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Physics Letters A 315 (2003) 101–108

www.elsevier.com/locate/pla

Coexistence and switching of anticipating synchronization alag synchronization in an optical system

Liang Wua,b, Shiqun Zhua,b,∗

a China Center of Advanced Science and Technology (World Laboratory), PO Box 8730, Beijing 100080, PR Chinab School of Physical Science and Technology, Suzhou University, Suzhou, Jiangsu 215006, PR China 1

Received 22 January 2003; received in revised form 17 June 2003; accepted 18 June 2003

Communicated by C.R. Doering

Abstract

The chaotic synchronization between two bi-directionally coupled external cavity single-mode semiconductor linvestigated. Numerical simulation shows that anticipating synchronization and lag synchronization coexist and switcheach other in certain parameter regime. The anticipating time with different effects that were discussed quite differenprevious theoretical analysis and experimental observation is determined by the involved parameters in the system. 2003 Elsevier B.V. All rights reserved.

PACS: 05.45.Xt; 42.55.Px; 42.65.Sf

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1. Introduction

In recent years, much attention has been paid tochaotic synchronization in nonlinear systems [1–There is much prospect of chaotic synchronizationing applied in various ways, especially in secure comunication where many works have been conducand a lot of progress has been achieved [3–5]. Infew years, the theory of chaotic synchronizationbeing utilized in the research of neural networks [6

Chaotic synchronization was introduced as stuing the dynamics of several classical models [1–such as Lorenz model. Then the experimental dem

* Corresponding author.E-mail addresses: liangwu@suda.edu.cn (L. Wu),

szhu@suda.edu.cn (S. Zhu).1 Mailing address.

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/S0375-9601(03)01004-1

stration of the chaotic synchronization was giventhe electrical systems and optical systems success[3–5]. Particular emphasis is put upon the synchnization in the chaotic external cavity semicondutor lasers [5,7–15] because of their ability to genate high-dimensional chaos and their ease of option. Several years ago, the investigation in chasynchronization was confined tolag synchronization[7,8]. There is a retardation timeτc between the outputs of two lasers due to the finite time for the ligto travel from the master to the slave. Recently, ifound that there is alsoanticipating synchronization[9–15] other than lag synchronization, the intensitythe slave is synchronized to the future intensity ofmaster. That means the slave can anticipate the dynics of the master.

Although anticipating synchronization is counteintuitive, its existence can be illuminated both the

.

102 L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108

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retically and experimentally [9–15]. Voss discoveranticipating synchronization in some simple modfirst [9]. Following this discovery, Masoller predicteanticipating synchronization in unidirectionally copled laser system [10] and showed that the anpating timeτa should be equal to the difference btween round-trip timeτ of the light in the externacavity of the master laser and the retardation tiτc , i.e., τa = τ − τc. Recently Sivaprakasam et al. rported the experimental demonstration of anticipatchaotic synchronization and lag synchronization ibi-directionally coupled external cavity semicondutor laser system [11]. They found that the anticipattime is irrespective of the external-cavity round ttime τ , i.e., τa = τc [11]. At present there is no fullytheoretical explanation of the difference betweentheoretical anticipating time and the experimentalservation [12].

In this Letter, the synchronization between twbi-directionally coupled chaotic external cavity semconductor lasers is investigated. Numerical simutions show very interesting results. (1) Anticipatisynchronization and lag synchronization coexist aswitch between each other in a system. Sometithe system is inclined to anticipating synchronizatisometimes to lag synchronization. These two kindchaotic synchronization may also switch between eother. The long-term behavior is decided by thevolved parameters such as the coupling strengthsthe feedback rate in the master. (2) As an imporcharacter of the system, the anticipating time is adetermined by the parameters. The parametric spis divided into several zones. In one of these zonanticipating time is the same as derived in theoretanalysis reported in previous papers. In another zanticipating time is in agreement with experimenobservation.

2. Theoretical model

The dynamics of two bi-directionally coupled sigle-mode semiconductor lasers with only the malaser subjected to external optical feedback candescribed by the widely utilized Lang–Kobayas

equations [16]

(1)

dEm

dt= km(1+ iαm)[Gm − 1]Em(t)

+ ηsmEs(t − τc)× exp[−i(ωsτc + �ωt)]

+ γmEm(t − τ )× exp(−iωmτ)+ βmξm(t),

(2)dNm

dt= jm −Nm −Gm|Em|2

τnm,

(3)

dEs

dt= ks(1+ iαs)[Gs − 1]Es(t)+ ηmsEm(t − τc)

× exp[−i(ωmτc + �ωt)] + βsξs(t),

(4)dNs

dt= js −Ns −Gs |Es |2

τns.

