- Email: [email protected]
Dual and dual-cross synchronizations in chaotic systems
Optics Communications 216 (2003) 179–183 www.elsevier.com/locate/optcom
Dual and dual-cross synchronizations in chaotic systems E.M. Shahverdiev 1, S. Sivaprakasam, K.A. Shore * School of Informatics, University of Wales, Bangor, Dean Street, Bangor, LL57 1UT, Wales, UK Received 19 March 2002; received in revised form 23 May 2002; accepted 22 November 2002
Abstract In this paper we investigate dual synchronization and dual-cross synchronization between a transmitter and a receiver, each consisting of two master and slave systems. We demonstrate our approach using the example of synchronization of chaotic external cavity laser diodes. In the synchronization scheme under consideration the joint error signal, which is the difference between the sum of the outputs from the master systems in the transmitter and the sum of outputs from the slave systems in the receiver, is fed into each slave system and synchronizations between the master lasers and slave lasers under possible configurations are studied. By use of error dynamics we derive existence conditions for synchronization between the transmitter and receiver. We find that in all the studied cases synchronization between the transmitter and receiver occurs when the sum of the contributions from individual master (or slave) systems in the transmitter (or receiver) is equal to unity. For the first time we demonstrate that an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal. These findings are considered to be of interest in multichannel communication schemes. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.)a; 05.45.Xt; 42.65.Sf; 42.55.Ah
The seminal papers [1] on chaos synchronization have stimulated a wide range of research activity: a recent comprehensive review of such work is found in [2]. Application of chaos control theory can be found in secure communications, optimization of nonlinear system performance and modeling brain activity and pattern recognition *
Corresponding author. Fax: +44-1248-361429. E-mail address: [email protected] (K.A. Shore). 1 Present address: Institute of Physics, 370143 Baku, Azerbaijan.
phenomena [2]. Because communication schemes on hyperchaos are considered to be more reliable, a particular focus of such work is on the development of secure optical communication systems based on control and synchronization of laser hyperchaos [3]. For practical use of this approach particular emphasis is given to the use of chaotic external cavity semiconductor lasers, see e.g. [3], because laser systems with optical feedback are prominent representatives of time-delay systems, which can generate hyperchaos. Due to their practicality, investigation of multichannel communication schemes is of consider-
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02286-1
180
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
able research interest, see e.g. [4] and references therein. Recent work treats the problem of simultaneous synchronization of two different pairs of chaotic oscillators with a single scalar signal, which is referred to as dual synchronization [5]; the outputs of a pair of master oscillators are linearly coupled and fed to a pair of slave oscillators. The signals from the slave oscillators are coupled in a similar way and subtracted from the signal received from the masters. The difference signal (or joint error signal) is then injected into each slave oscillator. When the slaves are synchronized to their respective masters, the difference (error) signal is zero. We point out that there is no direct coupling between the two master oscillators and therefore the dual synchronization case is different from the problem of using a scalar signal to synchronize hyperchaotic or multidimensional oscillators. In [5], by evaluating the Lyapunov exponent of the receiver with respect to the synchronized state the stability condition of dual synchronization is derived as 1 þ 2 ¼ 1 (where 1 and 2 are coupling parameters: 0 6 1 , 2 6 1) for a number of different classes of maps and for oscillators modelled by delay-differential equations. In this paper by the use of error dynamics we derive the existence conditions of both dual and dual-cross synchronizations between the transmitter and receiver each one consisting of two master and slave systems. We demonstrate our approach using the example of synchronization of chaotic external cavity laser diodes. In dual synchronization, as in [5], we synchronize the slave lasers to their respective master lasers. In dual-cross synchronization one synchronizes the slave lasers to their complementary master lasers. We obtain that 1 þ 2 ¼ 1 is also an existence condition for dual synchronization. We also establish an existence condition for dual-cross synchronization: 1 ¼ 2 ¼ 1=2. We find that in all the cases studied here synchronization between the transmitter and receiver occurs when the sum of contributions from individual master (or slave) systems in the transmitter (or receiver) is equal to unity. For the first time we demonstrate that an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal.
