Table Of Content

Patterns of Chaos Syn hronization 1 2 2 1 Johannes Kestler, Evi Kopelowitz, Ido Kanter, and Wolfgang Kinzel 1 Institute for Theoreti al Physi s, University of Würzburg, Am Hubland, 97074 Würzburg, Germany 2 Department of Physi s, Bar-Ilan University, Ramat-Gan, 52900 Israel (Dated: De ember 2007℄) Small networks of haoti units whi h are oupled by their time-delayed variables, are investi- gated. In spite of the time delay, the units an syn hronize iso hronally, i.e. without time shift. Moreover, networks an not only syn hronize ompletely, but an also split into di(cid:27)erent syn hronized sublatti es. These syn hronization patterns are stable attra tors of the network dynami s. Di(cid:27)erent networks with their asso iated behaviors and syn hronization patterns are presented. In parti ular, we investigatesublatti e syn hronization, symmetrybreaking, spreading haoti motifs, 8 syn hronizationbyrestoringsymmetryand ooperativepairwisesyn hronizationofabipartitetree. 0 0 2 I. INTRODUCTION n a Chaos syn hronization is a ounter-intuitive phe- J nomenon. On one hand, a haoti system is unpre- FIG. 1: Two mutually oupledunits. 3 di table. Two haoti systems,startingfromalmostiden- ] ti al initial states, end in ompletely di(cid:27)erent traje to- D ries. Ontheotherhand,twoidenti al haoti unitswhi h theunitsinea hsublatti e ansyn hronizetoa ommon C are oupled to ea h other an syn hronize to a ommon haoti traje toryalthoughtheyarenotdire tly oupled. . haoti traje tory. The system is still haoti , but af- The oupling of one sublatti e is relayed by the haoti n ter a transient the two haoti traje tories are lo ked to traje tory of a di(cid:27)erent sublatti e. The traje tories of i l ea hother[1,2℄. Thisphenomenonhasattra tedalotof di(cid:27)erent sublatti es are only weakly orrelated, but not n resear ha tivities, partly be ause haos syn hronization syn hronized [21℄. [ has the potential to be applied for novel se ure ommu- In this paper we want to investigate patterns of haos 1 ni ation systems [3, 4℄. In fa t, syn hronization and bit syn hronization for several latti es with uni- and bi- v ex hange with haoti semi ondu tor lasers has re ently dire tional ouplings with time delay. There exists a 8 been demonstratedovera distan eof120km in apubli mathemati altheoryto lassifypossiblesolutionsofnon- 8 4 (cid:28)ber-opti ommuni ation network [5℄. In this ase, the linear di(cid:27)erential or di(cid:27)eren e equations for a given lat- 0 oupling between the haoti lasers was uni-dire tional, ti e [22℄. However, this theory does not determine the . the sender was driving the re eiver. For bi-dire tional stabilityofthesesolutions. Butinordertodes ribephys- 1 ouplings, when two haoti units are intera ting, addi- i al or biologi al dynami networks, we are interested in 0 8 tional interesting appli ations have been suggested. In stable patterns of haoti networks. The patterns whi h 0 this ase, a se ure ommuni ation over a publi hannel aredis ussed in this paperareattra torsin phase spa e, : may be established. Although the algorithm as well as any perturbation of the system will relax to these pat- v alltheparametersarepubli ,itisdi(cid:30) ultforanatta ker terns whi h move haoti ally on some high dimensional i X to de ipher the se ret message [6, 7, 8℄. syn hronizationmanifold. r Typi ally, the oupling between haoti units has a Ourresultsaredemonstratedforiteratedmaps,forthe a time delay due to the transmissionof the ex hanged sig- sake of simpli ity and sin e we an al ulate the stabil- nal. Nevertheless, haoti units an syn hronize without ity of these networks analyti ally. But we found these timeshift,iso hroni ally,althoughthedelaytimemaybe patterns for other systems, as well, for example