A Function Projective Synchronization Control for Complex Networks with Proportional Delays

Transcription

1 Modelg, Smulao ad Opmzao echologes ad Applcaos MSOA 06 A Fuco Projecve Sychrozao Corol for Comple eworks wh Proporoal Delays Xulag Qu, Hoghua B,* ad Lca Chu Chegy Uversy College, Jme Uversy, Xame 60, Cha School of Scece, Jme Uversy, Xame 60, Cha * Correspodg auhor Absrac I hs paper, we vesgae fuco projecve sychrozao for geeral comple eworks wh proporoal delays. Wh esece of proporoal delays, we desg a effecve error feedback corol o aa fuco projecve sychrozao of eworks. umercal eample s provded o show effecveess of our resul. eworks; fuco projecve Keywords-comple sychrozao; proporoal delays; error feedback corol I. IRODUCIO Comple dyamcal eworkscds [-4] are ubquous real world, such as Iere, world wde web, ecologcal eworks, socal eworks, rasporao eworks, commucao eworks, eural eworks, power grd eworks, ad so o. I rece decades, cosderable aeo has bee pad o sudy sychrozao of CDs because of r poeal applcaos areas such as secure commucao, mage processg, ad harmoc oscllao geerao [5-7]. May sychroous ypes ad relave resuls are advaced, such as complee sychrozao [8], lag sychrozao [9], geeralzed sychrozao [0], phase sychrozao [], a-sychrozao [], projecve sychrozao []. Fuco projecve sychrozaofps has bee proposed ad eesvely vesgaed [4-4] laes. FPS meas ha drve ad respose sysems could be sychrozed up o a scalg fuco [4,5]. Because upredcably of scalg fuco FPS ca addoally ehace secury of commucao [6,7], FPS has araced eres of may researchers varous felds. O bass of a acve corol scheme, a geeral mehod of FPS was vesgaed [8]. FPS of a geeral class of comple eworks wh me delay was vesgaed by adapve corol scheme [9]. he work [0] gves FPS of comple eworks wh me-varyg delay va med feedback corol. Ref.[] vesgae FPS comple eworks wh swchg opology ad sochasc effecs. Ref.[] vesgae FPS comple eworks wh or whou eeral dsurbaces va error feedback corol. Ref.[] vesgae FPS comple eworks wh asymmerc couplg va adapve ad pg feedback corol. Ref.[4] vesgae FPS bewee eger-order ad sochasc fracoal-order olear sysems. I [5], a hybrd feedback corol mehod was proposed for achevg FPS CDs wh dsrbued delays. So far, mos suded models of FPS are CDs wh cosa delays [9,,], me-varyg ad bouded delays [0,4], dsrbued delays [5],ec. However, proporoal delay s oe of may delay ypes ad objecvely ese. Ulke cosa delay or bouded me delay, proporoal delay [6,7] s me-varyg ad ubouded, less coservave, ad more wdely appled. For eample, Web qualy of servceqos roug decso, proporoal delay s usually requred [8,9]. From above dscusso movao, we creavely ake eleme of proporoal delays o model of CDs o realze FPS. A smple geeral scheme of FPS CDs s vesgaed hs paper, whch coas oly error feedback erms. Compared wh prevous proposed corol mehod s a smpler ad more easly mplemeed corol echque for FPS. Cosderg ha eeral dsurbaces ad umodeled dyamcs are always uavodably praccal evoluoary processes of sychrozao, FPS CDs wh proporoal delays ad dsurbaces wll be vesgaed by proposed scheme. Fally, a eample s gve o llusrae effecveess of our resul. he res of hs paper s orgazed as follows. I Seco, we shall make some preparaos by gvg some defos ad a basc lemma. I Seco, by way of equvale sysem, we dscuss sychrozao of comple eworks by pg corol mehod. Fally, eample s performed o llusrae our resul. II. PRELIMIARIES Cosder a geerally corolled comple dyamcal ework cossg of decal learly coupled odes wh proporoal delays by followg equaos: f g j j q u j,,,,,,,, R deoes sae vecor of h ode, f : R R s a couously dffereable vecor fuco deermg dyamc behavor of odes, u R s corol Copyrgh 06, Auhors. Publshed by Alas Press. hs s a ope access arcle uder CC BY-C lcese hp://creavecommos.org/lceses/by-c/4.0/. 7