External cavity laser diodes are commonly modelled with the Lang–Kobayashi equations [6]. Consider the following systems of the Lang–Kobayashi equations for the real electric field amplitude EðtÞ, slowly varying phase UðtÞ and the carrier number nðtÞ for: the master laser pair M1, M2 dEM1 1 ¼ GN nM1 EM1 2 dt þ kM1 EM1 ðt sÞ cosðx0 s þ UM1 ðtÞ UM1 ðt sÞÞ ¼ F ðtÞ dUM1 1 ¼ aGN nM1 2 dt EM1 ðt sÞ sinðx0 s þ UM1 ðtÞ kM1 EM1 ðtÞ UM1 ðt sÞÞ ¼ F1 ðtÞ;
ð1Þ
dnM1 2 ¼ ðp 1ÞJth cnM1 ðtÞ ðC þ GN nM1 ÞEM1 ; dt dEM2 1 ¼ GN nM2 EM2 2 dt þ kM2 EM2 ðt sÞ cosðx0 s þ UM2 ðtÞ UM2 ðt sÞÞ ¼ GðtÞ; dUM2 1 ¼ aGN nM2 2 dt EM2 ðt sÞ sinðx0 s þ UM2 ðtÞ kM2 EM2 ðtÞ UM2 ðt sÞÞ ¼ G1 ðtÞ;
ð2Þ
dnM2 2 ¼ ðp 1ÞJth cnM2 ðtÞ ðC þ GN nM2 ÞEM2 ; dt and slave laser pair S1, S2 dES1 1 ¼ GN nS1 ES1 2 dt þ kS1 ES1 ðt sÞ cosðx0 s þ US1 ðtÞ; US1 ðt sÞÞ þ eðtÞ ¼ f ðtÞ þ eðtÞ; dUS1 1 ¼ aGN nS1 2 dt ES1 ðt sÞ sinðx0 s þ US1 ðtÞ kS1 ES1 ðtÞ US1 ðt sÞÞ þ e1 ðtÞ ¼ f1 ðtÞ þ e1 ðtÞ; dnS1 2 ¼ ðp 1ÞJth cnS1 ðtÞ ðC þ GN nS1 ÞES1 ; dt
ð3Þ
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
dES2 1 ¼ GN nS2 ES2 2 dt þ kS2 ES2 ðt sÞ cosðx0 s þ US2 ðtÞ US2 ðt sÞÞ þ eðtÞ ¼ gðtÞ þ eðtÞ; dUS2 1 ¼ aGN nS2 2 dt ES2 ðt sÞ sinðx0 s þ US2 ðtÞ kS2 ES2 ðtÞ US2 ðt sÞÞ ¼ g1 ðtÞ þ e1 ðtÞ;
ð4Þ
dnS2 2 ¼ ðp 1ÞJth cnS2 ðtÞ ðC þ GN nS2 ÞES2 ; dt where GN is the differential optical gain; s the lasersÕ external cavity round-trip time; a the linewidth enhancement factor; c the carrier decay rate; C the cavity decay rate; p the pump current relative to the threshold value Jth of the solitary laser; x0 the angular frequency of the solitary laser; and k is the feedback rate. We define the joint error signal eðtÞ to be fed into each slave laser as the difference signal between the sums of the master and slave lasersÕ outputs uðtÞ and vðtÞ (Fig. 1), respectively, eðtÞ ¼ uðtÞ vðtÞ ¼ 1 F ðtÞ þ 2 GðtÞ ð1 f ðtÞ þ 2 gðtÞÞ; e1 ðtÞ ¼ u1 ðtÞ v1 ðtÞ
ð5Þ
¼ 1 F1 ðtÞ þ 2 G1 ðtÞ ð1 f1 ðtÞ þ 2 g1 ðtÞÞ; where 1 and 2 are coupling parameters or contribution weights from the master lasers (or slave
Fig. 1. Schematic diagram of dual synchronization: EM1 ¼ ES1 , EM2 ¼ ES2 and dual-cross synchronization: EM1 ¼ ES2 , EM2 ¼ ES1 . In the synchronization scheme under consideration the joint error signal eðtÞ, which is the difference between the sum of the outputs from the master systems M1, M2 in the transmitter and the sum of outputs from the slave systems S1, S2 in the receiver, is fed into each slave system and synchronizations between the master lasers and slave lasers under possible configurations are studied.