for the extremelylong omparedtothetimes alesofthe haoti Lang-Kobayashirateequationsdes ribingsemi ondu tor units. This (cid:21) again ounter-intuitive (cid:21) phenomenon has lasers. Hen e we think that our results are generi . re ently been demonstrated with haoti semi ondu tor lasers[9, 10, 11, 12℄, and it is dis ussed in the ontextof orresponding measurements on orrelated neural a tiv- II. TWO INTERACTING UNITS ity [13, 14, 15, 16℄. Several haoti unitsmaybe oupledtoanetworkwith We start with the simplest network: Two units with delayed intera tions. Su h a network an syn hronize delayed ouplings and delayed self-feedba k, as sket hed ompletely to a single haoti traje tory, or it may end in Fig. 1. inastateofseveral lusters,dependingonthetopologyof For iterated maps, this network is des ribed by the thenetworkorthedistributionofdelaytimes[17,18,19, following equations: 20℄. Re entlyanotherphenomenonhasbeenreportedfor at =(1−ε)f(at−1)+εκf(at−τ)+ε(1−κ)f(bt−τ) haoti networks: Sublatti e syn hronization. If a small bt =(1−ε)f(bt−1)+εκf(bt−τ)+ε(1−κ)f(at−τ) (1) network an be de omposed into two sublatti es, then 2 1 further identi al units re eiving the same drive would relax to the same traje tory. Thus Eq. (5) de(cid:28)nes the 0.8 region where identi al units whi h get the same input syn hronize to a ommon traje tory. I Now let us rewrite Eq. (1): 0.6 κ at =(1−ε)f(at−1)+ε(2κ−1)f(at−τ)+ 0.4 II +ε(1−κ)f(at−τ)+ε(1−κ)f(bt−τ), bt =(1−ε)f(bt−1)+ε(2κ−1)f(bt−τ)+ (6) 0.2 +ε(1−κ)f(bt−τ)+ε(1−κ)f(at−τ). III Both systems are driven by the identi al signal 0 0 0.2 0.4 ε 0.6 0.8 1 st−τ =ε(1−κ) f(at−τ)+f(bt−τ) . (7) (cid:2) (cid:3) α=3/2 FIG. 2: Phase diagram for (analyti al result). Comparing (4) with (6), one (cid:28)nds that for κ˜ =2κ−1 f(x) (8) where is some haoti map, for example the Bernoulli shift, theintera tingunits (6)aredes ribed bythe drivenunit f(x)=αx mod 1 (4). The phase boundary of the driven system, region II (2) +III[Eq.(5)℄,andthephaseboundaryoftheintera ting α > 1 with . In this ase, the system is haoti for all system, region I + II [Eq. (3)℄, are onne ted with ea h 0 < ε < 1 0 < κ < 1 ε parameters and . measures the other: WiththemappingofEq.(8), onephaseboundary κ total strength of the delay terms and the strength of τ a→nb∞eobtainedfrom theothτer. Thisisnotonlytrue for the self-feedba k relative to the delayed oupling. but for any value of . a =b t t Obviously, the syn hronized haoti traje tory Additionally, this mapping does not only hold for the τ is a solution of Eq. (1). Its stability is determined by Bernoullishift but forany haoti system,providedthat onditional Lyapunov exponents whi h des ribe the sta- the signal does not hange the Lyapunov exponents of bility ofperturbations perpendi ular to the syn hroniza- the driven system. Simulations with other maps, e.g. tion manifold. For the Bernoulli map, these Lyapunov with the Tent map or the Logisti map, on(cid:28)rm the ap- exponentshavebeen al ulatedanalyti ally[21, 23℄, and pli ability of the mapping (8). Sin e the slopes of these τ → ∞ for in(cid:28)nitely long delay, , one obtains the phase maps are not onstant, the Lyapunov exponents depend diagram of Fig. 2. on the traje tories;in orderto gettraje torieswhi h are In region I and II, i.e. for omparable to the traje tories of two mutually oupled α−1 2αε+1−α units,thedrivingsignalinthedriver/re eiversetupmust <κ< , ct 2αε 2αε (3) ome from an identi al unit (whi h has an in reased self- oupling due to the la k of external ouplings), i.e. tjeh etotwryoautn=itsbat.reAslytnh ohurgohnitzheedttwooanuniditesnatri ea lo uhpaloetdi wtritah- st−τ = ε(1−κ˜)f(ct−τ) with (9) τ ct = (1−ε)f(ct−1)+εf(ct−τ). a long delay , they are ompletely syn hronized with- (10) τ →∞ out any time