2 pu. G g j R s couplg cofgurao mar represeg opologcal srucure of ework, g j 0 f re s a coeco bewee ode ad ode j; orwse gj g j 0, ad dagoal elemes of represeg opologcal srucure of ework, ad dagoal elemes of mar G are defed by Eq.. Assumpo. he dervave of scalg fuco bouded, ha s mar G are defed by g a* j, j gj,,,,, q s proporoal delay coeffce ad sasfy 0 q. Furrmore, comple ework descrbed possess al s codos 0, [q,], 0,,, are cosas. lm 0,,,,, for all R, a R s upper lm of. * Assumpo. he orm of msmached erms d,,, are bouded, ha s of Defo. FPShe ework wh proporoal delays s sad o acheve fuco projecve sychrozao f re ess a couously dffereable scalg fuco such ha 5 d d* 6 d R s upper lm of orm of d * I hs paper, we deoe M, M, M are upper lm of orm of respecvely. f z, Q G I, produc, f y, y, represe krocecker J dag H z, H z,, H z, sads for Eucldea vecor orm ad R ca be a equlbrum po, or a perodc orb, or a orb of a chaoc aracor, whch sasfes f. I hs paper, our goal s o desg some smple corollers H z f z / z f z wh s respec Jacoba o mar z, of z y. ma M deoes mamum egvalue for symmerc mar M. u,,,, so ha CDs ca reach FPS. Because Lemma [0]. For ay vecor, y R ad posve ose of commucao bewee coeced odes durg sgal rasmsso s uavodable a real world, such as eurorasmers ad packe loss. I addo, re are umodeled dyamcs ad here dsurbaces may praccal sysems. So order o reflec more realsc dyamcal behavors, we wll furr cosder CDs wh umodeled dyamcs as well as eeral dsurbaces as follows: defe mar C R holds: f gj j q d u j,,,,,,,, R deoes sae vecor of h ode, f : R R s a couously dffereable vecor fuco deermg dyamc behavor of odes, u R s corol pu ad d R s msmached erms, whch could es may perurbao, ose dsurbace. G gj R s couplg cofgurao mar, followg mar equaly y C y C y III. FPS I COMPLEX EWORKS WIH PROPORIOAL DELAYS I hs seco, we propose a error feedback corol mehod for realzg fuco projecve sychrozao comple dyamcal eworks wh proporoal delays. Le y e, a couple of eworks ad 4 s equvalely rasformed o followg couple of comple eworks wh cosa delay ad me varyg coeffces y e f y gj y j U j 8

3 y e f y gj y j D U j,,,. he vecor fuco f y s learzed as follows eghborhood of goal value va aylor epasos,,,, 0, l q 0, D d e, U u e, ad y s C [, 0],, whch s 0, s [, 0]. Defo. he ework 9 s sad o acheve fuco projecve sychrozao f re ess a couously dffereable scalg fuco such ha f y f z,,,, z y. Keepg frsorder erms Eq.4 ad subsug Eq., we have e e f z H z e gj e j lm y y 0,,,,. sads for Eucldea vecor orm ad y R heorem. Suppose Assumpos ad hold. For a gve sychrozao scalg fuco, f re es posve cosas k, k, k whch j D U y e f y ca be a equlbrum po, or a perodc orb, or a orb of a chaoc aracor, whch sasfes y e f y. sasfy QQ, CDs wh dsurbace 9 ca relze fuco projecve sychrozao va corol law: f z y z 4 z 5,,,. Cosruc Lyapuov fuco V e e e e ve vdv 6 k M M d*, k M, k ma J he me dervave of V alog rajecores of Eq.5 s U k k e sge ke,,,, Proof. Defe sg deoes sg fuco. e y y,,,,. s a couously dffereable fuco. I follows from Eq.9 ad Eq. ha e e f y gj e j D U j y e f y 9