181
lasers) to the joint error signal eðtÞ. In the following derivation of the existence conditions for dual and dual-cross synchronizations between chaotic laser pairs we will follow the approach developed in [7,8] of investigating the error dynamics. Earlier we have used this approach to derive a necessary (existence) condition for lag synchronization between unidirectionally coupled chaotic external cavity semiconductor lasers [7] and the Ikeda model [8]. First consider the case of dual synchronization. We write the complete dual synchronization manifold as EM1 ðtÞ ¼ ES1 ðtÞ;
nM1 ðtÞ ¼ nS1 ðtÞ;
UM1 ðtÞ ¼ US1 ðtÞðmod 2pÞ; EM2 ðtÞ ¼ ES2 ðtÞ;
ð6aÞ
nM2 ðtÞ ¼ nS2 ðtÞ;
UM2 ðtÞ ¼ US2 ðtÞðmod 2pÞ:
ð6bÞ
In general, sychronization between master (driving) and slave (response) systems strongly depends on the driving variable. For example, as shown in the seminal paper by Pecora–Carroll for the paradigm Lorenz model, driving by the x and y variables results in synchronization between master and slave systems, as the so-called conditional Lyapunov exponents for the response system are negative [1]; driving with the z variable gives rise only to conditional synchronization, as one of the conditional Lyapunov exponents in this case is equal to zero, see, e.g. [9]. Our analysis shows that the electric field amplitude is a good driving variable; indeed as can be seen from the third equations in Eqs. (1)–(3) and (2)–(4), when the differences EM1 ES1 and EM2 ES2 approach zero so do the differences between the corresponding carrier densities. Now consider the deviation from the synchronization manifold e2 ¼ EM1 ES1 and e3 ¼ EM2 ES2 . Then writing Eqs. (1)–(5) in the concise form (which indicate the applicability of the approach developed here to a wide class of continuous dynamical systems) of dEM1 dEM2 ¼ F; ¼ G; dt dt dES1 dES2 ¼ f þ u v; ¼gþuv dt dt
ð7Þ
182
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
and calculating the error dynamics for e2 and e3 we arrive at the system of equations: de2 ¼ F f ðu vÞ; dt de3 ¼ G g ðu vÞ: dt
ð8Þ
Our goal is to achieve synchronization e2 ¼ 0 and e3 ¼ 0. Then with u ¼ 1 F þ 2 G;
v ¼ 1 f þ 2 g;
ð9Þ
one finds that ð1 1 ÞðF f Þ ¼ 2 ðG gÞ; ð1 2 ÞðG gÞ ¼ 1 ðF f Þ:
ð10Þ
Using the expressions for the F , G, f and g from Eqs. (1)–(4) on the synchronization manifolds (6a) and (6b), Eq. (10) can be modified to ð1 1 ÞðkM1 kS1 ÞEM1 ¼ 2 ðkM2 kS2 ÞEM2 ; ð1 2 ÞðkM2 kS2 ÞEM2 ¼ 1 ðkM1 kS1 ÞEM1 :
ð11Þ
From Eq. (11) we obtain that ½ð1 1 Þð1 2 Þ 1 2 ðkM1 kS1 ÞEM1 ¼ 0:
ð12Þ
Thus for kM1; M2 6¼ kS1; S2 we establish the relationship between the coupling weights of the master or slave lasers outputs to achieve dual synchronization: 1 þ 2 ¼ 1:
ð13Þ
We also observe that the condition (13), as mentioned above is fairly general for different types of continuous dynamical systems. Indeed as it follows from Eq. (10) for the system of equations: dx1 dx2 ¼ F; ¼ G; dt dt dy1 dy2 ¼ f þ u v; ¼ g þ u v; dt dt
ð14Þ
with u and v defined above (9), for F 6¼ f and G 6¼ g on the synchronization manifold x1 ðtÞ ¼ y1 ðtÞ, x2 ðtÞ ¼ y2 ðtÞ one can easily arrive at the dual synchronization condition (13). Next we consider dual-cross synchronization, when one synchronizes the slave lasers to their complementary master lasers: EM1 ðtÞ ¼ ES2 ðtÞ and EM2 ðtÞ ¼ ES1 ðtÞ. Following the procedure above
leading to Eq. (10) and to condition (13), we find the existence condition for dual-cross synchronization: 1 ¼ 2 ¼ 1=2:
ð15Þ
Notice that condition (13) also holds for the dual-cross synchronization case. Thus we conclude that both dual and dual-cross synchronizations are possible if the coupling parameters (or contribution weights) to the joint error signal from the oscillators to be synchronized become equal. In other words symmetry in contributions to the coupling plays a crucial role in the synchronization phenomenon studied here. This conclusion is also valid for larger number of constituent parts of the transmitter and receiver. Indeed if we have three master ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ and three slave