shκif=t. 1For , this region is symmetri about the line 2. EvenfortheLang-Kobayashiequations[24,25℄,whi h Complete syn hronization an be understsood by on- des ribe oupled semi ondu tor lasers, we found a good t sidering a single unit driven by some signal : agreement for the mapping (8), see se tion VII. at =(1−ε)f(at−1)+εκ˜f(at−τ)+st−τ. Now let us ome ba k to the iterated equations (1). (4) Let us assume that we re ord the syn hronized traje - a = b t t tory of two intera ting haoti units, regions I Ifthesystemisnot haoti ,i.e.ifitsLyapunovexponents a t and II of Fig. 2. Now we insert the re orded traje tory are negative, then the traje tory srelaxes to a unique bt at t into Eq. (1). How will respond to this drive? We traje tory determined by the drive . For the Bernoulli (cid:28)nd that in region II the unit A will syn hronize om- shift, this region is de(cid:28)ned by the inequality bt pletely to the re orded traje tory , whereas in region I 1+αε−α 1+αε−α − < κ˜ < theunitAdoesnotsyn hronize. Althoughthetwointer- (cid:18) αε (cid:19) αε (5) a tingunitsAandBdosyn hronize,theunitAdoesnot followthere ordedtraje toryinregionI.Thisshowsthat and indi ated by II + III in Fig. 2. Sin e the traje tory bi-dire tionalintera tionisdi(cid:27)erentfromuni-dire tional whi h A relaxes to is uniquely determined by the drive, drive. 3 FIG. 3: Ring of 4 units. FIG.4: Sublatti esyn hronizationinatriangularlatti ewith periodi boundaries. The double lines signify bi-dire tional ouplings. The self-feedba k is not drawn to simplify the il- lustration. III. SUBLATTICE SYNCHRONIZATION The response of a single haoti unit to an external fromrandominitialstatesitrelaxestothestateofsublat- drive, Fig. 2, determines also the phase diagram of a ti e syn hronization(regionIII of Fig. 2forthe ringof 4 ring of four haoti units. Additionally, it shows a new units). The haoti traje toriesofsublatti esyn hroniza- phenomenon: sublatti e syn hronization [21℄. Consider AB tion may be depi ted as BA. Note that this stru ture the ring of four identi al units of Fig. 3. The dynami s does not breakthe symmetryof the ring: The statisti al of unit A is des ribed by properties of the haoti traje tory of A is identi al to those of B. at =(1−ε)f(at−1)+εκf(at−τ)+ However, there are other solutions of the dynami 1 1 +ε(1−κ) f(bt−τ)+ f(dt−τ) ; equations, as well. These solutions are lassi(cid:28)ed a ord- (cid:20)2 2 (cid:21) (11) ing to the theory of Golubitsky et al [22℄. For example, AA for the ring, the state B B is a solution as well. Su h a state breaks the symmetry of the latti e. But we (cid:28)nd thedynami sofB,Cand1Darede(cid:28)nedanalogously. Note theadditionalweighting 2 fortheexternal oupling(to2 thatthisstateisunstable. Anytinyperturbationrelaxes A B AA AB neighbors)whi h ausesthetotalstrengthoftheexternal to the states BA in regionIII, AA in regionII and DC ε(1−κ) outside of II and III. In fa t, we have never found a sta- oupling to remain . blestatewhi hbreaksthesymmetryofthelatti e. Only Obviously, the two units A and C re eive identi al in- when the ouplings in the two dire tions of the ring are put from the units B and D. Consequently, they will re- AA AB spond with an identi al traje tory in the regions II and di(cid:27)erent,thestates B B and AB arestableinsome(dif- ferent) parts of the parameter spa e, see Fig. 5. Hen e III of Fig. 2. The sameargumentholdsforthe twounits wepostulatethatspontaneoussymmetrybreakingisnot B and D, whi h both get the same input from A and possible for (cid:28)nite latti es of haoti units. C. That leads to sublatti e syn hronization in region III of Fig. 2: A and C have an identi al haoti traje tory and B and D have a di(cid:27)erent one. Although there is a delay of arbitrary long time of