Where subscriptsm ands denote the master and thslave respectively,E is the slowly varying complexfield, andN is the normalized carrier density. Thsecond term on the right-hand side in both Eqs.and (3) corresponds to the bi-directional couplingkis the cavity loss,α the linewidth enhancement factoG = N/(1 + ε|E|2) is the optical gain,ε is the gainsaturation coefficient,γ is the feedback rate,η is thecoupling strength and subscriptsm indicates that thecoupling is from the slave to the master,ms is from themaster to the slave,ω is the optical frequency withoufeedback,�ω = ωs − ωm is the frequency detuninbetween two lasers,τ is the external cavity round tritime in the master, andτc is the time for the light totransmit from the master to the slave,ξ is independencomplex Gaussian white noise, andβ measures thenoise intensity,j is the normalized injection currenandτn is the carrier lifetime. For simplicity, we machooseβm = βs = 0.

3. Coexistence and switching of anticipating andlag synchronization

According to Eqs. (1)–(4), we numerically simulathe dynamics of two bi-directionally coupled semconductor lasers. On the plane of field intensitiesIs(= |Es |2) of the slave versusIm (= |Em|2) of the mas-ter, the quantity arctan(Is/Im) can be used to check thquality of chaotic synchronization betweenIs andIm.If there is a straight line with constant slope onIs–Implane, arctan(Is/Im) is a constant. If there is a spindshape onIs–Im plane, arctan(Is/Im) is not a constant

L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108 103

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There is certain deviation about the straight line. Ththe varianceQ = σ 2(arctan(Is/Im)) of the quantityarctan(Is/Im) can be used to represent the synchnization quality [11]. Perfect synchronization is reresented byQ = 0. On the other hand, a high valuof varianceQ represents a poor synchronization. Bcause of the effect of anticipating synchronizationlag synchronization, we would derive a good synchnization represented by a low varianceQ if the masterlaser output is shifted relatively to the slave outputan appropriate timeτs . In this Letter,τs > 0 if the mas-ter laser output is shifted forward relative to the slaoutput. In this case, a low varianceQ represents an anticipating synchronization. If the master laser outpushifted backward relative to the slave output,τs < 0,a low varianceQ in this case represents a lag synchnization.

We find that anticipating synchronization and lsynchronization coexist in the bi-directionally couplsemiconductor laser system. The coexistence caseen from Fig. 1. Fig. 1 shows the dependence ofvarianceQ on the time shiftτs of the master laserThere are two dips of very low variance on each saboutτs = 0. The low variance on the right (τs > 0)indicates the anticipating synchronization while tlow variance on the left (τs < 0) indicates the lagsynchronization.

The coexistence of anticipating and lag synchnization is plotted in detail in Fig. 2. Fig. 2(a-1

Fig. 1. Varianceσ2(arctan(Is/Im)) as a function of the timeshift τs . The parameters are chosen askm = ks = 500 ns,αm = αs = 3, εm = εs = 0.1, τnm = τns = 1.0 ns,jm = js = 1.01,ωm = ωs = 0.3 rad · ns−1, γm = 2 ns−1, ηms = 5 ns−1,ηsm = 3 ns−1, τ = 6.7 ns,τc = 3.5 ns.

and (a-2) are plots of the time traces of the fiintensities of the master (Fig. 2(a-1)) and the sl(Fig. 2(a-2)) lasers respectively from 200 ns to 350From the whole traces, it seems that on the left pthe lag synchronization dominates. While on the ripart, the anticipating synchronization dominates.the middle, the chaotic synchronization is switchfrom lag to anticipating synchronization. From tboxes marked by ‘A’, ‘a’ and ‘B’, ‘b’ in Fig. 2(a-1)to (a-4), it seems that the anticipating and lag schronization exist simultaneously. This can be fther illustrated when the segment of the time trafrom 250 ns to 300 ns is plotted in Fig. 2(a-3)(a-5). From Fig. 2(a-3) and (a-4), it seems thatlag synchronization dominates in boxes marked byand ‘a’. While in boxes marked by ‘B’ and ‘b’, thtime traces show anticipating synchronization. This a transition from lag synchronization to anticiping synchronization with a dropout as a connectiIn Fig. 2(a-3) to (a-5), each trace can be naturallyvided into many small zones, labelled by M1, M2, MM4, . . . in the master laser, and S1, S2, S3, S4, . . .

in the slave laser. We find the similaritiesM1 ⇒ S2,M2 ⇒ S3,M3 ⇒ S4 before the dropout,M9 ⇒ S10,M10⇒ S11,M11⇒ S12 after the dropout (solid linin Fig. 2(a-5)). All the similaritiesM(i)⇒ S(i + 1)form the lag synchronization. On the other hand,fore and after the dropout, the similaritiesS(i) ⇒M(i + 1), i.e.,S1 ⇒M2, S2 ⇒M3 and so on (dasline in Fig. 2(a-5)) form the anticipating synchroniztion. These two kinds of synchronization coexistthe same time. Before the dropout the lag synchnization is better than the anticipating one. Whileter the dropout, the anticipating synchronization doinates the dynamics. There is a transition from laganticipating synchronization in the area of dropoAmong M6, M7, M8 and S6, S7, S8 in the areadropout, neither lag nor anticipating synchronizatoccurs (Fig. 2(a-5)).