ðy1 ðtÞ; y2 ðtÞ; y3 ðtÞÞ oscillators in the transmitter and receiver, respectively, it is possible to establish that the triple synchronization x1 ðtÞ ¼ y1 ðtÞ, x2 ðtÞ ¼ y2 ðtÞ, x3 ðtÞ ¼ y3 ðtÞ existence condition is 1 þ 2 þ 3 ¼ 1. Triple-cross synchronization, e.g. x1 ðtÞ ¼ y3 ðtÞ; x2 ðtÞ ¼ y1 ðtÞ; x3 ðtÞ ¼ y2 ðtÞ, takes place if 1 ¼ 2 ¼ 3 ¼ 1=3. In the case of three and more oscillators we also obtain the case of mixed synchronization, when some of the oscillators are coupled in crossed configuration, others in the uncrossed configuration. For example, in the case of three oscillators in both the transmitter and receiver we can define the following synchronization manifold as a mixed synchronization manifold: x1 ðtÞ ¼ y2 ðtÞ, x2 ðtÞ ¼ y1 ðtÞ, x3 ðtÞ ¼ y3 ðtÞ. Again by use of error dynamics approach one obtains that the mixed synchronizationÕs existence condition is: 2 þ 3 ¼ 1 with 1 ¼ 2 ¼ . In other words, we find that all types of synchronizations exist if the contribution weights to the joint error signal from the oscillators to be synchronized are equal. Generalization to the case of N master and N slave oscillators is straightforward. To summarize, we have investigated dual synchronization and dual-cross synchronization between a transmitter and receiver each consisting of two master and slave systems. By use of error dynamics we have derived the existence conditions for dual and dual-cross synchronizations. For the
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
first time we have established that in all the cases studied synchronization occurs, when the sum of the contribution weights from individual master (or slave) systems in the transmitter (or receiver) is equal to unity; an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal. The synchronization conditions identified here are those which must be satisfied to obtain exact synchronization. For practical applications in chaotic communications it is not essential to meet such a stringent condition. Experimental demonstrations of chaos synchronization using chaotic external cavity laser diodes [3,10] are remarkably robust with synchronization having been maintained for long periods of time using commercially available laser diodes where parameter mismatches are the expectations rather than exceptions. Moreover, these experiments have demonstrated that exact synchronization is not required for successful gigahertz message transmission and extraction [11]. With this experimental experience we consider that practical demonstration of the phenomena examined here is eminently feasible. As such we believe that the present findings will be of interest in the context of multichannel chaotic communication schemes where message decoding is dependent on the synchronization of several chaotic transmitters and receivers.
183
Acknowledgements This work is supported by the UK Engineering and Physical Sciences Research Council Grant GR/R22568/01 and GR/N63093/01. References [1] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821; E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [2] G. Chen, X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, 1998; H.G. Schuster (Ed.), Handbook of Chaos Control, WileyVCH, Weinheim, 1999. [3] R. Roy, K.S. Thornburg Jr., Phys. Rev. Lett. 72 (1994) 2009; S. Sivaprakasam, K.A. Shore, Opt. Lett. 24 (1999) 466; S. Sivaprakasam, K.A. Shore, Opt. Lett. 24 (1999) 1200; S. Sivaprakasam, K.A. Shore, IEEE J. Quantum Electron. 36 (2000) 35. [4] S. Sundar, A.A. Minai, Phys. Rev. Lett. 85 (2000) 5456. [5] Y. Liu, P. Davis, Phys. Rev. E 61 (2000) R2176. [6] R. Lang, K. Kobayashi, IEEE J. Quantum Electron. 16 (1980) 347. [7] S. Sivaprakasam, E.M. Shahverdiev, K.A. Shore, Phys. Rev. E 62 (2000) 7505. [8] E.M. Shahverdiev, S. Sivaprakasam, K.A. Shore, Phys. Lett. A 292 (2002) 320. [9] E.M. Shahverdiev, Phys. Rev. E 60 (1999) 3905. [10] S. Sivaprakasam, E.M. Shahverdiev, P.S. Spencer, K.A. Shore, Phys. Rev. Lett. 87 (2001) 154101. [11] J. Paul, S. Sivaprakasam, P.S. Spencer, P. Rees, K.A. Shore, Electron. Lett. 38 (2002) 28.