the transmitted signal, IV. SPREADING CHAOTIC MOTIFS syn hronizationis omplete,withoutanytimeshift. The syn hronization of A and C is mediated by the haoti The response of a haoti unit to an external drive, traje tory of B and D and vi e versa. But the two tra- Fig. 2, points to another interesting phenomenon. Con- je tories have only weak orrelations, they are not syn- sider a triangle of haoti units with bi-dire tional ou- hronized. Numeri alτsimulationsoftheBernoullisystem plings as sket hed in Fig. 6(a). Choose the parameters with small values of have shown that there is no gen- su h that the triangle is ompletely disordered, but ea h eralized syn hronizationeither [21, 26, 27℄. unit has negative Lyapunov exponents when it is sepa- Sublatti e syn hronization has been found for rings rated from the two others. (Both onditions are ful(cid:28)lled with an even number of units, for hains and also for in region III of Fig. 6(b), whi h shows analyti al results other latti es whi h an be de omposed into identi al fortheBernoullisystem.) Whenwere ordthethreetime a b c t t t sublatti es [21℄. Forexample,thelatti eof Fig. 4 anbe series , and we (cid:28)nd three di(cid:27)erent weakly orre- de omposed into three sublatti es. For some parameters lated haoti traje tories. Nowwefeed the twotraje to- b c t t oftheBernoullisystemwe(cid:28)ndsublatti esyn hronization ries and into an in(cid:28)nitely large latti e of identi al with three haoti traje tories. Again, the syn hronized unitswithuni-dire tional ouplingsasshowninFig.6(a). units are not dire tly oupled, but they are indire tly Ea hunitre eivestwoinputsignalsfromtwootherunits. onne ted via the traje tories of the other sublatti es. But sin e all Lyapunov exponents are negative, the sys- a b t t The sublatti e traje tories, des ribed before, are sta- tem responds with the three haoti traje tories , c t ble, i.e. the onditional Lyapunov exponents whi h de- and . Although the units of the initial triangle are s ribe perturbations perpendi ular to the syn hroniza- not syn hronized, their pattern of haos is transmitted tion manifoldarenegative. Even when thesystem starts to the in(cid:28)nite latti e. All units of the same sublatti e 4 (a) Twodi(cid:27)erent oupling κ1 κ2 parameters and . 1 0.8 (a) Syn hronization pattern (Sublatti e AA − syn hronization) ofregionIII. BB 1 0.6 κ 2 − 0.8 0.4 AA AA 0.6 I AB AB κ 0.2 BA AB − 0.4 0 II 0 0.2 0.4 κ 0.6 0.8 1 1 0.2 α(b=)3P/h2asediεag=ra1m1/f2o0rBernoullisystemwith III and (analyti alresult). The 0 regionsofnosyn hronisat−ion are labeledwitha 0 0.2 0.4 ε 0.6 0.8 1 dash( ). α(b=)3P/h2asediagramforBernoullisystem, FIG. 5: Ring of 4 units with two di(cid:27)erent oupling parame- . Analyti alresult ombiningthe ters. analyti alsyn hronization regionforaringof3 units((cid:22))andtheregionwhere identi alunits whi hare drivenbyanidenti alsignal, syn hronize (--). are ompletelysyn hronizedwithouttimeshift,although τ the oupling has a long delay time . Hen e the haoti FIG. 6: Triangle (three bi-dire tionally oupled units) with motif, three weakly orrelated haoti traje tories, an a uni-dire tionally atta hed in(cid:28)nitely large latti e. Double be imposed on an arbitrarilylargelatti e. Note that the linessignifybi-dire tional ouplingswhereasarrowsshowuni- time for spreading a motif on a large latti e in reases dire tional ouplings. The self-feedba k is not drawn to sim- only linearly with the number of units be ause of the plify theillustration. unidire tional ouplings. κ ε For some parameters and , namely in regions I and II of Fig. 6(b), the three units of the triangle are unidire tional ouplings. ompletely syn hronized. In region I, only the three units are syn hronized while the other units remain un- syn