It is also very interesting to note that there is a ctinuous switching process between lag and anticiing synchronization. For a segment, such asM1 ⇒S2 ⇒M3,M1 ⇒ S2 indicates that the master leathe slave, whileS2 ⇒M3 indicates that the slave bcomes the leader. The leader role switches frommaster to the slave at S2 and then switches bacthe master at M3 withinS2 ⇒ M3 ⇒ S4. It seemsthat the coexistence of two kinds of synchronizat

104 L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108

Fig. 2. The time traces of the master laser field intensity (a-1), (b-3) and the slave laser field intensity (a-2), (b-4). (a-1) and (a-2) are in therange of 200–350 ns; (b-3) and (b-4) are in the range of 500–1000 ns. The parameters areγm = 10 ns−1, ηms = 10 ns−1, ηsm = 2 ns−1. Otherparameters are the same as that in Fig. 1.

L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108 105

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is based on the continuous switching process in bihelix structure showed in Fig. 2(a-5).

Fig. 2(b-3) and (b-4) are plots of the intensity timtraces of the master laser (Fig. 2(b-3)) and slave la(Fig. 2(b-4)) from 500 ns–1000 ns. There are tdropouts at about 570 and 915 ns, respectively.clear that the master laser leads the slave laser at a570 ns (Fig. 2(b-1) and (b-2)) while the master lagshind the slave at about 915 ns (Fig. 2(b-5) and (b-This is consistent with recent experimental obsertions performed by Sivaprakasam et al. [15]. In thexperiment, the transition from anticipating to lag sychronization can be achieved by increasing the opeing temperature in the slave laser. At an intermedtemperature the slave laser toggles between antiction and lag synchronization. The theoretical analyshown in Fig. 2(b-3) and (b-4) is almost exactly tsame as those shown in Fig. 4 in Ref. [15].

In addition, the similarities betweenmi and si+1,or si andmi+1 fluctuate. The fluctuations indicate thsometimes the system behavior is inclined to antpating synchronization orientation, sometimes it isclined to lag synchronization orientation. Four tracin Fig. 2 share one dynamical trajectory. Fig. 2(aand (a-2) are plots of the intensities in the masterslave lasers respectively in the range of 200–350Fig. 2(b-3) and (b-4) are plots in the range of 501000 ns, typically showing lag synchronization orietation between 560–590 ns and anticipating synchnization orientation in the range of 900–930 ns.

It is seen from Fig. 2 that the switching procefrom lag to anticipating chaotic synchronization occdeterministically since there is no noise in numerisimulations. This self-induced switching processtween lag and anticipating synchronization may beexample of the generic phenomenon “chaotic itinercy” in complex systems [17]. It seems that the perioswitching process is mainly induced and enhancedthe feedback in the master laser.

Whether the long term behavior in the systemanticipating or lag synchronization is decided byinvolved laser parameters. In this Letter the couplstrengths in both directions and the feedback ratthe master are discussed. We use"Q to indicate theorientation of the synchronization in a sufficient loterm with

(5)Qa =Qmin (τs > 0),

t

(6)Ql =Qmin (τs < 0),

(7)"Q=Qa −Ql,

where the subscriptsa and l denote anticipating anlag synchronization respectively. The anticipating schronization orientation is characterized by positvalue of"Q ("Q > 0), while lag synchronizationorientation is characterized by negative value of"Q("Q< 0).

In Fig. 3(a) we plot normalized"Q as a function ofthe feedback rate in the master (γm) and the couplingstrength from the slave to the master (ηsm). The cou-pling strength from the master to the slave is choto beηms = 5.0. We can find"Q> 0 in two regionslabelled byI1 andI2, where the long-term system bhavior is inclined to anticipating synchronization oentation. In regionI1, the feedback rate in the masterrelatively small, while the injectionηsm into the mas-ter from the slave is strong. In regionI2, the injectionηsm is small. It is approximately equivalent to unidrectional coupling, the condition assumed in the thretical papers [5,10]. In another region labelled by"Q< 0 indicates the long-term system behavior isclined to lag synchronization orientation. In region Ithe long-term system dynamics is adiaphorous, resented by"Q≈ 0.

In Fig. 3(b) we plot"Q as a function of thecouplings between slave and master in both direct(ηms , ηsm). We find that almost all the plane has"Q�0, which indicates the long-term system behavioinclined to lag synchronization in most cases. It is sthat only the master has feedback which breakssymmetry of the system.