Dual and dual-cross synchronizations in chaotic systems E.M. Shahverdiev 1, S. Sivaprakasam, K.A. Shore * School of Informatics, University of Wales, Bangor, Dean Street, Bangor, LL57 1UT, Wales, UK Received 19 March 2002; received in revised form 23 May 2002; accepted 22 November 2002
Abstract In this paper we investigate dual synchronization and dual-cross synchronization between a transmitter and a receiver, each consisting of two master and slave systems. We demonstrate our approach using the example of synchronization of chaotic external cavity laser diodes. In the synchronization scheme under consideration the joint error signal, which is the difference between the sum of the outputs from the master systems in the transmitter and the sum of outputs from the slave systems in the receiver, is fed into each slave system and synchronizations between the master lasers and slave lasers under possible configurations are studied. By use of error dynamics we derive existence conditions for synchronization between the transmitter and receiver. We find that in all the studied cases synchronization between the transmitter and receiver occurs when the sum of the contributions from individual master (or slave) systems in the transmitter (or receiver) is equal to unity. For the first time we demonstrate that an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal. These findings are considered to be of interest in multichannel communication schemes. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.)a; 05.45.Xt; 42.65.Sf; 42.55.Ah
The seminal papers [1] on chaos synchronization have stimulated a wide range of research activity: a recent comprehensive review of such work is found in [2]. Application of chaos control theory can be found in secure communications, optimization of nonlinear system performance and modeling brain activity and pattern recognition *
Corresponding author. Fax: +44-1248-361429. E-mail address: [email protected] (K.A. Shore). 1 Present address: Institute of Physics, 370143 Baku, Azerbaijan.
phenomena [2]. Because communication schemes on hyperchaos are considered to be more reliable, a particular focus of such work is on the development of secure optical communication systems based on control and synchronization of laser hyperchaos [3]. For practical use of this approach particular emphasis is given to the use of chaotic external cavity semiconductor lasers, see e.g. [3], because laser systems with optical feedback are prominent representatives of time-delay systems, which can generate hyperchaos. Due to their practicality, investigation of multichannel communication schemes is of consider-
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02286-1
180
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
able research interest, see e.g. [4] and references therein. Recent work treats the problem of simultaneous synchronization of two different pairs of chaotic oscillators with a single scalar signal, which is referred to as dual synchronization [5]; the outputs of a pair of master oscillators are linearly coupled and fed to a pair of slave oscillators. The signals from the slave oscillators are coupled in a similar way and subtracted from the signal received from the masters. The difference signal (or joint error signal) is then injected into each slave oscillator. When the slaves are synchronized to their respective masters, the difference (error) signal is zero. We point out that there is no direct coupling between the two master oscillators and therefore the dual synchronization case is different from the problem of using a scalar signal to synchronize hyperchaotic or multidimensional oscillators. In [5], by evaluating the Lyapunov exponent of the receiver with respect to the synchronized state the stability condition of dual synchronization is derived as 1 þ 2 ¼ 1 (where 1 and 2 are coupling parameters: 0 6 1 , 2 6 1) for a number of different classes of maps and for oscillators modelled by delay-differential equations. In this paper by the use of error dynamics we derive the existence conditions of both dual and dual-cross synchronizations between the transmitter and receiver each one consisting of two master and slave systems. We demonstrate our approach using the example of synchronization of chaotic external cavity laser diodes. In dual synchronization, as in [5], we synchronize the slave lasers to their respective master lasers. In dual-cross synchronization one synchronizes the slave lasers to their complementary master lasers. We obtain that 1 þ 2 ¼ 1 is also an existence condition for dual synchronization. We also establish an existence condition for dual-cross synchronization: 1 ¼ 2 ¼ 1=2. We find that in all the cases studied here synchronization between the transmitter and receiver occurs when the sum of contributions from individual master (or slave) systems in the transmitter (or receiver) is equal to unity. For the first time we demonstrate that an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal.