hronized. In region II, the other units, too, get syn- V. SYNCHRONIZATION BY RESTORING hronized to the triangle (be ause all Lyapunov expo- SYMMETRY nents are negative), so the whole latti e is ompletely syn hronized. In general we expe t that the larger the network is, The phenomenon of spreading motifs is not restri ted the smaller the region in the parameter spa e is where N =6 to a triangle. For example, a ring of 6 mutually ou- the network syn hronizes. For example, a ring of (ε,κ) pled units has also a region in the orresponding units has a smaller region of syn hronization than the N = 4 phase diagram where there is no syn hronization, but region II and III of Fig. 2 for . With in reasing N α where isolated units have negative Lyapunov exponents, (andtheslope beingheld onstant)syn hronization omparableto region III in Fig. 6(b) [21℄. Consequently, (cid:28)nally disappears ompletely [21℄. However, we found the pattern of six haoti traje tories an lead to sub- a ounterexample where adding a unit restores syn hro- latti e syn hronization with 6 di(cid:27)erent sublatti es on a nization. orresponding latti e of haoti units with short-range Consider the hain of 5 units shown in Fig. 7(a). The 5 (a) Chainof(cid:28)veunitswithtwodi(cid:27)erent timedelays andwithout self-feedba k. (b) Thelastunithasbeenremoved. N (a) There are di(cid:27)erentdelaytimeswhi h are pairwiseidenti alforoneunitofsideA FIG. 7: Symmetri and asymmetri hain without self- andoneunitofsideB. feedba k. oupling to the two outer units has a longer delay time than the internal ouplings. There is no self-feedba k, κ = 0 . Now remove unit E, Fig. 7(b), and res ale the oupling to unit D. (Unit D doubles the input from unit C to ompensate for the missing se ond neighbor.) In N this ase, a syn hronized solution does not exist. Nu- (b) There are di(cid:27)erentshiftsinthe Bernoullimapwhi hare pairwise meri alsimulationsoftheBernoullisystemandthelaser identi alforoneunitofsideAandone equations show∆hig=hτ or−reτlationsbetween units A and C unitofsideB. 1 2 with time shift , and between B and D with zero time shift, but the orrelation oe(cid:30) ient does not FIG. 8: Ea h unit on one side is oupled to all units of the a hieve the value one. On the other side, if we add unit otherside. E we restore the symmetry of the hain. In this ase we (cid:28)nd sublatti e syn hronization with time shift between k the outer units and the entral one: and vi e versa. Note tha1t/tNhe unit A re eives only a k weak signal of the order from its ounterpart B . at =et =ct−∆; bt =dt . (12) Nevertheless, we (cid:28)nd that the network syn hronizes to a = k,t τ τ a state of pairwise identi al haoti traje tories, 1 2 b ;k =1,...,N If is greater than , the entral unit is earlier than k,t . FortheBernoullisystem,theregionof the haoti traje toryoftheouterones,itleads,whereas pairwise syn hronizationis similar to region II of Fig. 2. for the opposite ase it lags behind. Thereisnosyn hronizationamongunitsofthesameside. Ea hunitre eivesthesumofall haoti traje tories,but it responds only to the tiny part whi h belongs to its VI. COOPERATIVE PAIRWISE ounterpart. The syn hronizationisa ooperativee(cid:27)e t. SYNCHRONIZATION As soon as a single unit is detuned, the whole network loses syn hronization. Is it possible to syn hronize two sets of haoti units The se ond setup, Fig. 8(b), is similar to the (cid:28)rst withasingle oupling hannel? Infa t,wefoundsu hex- one but allows analyti al al ulations. In this se ond ampleswheretwosetsof haoti unitsarebi-dire tionally setup, all delay times are identi al, while ea h unit of β k = 1,...,N onne tedbythesumoftheirunits,asindi atedinFig.8. k 2N one side has a di(cid:27)erent shift , , in its Thereare units, i.e.the