The synchronization quality represented byvarianceQ = σ 2 of two particular points on theplane of (ηms , γm) is plotted in Fig. 4(a) and (b)In Fig. 4(a), the parameters are chosen to beγm =2.0 andηms = 2.0. This is a point located in regioII shown in Fig. 3(a). It is clear that the quality olag synchronization is better than that of anticipatsynchronization with a smaller value of the left dThis means that the long term behavior of the sysis inclined to lag synchronization. In Fig. 4(b), thparameters are chosen to beγm = 5.0 andηms = 0.5.This is a point in regionI2. It is seen that the quality oanticipation synchronization is better than that ofsynchronization with a smaller value of the right d

106 L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108

the

Fig. 3. (a) The calculated"Q as a function ofγm andηsm. The coupling strength from the master to the slave is chosen to beηms = 5 ns−1.(b) The calculated"Q as a function of the couplings in both directions (ηms , ηsm). The feedback rate isγm = 5 ns−1.

Fig. 4. Varianceσ2 as a function of the time shiftτs . The parameters are chosen as (a)γm = 2.0 ns−1, ηsm = 2.0 ns−1. This a point located inregion II shown in Fig. 3(a). (b)γm = 5.0 ns−1, ηsm = 0.2 ns−1. This a point located in regionI2 shown in Fig. 3(a). Other parameters aresame as those in Fig. 3(a).

tem

an

o-t

by

tici-per-m-bi-n-

the

This means that the long term behavior of the sysis inclined to anticipation synchronization.

Similar discussions corresponding to Fig. 3(b) calso be obtained.

4. Anticipating time

In Fig. 1, the left dip corresponding to lag synchrnization is located atτs = −3.66 ns. It is noted tha

the lag timeτl = 3.66 ns is not identical toτc = 3.5 nsperfectly because the lag time is slightly influencedthe coupling strength ofηsm [18].

There is a difference between the theoretical anpating time reported in previous papers and the eximental observation [5,8,10,11]. Our numerical siulation shows that the two cases both occur in adirectionally coupled chaotic external cavity semicoductor laser system. In our numerical simulation,parameters are chosen to beτc = 3.5 ns,τ = 6.7 ns.

L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108 107

Fig. 5. (a) The calculated anticipating time as a function ofγm andηsm. (b) The calculated anticipating time as a function ofηms andηsm.Parameters are the same as that in Fig. 3.

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ncetherthe

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Fig. 5(a) plots the calculated anticipating time (τa =τs , when τs > 0 andQτs = Qmax) as a function ofthe feedback rate in the master (γm) and the couplingstrength from the slave to the master (ηsm). In termsof τa one can clearly distinguish two regions. In rgion I of Fig. 5(a), the coincidence with the expemental results of Ref. [11] (τa = τc = 3.5 ns) is due tothe fact that in such region the injectionηsm (i.e., fromslave to master) is strong, which makes operating cditions similar to those of the experiments reportedthat paper (bidirectional coupling) [11]. In regionof Fig. 5(a), the coincidence with previous theorecal papers (τa = τ − τc = 3.2 ns) is due to the facthat in such region the injectionηsm is small. It is ap-proximately equivalent to unidirectional coupling, tcondition assumed in such theoretical papers [5,10

Fig. 5(b) shows the dependence of anticipating tion the couplings in both directions. There are tregions that can be distinguished clearly. In regiothe anticipating time is about 3.5 ns, i.e.,τa = τc. Inregion II τa is about 3.2 ns, i.e.,τa = τ − τc. Wenoticed that from the point (ηms = 4, ηsm = 4), ifone of the couplingηms andηsm is increased or bothηms andηsm are increased simultaneously, a transitfrom region II to region I can occur. This means thatransition of anticipating timeτ − τc to τc can happenby increase the coupling of the system. We also fo

that the transition does not happen when the coupis weak withηms < 3.

5. Discussion

This Letter demonstrates numerically the coextence of anticipating and lag chaotic synchronizatin coupled laser diodes, in which only one lasersubject to delayed feedback. The coexistence appsimultaneously at any moment. The experimentalticipating time is different from the theoretical resusince the conditions of the theoretical analysis andperimental demonstration are in different regions.the laser system, there is not a full symmetry, sione of the lasers is subject to feedback, and the olaser is not subject to feedback, fact which breakssymmetry. Due to the asymmetry appeared in thetem, the quality of chaotic synchronization is not vegood.

Acknowledgements

It is a pleasure to thank Weijian Gao for his mahelpful discussions of numerical calculations. Thenancial support from the Natural Science Founda

108 L. Wu, S. Zhu / Physics Letters A 315 (2003) 101–108

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