External cavity laser diodes are commonly modelled with the Lang–Kobayashi equations [6]. Consider the following systems of the Lang–Kobayashi equations for the real electric field amplitude EðtÞ, slowly varying phase UðtÞ and the carrier number nðtÞ for: the master laser pair M1, M2 dEM1 1 ¼ GN nM1 EM1 2 dt þ kM1 EM1 ðt sÞ cosðx0 s þ UM1 ðtÞ UM1 ðt sÞÞ ¼ F ðtÞ dUM1 1 ¼ aGN nM1 2 dt EM1 ðt sÞ sinðx0 s þ UM1 ðtÞ kM1 EM1 ðtÞ UM1 ðt sÞÞ ¼ F1 ðtÞ;
ð1Þ
dnM1 2 ¼ ðp 1ÞJth cnM1 ðtÞ ðC þ GN nM1 ÞEM1 ; dt dEM2 1 ¼ GN nM2 EM2 2 dt þ kM2 EM2 ðt sÞ cosðx0 s þ UM2 ðtÞ UM2 ðt sÞÞ ¼ GðtÞ; dUM2 1 ¼ aGN nM2 2 dt EM2 ðt sÞ sinðx0 s þ UM2 ðtÞ kM2 EM2 ðtÞ UM2 ðt sÞÞ ¼ G1 ðtÞ;
ð2Þ
dnM2 2 ¼ ðp 1ÞJth cnM2 ðtÞ ðC þ GN nM2 ÞEM2 ; dt and slave laser pair S1, S2 dES1 1 ¼ GN nS1 ES1 2 dt þ kS1 ES1 ðt sÞ cosðx0 s þ US1 ðtÞ; US1 ðt sÞÞ þ eðtÞ ¼ f ðtÞ þ eðtÞ; dUS1 1 ¼ aGN nS1 2 dt ES1 ðt sÞ sinðx0 s þ US1 ðtÞ kS1 ES1 ðtÞ US1 ðt sÞÞ þ e1 ðtÞ ¼ f1 ðtÞ þ e1 ðtÞ; dnS1 2 ¼ ðp 1ÞJth cnS1 ðtÞ ðC þ GN nS1 ÞES1 ; dt
ð3Þ
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
dES2 1 ¼ GN nS2 ES2 2 dt þ kS2 ES2 ðt sÞ cosðx0 s þ US2 ðtÞ US2 ðt sÞÞ þ eðtÞ ¼ gðtÞ þ eðtÞ; dUS2 1 ¼ aGN nS2 2 dt ES2 ðt sÞ sinðx0 s þ US2 ðtÞ kS2 ES2 ðtÞ US2 ðt sÞÞ ¼ g1 ðtÞ þ e1 ðtÞ;
ð4Þ
dnS2 2 ¼ ðp 1ÞJth cnS2 ðtÞ ðC þ GN nS2 ÞES2 ; dt where GN is the differential optical gain; s the lasersÕ external cavity round-trip time; a the linewidth enhancement factor; c the carrier decay rate; C the cavity decay rate; p the pump current relative to the threshold value Jth of the solitary laser; x0 the angular frequency of the solitary laser; and k is the feedback rate. We define the joint error signal eðtÞ to be fed into each slave laser as the difference signal between the sums of the master and slave lasersÕ outputs uðtÞ and vðtÞ (Fig. 1), respectively, eðtÞ ¼ uðtÞ vðtÞ ¼ 1 F ðtÞ þ 2 GðtÞ ð1 f ðtÞ þ 2 gðtÞÞ; e1 ðtÞ ¼ u1 ðtÞ v1 ðtÞ
ð5Þ
¼ 1 F1 ðtÞ þ 2 G1 ðtÞ ð1 f1 ðtÞ þ 2 g1 ðtÞÞ; where 1 and 2 are coupling parameters or contribution weights from the master lasers (or slave
Fig. 1. Schematic diagram of dual synchronization: EM1 ¼ ES1 , EM2 ¼ ES2 and dual-cross synchronization: EM1 ¼ ES2 , EM2 ¼ ES1 . In the synchronization scheme under consideration the joint error signal eðtÞ, which is the difference between the sum of the outputs from the master systems M1, M2 in the transmitter and the sum of outputs from the slave systems S1, S2 in the receiver, is fed into each slave system and synchronizations between the master lasers and slave lasers under possible configurations are studied.