numberofunitsonea hside N shifted Bernoulli map, see Eq. (14); the shifts are pair- is . Ea h side is the mirror image of the other. Only k k N wise identi alforunit A and the orrespondingunit B onesinglebi-dire tionalsignal omposedof all signals so one side is again the mirror image of the other side. from ea hsideisdrivingthe otherside,and this leadsto The shifted Bernoulli map is de(cid:28)ned by ooperative pairwise syn hronization. f (x)=[α(x+β)] mod 1. In the (cid:28)rst setup, Fig. 8(a), all units areidenti al, but β (14) thedelaytimesoftheir ouplingsaredi(cid:27)erent. Theunits k k have pairwise identi al delay times, i.e. A and B have Ea h A unit re eives the signal 2τ + τ k a oupling delay time τsw=hi2 hτki+s eτnfor ed by a 1 N self-feedbaN k=w1ith delay time . Hen e, for st =ε(1−κ)N fβk(bk,t−τ) (15) one pair, , we obtain the phase diagram of Fig. 2, Xk=1 where the two units are ompletely syn hronized in re- N gions I and II. For a large number of units, ea h A and vi e versa. Analyti al al ulations are possible for unit re eives the identi al signal this se ond setup be ause the delay times are equal (cid:21) in N 1 ontrast to the (cid:28)rst setup (cid:21) and the di(cid:27)erent shifts in st =ε(1−κ)N f(bk,t−(2τk+τ)) (13) themapdonothamperthe al ulations. IftheBernoulli kX=1 maps were not shifted, the units would all be identi al. 6 N Then in regions II and III ofaFig=. 2a, all=u.n.it.s=ofathe of lasers oupled semi ondu tor lasers, where the net- 1,t 2,t N,t same side would syn hronize, ; workstru tureand ouplingstrengthsaredeterminedby b = b = ... = b w 1,t 2,t N,t j,k be ause they re eive the same the weightings , whi h are normalized so that the input. Then one would e(cid:27)e tively get two oupled units strNengthwof th=e1total input for jea h laser is the same, rAegaionndIBIa(lwl2hNi hunsyitns whroounlidzebeinsyrneg hiornosniIzeadn.dFoIIr),shsiofteind PNsko=ll,a1sGersN,jτ,k,αlef,γf,oΓr,epa, ωh0laasnedrCsp[2a8r℄e. Thohseenpaar a moredtienrgs Bernoullimaps,theunitsofthesamesidearenotallowed to Ref. [25℄. to syn hronize by de(cid:28)nition. Nevertheless, the stability In a very simpli(cid:28)ed form, the equations above read analysisregardingthelinearizedequationsisnota(cid:27)e ted by the di(cid:27)erent shifts, meaning that the same perturba- d x=[ ]+λ[ ] tions still are damped in the same regions; the only dif- dt internal dynami s self-feedba k feren e is that due to the shifts, most of the traje tories +σ[ ]. external oupling (19) arenotallowedto ome losetogether. Onlythepairsof a = b k,t k,t orresponding units an syn hronize, , and so they do in region II. Comparing equations (1) and (19) yields a relation between the parameter spa e of the Lang-Kobayashiequa- {λ,σ} tions, , and the parameter spa e of the maps, {ε,κ} VII. ANALOGY OF LASER EQUATIONS TO , if the two following onditions are onsidered: ITERATED MAPS (i) The ratio of the self-feedba k to the external oupling should be the same in both ases. Besides iterated maps, we onsidered the Lang- Kobayashi equations, whi h des ribe the dynami s of (ii) The ratioof the time-delayed terms to the internal semi ondu tor lasers opti ally oupled to their own or/and to the light of other semi ondu tor lasers. We dynami s should be the same in both ases. used them for simulations in the following form, a ord- ing to Ref. [25℄: These two onditionsyield the followingtransformation: λ λ+σ d 1 C γ[N +nj(t)] κ= , ε= . E0,j(t)= GNnj(t)E0,j(t)+ sp Sol λ+σ v (20) dt 2 2E (t) 0,j +λE (t−τ)cos[ω τ +φ (t)−φ (t−τ)] 0,j 0 j j These ond ondition(ii)isnotproperlyde(cid:28)ned be ause N lasers the Lang-Kobayashi equations (19) (cid:21) in ontrast to the +σ w E (t−τ)cos[ω τ +φ (t)−φ (t−τ)], j,k 0,k 0 j k iteratedequations(1)(cid:21)aredi(cid:27)erentialequationsandthe Xk=1 internal dynami s is not only the (cid:28)rst termon the right- (16) handside ofEq. (19)but isalso ontainedin the urrent x v state . Thereforethedenominator in(20v)i=sn1o8t0giv−en1 and has to be hosen reasonably. We took ns d 1 ε 0 1 λ + σ φ (t)= α G n (t) so is between and [be ause the sum (the j N j dt E2 (ltef−τ) strengthvo=f t1h8e0re-i−n1je ted light) should not ex eed the −λ 0,j sin[ω τ +φ (t)−φ (t−τ)] value of ns in our ase℄. 