys. 24, (29) 6. Mahmoud, G.M., Mahmoud, E.E.: Complee synchonzaon of chaoc complex nonlnea sysems wh uncean paamees. Nonlnea Dyn. 62, (2) 7. Lu, S., Chen, L.Q.: Second-ode emnal sldng mode conol fo newoks synchonzaon. Nonlnea Dyn. 79, 2 23 (2) 8. Lu, S.T., Lu, P.: Adapve an-synchonzaon of chaoc complex nonlnea sysems wh uncean paamees. Nonlnea Anal. RWA 2, (2)

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