4 V e e e e e e e e e e e e e gj e j e e e e e e e V e H z e k e e e e 9 Where e f z D k k e sge e y f y ke e 7 Le e e, e,, e R Lemma, we have e H z e e gj e j k mk, k,, k.. he by j e e e e V e Je k e e e Qe e e e e e Je k e e e QQ e e e QQ e k e e e e e J QQ k e e ma J e f z D k k e e y f y ke e j e H z e e gj e j e e e e Because chaos sysems ad scalg fuco are bouded, y, ad z are bouded. Furrmore, f s a couously vecor fuco, re es posve cosas M ad M sasfyg f y M. Because Assumpo holds, re V e M M d* k M k e k e e e H z e e e e g j e j j akg k ma [ J QQ / ] /, we oba y M. Because Assumpo holds, re ess a posve cosa d* sasfyg D d*,,,. 0 f z M ad ess a posve cosa M sasfyg e e V 0 Accordg o Lyapuov sably ory, error asympocally sable. hs complees proof. 5 s Corollary. Suppose Assumpos hold. For a gve sychrozao scalg fuco, f re es posve cosas k, k, k k M M, k M, k ma J whch sasfy QQ, CDs whou 8 ca relze dsurbace sychrozao va corol law: fuco projecve 8 U k ke sge ke akg k M M d * ad k M,,,,, we oba,,,, sg deoes sg fuco. 40

5 By heorem, s easy o see ha a smlar proof holds for D 0,,,. hus, proof s omed here. he me dervave of V alog rajecores of Eq.5 s A adapve scheme s esablshed order o selec approprae gas k, k ad k o realze FPS CDs wh or whou dsurbaces followg. heorem. Suppose Assumpos ad hold. For a gve sychrozao scalg fuco, CDs wh dsurbace 9 ca relze fuco projecve sychrozao va corol law: U k k e sge k e V e e e e e e e e e e k k k k k k l l k k k l e f z f y D k k l e sge e y k e H z e e gj e j j k l e e sge 5 k l e e 6,,,, sg deoes sg fuco. l 0, l 0 ad l 0 are arbrary posve e e e e k e e 8 Because chaos sysems ad scalg fuco are bouded, y, ad z are bouded. Furrmore, f s a couously vecor fuco, re es posve cosas M ad M sasfyg f z M ad f y M. Because Assumpo holds, re y M. cosas. ess a posve cosa M sasfyg Proof. Cosruc Lyapuov fuco Because Assumpo holds, re ess a posve cosa d * sasfyg D d*,,,. V e e e e v e v dv k k k k l l k k l V e M M d* k e M k 7 j k e e e gj e j e H z e e e e e 9 akg k M M d * ad k M,,,,, oba 4 we

6 j V k e e e gj e j e e e H z e e e 0 mar G gj s chose o be 0 G 0 Le e e, e,, e R Lemma, we have f d f e d 5, e, f 8. he couplg cofgurao k m k, k,, k. d. he by. Comple eworks wh dsurbaces ad proporal delays ca be descrbed as follows: V e Je k e e e Qe 5 d 7 8 d d e e e e e Je k e e e QQ e e e QQ e k e e e e e J QQ k e e ma J gj j q u,,,. j 4 corollers u sasfed: U u e, U ca be desged by usg heorem as follow: akg k ma [ J QQ / ] /, we oba V e e Accordg o Lyapuov sably ory, error 5 s asympocally sable. hs complees proof. Corollary. Suppose Assumpos hold. For a gve sychrozao scalg fuco, CDs whou sge e U k k e sge k e sge e 5 dsurbace 8 ca relze fuco projecve sychrozao va corol law 6. By heorem, s easy o see ha a smlar proof holds for D 0,,,. hus, proof s omed here. k l ej sgej j 6 IV. EXAMPLE I hs subsceo, we wll ake chaoc Che sysem as odes of CDs o verfy effecveess of proposed scheme heorem. Cosder followg sgle Che sysem: k l e ej sgej 7 j k l ej 8 j 4