0 j j E (t) 0,j In order to measure syn hronization, we averaged the E 1 0,j N amplitudes of the ele tri (cid:28)elds, , in ns intervalls lasers E0,k(t−τ) ∆ = 0 −σ wj,k sin[ω0τ +φj(t)−φk(t−τ)], and al ulated the un-shifted, , ross orrelation E (t) Xk=1 0,j fun tion de(cid:28)ned by (17) C∆ = hxt·yt−∆i−hxihyi . xy hx2i−hxi2 · hy2i−hyi2 (21) d p p dtnj(t)=(p−1)γNsol−γnj(t)−[Γ+GNnj(t)]E02,j(t), A ross orrelationover0.99 anberegardedassyn hro- (18) nization in the ase of lasers. Figure 9 shows both the parameter transformation, E (t) φ (t) {λ,σ} {ε,κ} 0,j j where and are the amplitude and the Eq.(20), from to andthemapping,Eq.(8), E (t) = j slowly varying phase of the ele tri (cid:28)eld from two mutually oupled units to driven ones. From E (t)exp{[ω τ +φ (t)]} n (t) 0,j 0 j j i and is the arrier num- the omparison of this (cid:28)gure with Fig. 2, the analogy of j =1,...,N berabovethevalueforasolitarylaser, laserλs. laserequationstoiteratedmaps anbeseen. Besidesthe The strength of the self-feedba k is determined by , analogy of features des ribed in se tion II, one an also while the strength of the external oupling is de(cid:28)ned (cid:28)nd this analogy for the features des ribed in se tions σ by . The equations above over the general ase III, V and VI. 7 140 1 120 0.8 100 0.6 80 σ κ / n s −1 60 0.4 40 0.2 20 00 20 40 60 λ /8n 0 s −1 100 120 140 00 0.2 0.4 ε 0.6 0.8 1 ⇆ ⇆ (a) A Bbeforetransformation(20) (b) A B 1 1 0.8 0.8 0.6 0.6 ∼ κ κ 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 ε 0.6 0.8 1 0 0.2 0.4 ε 0.6 0.8 1 ⇆ κ→κ˜=2κ−1 ′← → ( ) A Baftermapping (d) A S A F◦IG. 9: Phase diagrams forCth>e L0.a9n9g-Kobayashi laser equations. Every ir le/point represents one simulation. Open ir les ( ) show a ross orrelation , whi h an be regarded as syn hronization in the ase of the laser equations. Figure (a) shows results for two mutually oupled lasers before applying transformation (20), wheras (b) shows the same diagram after theparametertransformation. Thesyn hronizationregion oftwomutually oupledunits(b) anbemapped[usingEq.(8)℄to a region ( )whi h issimilar to thesyn hronization region ofa driver/re eiver system(d). Aftertheparametertransformation (20), the syn hronization regions for the Lang-Kobayashi equations, (cid:28)gures (b), ( ) and (d), look very similar to the ones for theBernoulli maps,Fig. 2. VIII. SUMMARY an be de omposed into a few sublatti es. Ea h sublatti e is ompletely syn hronized, but di(cid:27)erent sublatti es are only weakly orrelated. Syn hronization is relayed Small networks of haoti units with time-delayed by di(cid:27)erent haoti traje tories. The traje toriesofea h ouplings show interesting patterns of haos syn hrony. sublatti e have identi al statisti al properties. Thus the These patterns are stable attra tors of the network dy- symmetry of the latti e is not broken. There are solu- nami s. tions of the dynami equations whi h break the symme- The phase diagram of two and four units with mu- try of the latti e. However, we always found that these tual ouplings has been related to the properties of a solutions are unstable. Hen e we postulate that stable single driven haoti unit. Two intera ting units with patternsof haossyn hronypossessthesymmetryofthe self-feedba k an syn hronize ompletely, without time