7 We ake,,. as , 0 8, al saes 0 7, We ake l 0, l 0, l 500,,, k 0, k 0, k 0, k 0 5, k 0 7, k 0 6, k 0 8, k 0 7, k 0 6, q 0.8, ad 4 cos / 6, we ake dsurbace vecors d d d e s cos s, d d d d e cos s cos, d d e s s cos, Accordg o heorem we have e 0 wh. he resul show ha fuco projecve sychrozao akes place wh desred scalg fuco comple eworks 4. ACKOWLEDGME hs work s joly suppored by aural Scece Foudao of Cha 087, , 600 ad Scec Research Fud of Fuja Provcal Educao Deparme of Cha JA6069. REFERECES [] [] [] [4] [5] [6] [7] [8] [9] [0] [] [] [] [4] [5] E O, C Grebog, JA Yorke, Corollg chaos, Phys. Rev. Le H. G. ad Q. Z., Corollg spaoemporal chaos coupled map lace sysems, Phys. Le. A S. Pad, R. Amrkar, Characerzao ad corol of small-world eworks, Phys. Rev. E S. Srogaz, Eplorg comple eworks, aure D. J. Was, S. H. Srogaz, Collecve dyamcs of 'small-world' eworks, aure M. E. J. ewma, D. J. Was, Reormalzao group aalyss of small-world ework model, Physcs Leers A X. Wag, G. Che, Pg corol of scale-free dyamcal eworks, Physcs A J. Q. Lu, J. D. Cao, Adapve complee sychrozao of wo decal or dffere chaochyperchaoc sysems wh fully ukow parameers, Chaos, E. M. Shahverdev, S. Svaprakasam, Lag sychrozao medelayed sysems, Phys. Le. A F. Rulkov, M. M. Sushchk, L. S. smrg e al., Geeralzed sychrozao of chaoc oscllaors, Phys. Rev. E M. G. Roseblum, Pkovsky A S, Kurhs J., Phase sychrozao of chaos drecoally coupled chaoc sysems, Phys. Rev. Le C. Km, S. Rm, W. Kye, J. Ryu, Y. Park,A-sychrozao of chaoc oscllaors, Phys. Le. A R. Maer, J. Rehacek, Projecve sychrozao hree-dmesoal chaoc sysems, Phys. Rev. Le X. ag, J. Lu, W. Zhag, he FPS of chaoc sysem usg backseppg desg, Ch.J.Dyam.Corol Y. Che, X. L, Fuco projecve sychrozao bewee wo decal chaoc sysems, Iera.J.Moder Phys.C [6] H. Du, Q. Zeg, C. Wag, M. Lg, Fuco projecve sychrozao coupled chaoc sysems, olear Aal. RWA [7] X. Wu, H. Lu, Hyperchaoc secure commucao va geeralzed fuco projecve sychrozao, olear Aal. RWA [8] H. Du, Q. Zeg, C. Wag,A geeral mehod for fuco projecve sychrozao, I.J.Iovave Compu.Iform.Corol [9] H. Du, P. Sh,. Lu, Fuco projecve sychrozao comple dyamcal eworks wh me delay va hybrd feedback corol, olear Aal. RWA [0] L. Sh, H. Zhu, Y. Zeg, Fuco projecve sychrozao of comple dyamcal eworks wh me-varyg delay va med feedback corol, Ieraoal Symposum o Compuaoal Iellgece ad DesgISCID [] Y. J, S. Zhog, Fuco projecve sychrozao comple dyamcal eworks wh swchg opology ad sochasc effecs, Appl. Mah. Compu [] H. Du, Fuco projecve sychrozao comple dyamcal eworks wh or whou eeral dsurbaces va error feedback corol, eurocompug [] L. Sh, H. Zhu e al. Fuco projecve sychrozao of comple dyamcal eworks wh asymmerc couplg va adapve ad pg feedback corol, ISA rasacos06. [4] L. Geg, Y. Yu, S. Zhag, Fuco projecve sychrozao bewee er-order ad sochasc fracoal-order olear sysems, ISA rasacos [5] X. Qu, J. Che, L. Chu, Fuco projecve sychrozao comple eworks wh dsrbued delays va hybrd feedback corol, upublshed. [6] A. Iserles, O eural fucoal-dffereal equao wh proporoal delays, J. Mah. Aal. Appl [7] L. Zhou, Dsspavy of a class cellular eural eworks wh proporoal delays, olear Dy [8] Y. Che, M. Hamd, D. H. K. sag, Proporoal QoS over OBS eworks, IEEE Globe elecommucaos Coferece [9] S. Kulkarm, R. Sharma, I. Mshra, ew QoS roug algorhm for MPLS eworks usg delay ad badwdh cosras. I. J. Iform. Commu. echol. Res [0] J. Lu, J. Cao, sychrozao-based approach for parameers defcao delayed chaoc eural eworks, Physcs A