orresponding latti e. shift,evenifthedelaytimeisextremelylarge. Whenthe Syn hronization depends on the symmetry of the net- haoti traje toryoftwointera tingunitsisre ordedand work. When the symmetry of an asymmetri hain is usedtodriveasingleidenti alunit, itturnsoutthatthe restoredbyaddingunits,sublatti eor ompletesyn hro- driven unit syn honizesonlyin asmall part ofthe phase nization is restored, too. diagram. Hen e intera tion is di(cid:27)erent from drive. Finally, a bi-partite network, where the two parts are Sublatti e syn hronization is found for latti es whi h oupled by a single mutual signal, shows pairwise om- 8 plete syn hronization, whereas the units of ea h part do ingasingleunitdestroysthesyn hronizationofthewhole not syn hronize. Ea h unit responds to the weak on- network. tribution of its partner in the other part of the network. The work of Ido Kanter is partially supported by the Pairwise syn hronization is a ooperative e(cid:27)e t: Detun- Israel S ien e Foundation. [1℄ A. Pikovsky, M. Rosenblum, and J. Kurths, Syn hro- [15℄ S.R.CampbellandD.Wang,Physi aD111,151(1998). nization,auniversal on eptinnonlinears ien es (Cam- [16℄ M. Rosenbluh, Y. Aviad, E. Cohen, L. Khaykovi h, bridge UniversityPress, Cambridge, 2001). W. Kinzel, E. Kopelowitz, P. Yoskovits, and I. Kanter, [2℄ H.G.S husterandW.Just,Deterministi Chaos (Wiley Phys. Rev. E 76, 046207 (2007). VCH, Weinheim,2005). [17℄ F.M.Atay,J.Jost, andA. Wende,Phys.Rev.Lett.92, [3℄ L.M.Pe oraandT.L.Carroll, Phys.Rev.Lett.64,821 144101 (2004). (1990). [18℄ I. Matskiv, Y. Maistrenko, and E. Mosekilde, Physi a D [4℄ K. M. Cuomo and A. V. Oppenheim, Phys. Rev. Lett. 199, 45 (2004). 71, 65 (1993). [19℄ C.MasollerandA.C.Martí,Phys.Rev.Lett.94,134102 [5℄ A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, (pages 4) (2005). P. Colet, I. Fis her, J. Gar ía-Ojalvo, C. R. Mirasso, [20℄ D.Topaj, W.-H.Kye,andA.Pikovsky,Phys.Rev.Lett. L. Pesquera, and K. A. Shore, Nature 438, 343 (2005). 87, 074101 (2001). [6℄ E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, [21℄ J. Kestler, W. Kinzel, and I. Kanter, Phys. Rev. E 76, L. Khaykovi h, W. Kinzel, and I. Kanter, Phys. Rev. 035202 (pages 4) (2007). E74, 046201 (2006). [22℄ M. Golubitsky and I. Stewart, Bull. Amer. Math. So . [7℄ E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, Phys. 43, 305 (2006). Rev. E 72, 016214 (2005). [23℄ S. Lepri, G. Gia omelli, A. Politi, and F. T. Are hi, [8℄ I. Kanter, N. Gross, E. Klein, E. Kopelowitz, Physi a D 70, 235 (1993). P. Yoskovits, L. Khaykovi h, W. Kinzel, and M. Rosen- [24℄ R. Lang and K. Kobayashi, IEEE J. QuantumEle tron. bluh,Phys.Rev. Lett. 98, 154101 (2007). 16, 347 (1980). [9℄ E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, [25℄ V. Ahlers, U. Parlitz, and W. Lauterborn, Phys. Rev. E L. Khaykovi h, and I. Kanter, Phys. Rev. E 73, 066214 58, 7208 (1998). (2006). [26℄ H. D. I. Abarbanel, N. F. Rulkov, and M. M. Sush hik, [10℄ I. Fis her, R. Vi ente, J. M. Buldu, M. Peil, C. R. Mi- Phys. Rev. E 53, 4528 (1996). rasso, M. C. Torrent, and J. Gar ía-Ojalvo, Phys. Rev. [27℄ S.Bo aletti,J.Kurths,G.Osipov,D.L.Valladares,and Lett. 97, 123902 (2006). C. S. Zhowuj,,jPh=ys0.Rep. 366, 1 (2002). [11℄ S. Sivaprakasam, J. Paul, P. S. Spen er, P. Rees, and [28℄ Usually be ause the self-feedba k is already λ K.A. Shore, Opt. Lett. 28, 1397 (2003). taken injto a ount by the term weighted bywj.,jO=nl1y if [12℄ M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, J. alaser getsnoinputfromotherlasers,then to Opt. So . Am.B 23, 846 (2006). ompensateforthemissingexternalinput.(Alternatively λ [13℄ A. Cho, S ien e 314, 37 (2006). one ould in rease the orresponding .) [14℄ A. K. Engel, P. König, A. K. Kreiter, and W. Singer, S ien e 252, 1177 (1991).