181
lasers) to the joint error signal eðtÞ. In the following derivation of the existence conditions for dual and dual-cross synchronizations between chaotic laser pairs we will follow the approach developed in [7,8] of investigating the error dynamics. Earlier we have used this approach to derive a necessary (existence) condition for lag synchronization between unidirectionally coupled chaotic external cavity semiconductor lasers [7] and the Ikeda model [8]. First consider the case of dual synchronization. We write the complete dual synchronization manifold as EM1 ðtÞ ¼ ES1 ðtÞ;
nM1 ðtÞ ¼ nS1 ðtÞ;
UM1 ðtÞ ¼ US1 ðtÞðmod 2pÞ; EM2 ðtÞ ¼ ES2 ðtÞ;
ð6aÞ
nM2 ðtÞ ¼ nS2 ðtÞ;
UM2 ðtÞ ¼ US2 ðtÞðmod 2pÞ:
ð6bÞ
In general, sychronization between master (driving) and slave (response) systems strongly depends on the driving variable. For example, as shown in the seminal paper by Pecora–Carroll for the paradigm Lorenz model, driving by the x and y variables results in synchronization between master and slave systems, as the so-called conditional Lyapunov exponents for the response system are negative [1]; driving with the z variable gives rise only to conditional synchronization, as one of the conditional Lyapunov exponents in this case is equal to zero, see, e.g. [9]. Our analysis shows that the electric field amplitude is a good driving variable; indeed as can be seen from the third equations in Eqs. (1)–(3) and (2)–(4), when the differences EM1 ES1 and EM2 ES2 approach zero so do the differences between the corresponding carrier densities. Now consider the deviation from the synchronization manifold e2 ¼ EM1 ES1 and e3 ¼ EM2 ES2 . Then writing Eqs. (1)–(5) in the concise form (which indicate the applicability of the approach developed here to a wide class of continuous dynamical systems) of dEM1 dEM2 ¼ F; ¼ G; dt dt dES1 dES2 ¼ f þ u v; ¼gþuv dt dt
ð7Þ
182
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
and calculating the error dynamics for e2 and e3 we arrive at the system of equations: de2 ¼ F f ðu vÞ; dt de3 ¼ G g ðu vÞ: dt
ð8Þ
Our goal is to achieve synchronization e2 ¼ 0 and e3 ¼ 0. Then with u ¼ 1 F þ 2 G;
v ¼ 1 f þ 2 g;
ð9Þ
one finds that ð1 1 ÞðF f Þ ¼ 2 ðG gÞ; ð1 2 ÞðG gÞ ¼ 1 ðF f Þ:
ð10Þ
Using the expressions for the F , G, f and g from Eqs. (1)–(4) on the synchronization manifolds (6a) and (6b), Eq. (10) can be modified to ð1 1 ÞðkM1 kS1 ÞEM1 ¼ 2 ðkM2 kS2 ÞEM2 ; ð1 2 ÞðkM2 kS2 ÞEM2 ¼ 1 ðkM1 kS1 ÞEM1 :
ð11Þ
From Eq. (11) we obtain that ½ð1 1 Þð1 2 Þ 1 2 ðkM1 kS1 ÞEM1 ¼ 0:
ð12Þ
Thus for kM1; M2 6¼ kS1; S2 we establish the relationship between the coupling weights of the master or slave lasers outputs to achieve dual synchronization: 1 þ 2 ¼ 1:
ð13Þ
We also observe that the condition (13), as mentioned above is fairly general for different types of continuous dynamical systems. Indeed as it follows from Eq. (10) for the system of equations: dx1 dx2 ¼ F; ¼ G; dt dt dy1 dy2 ¼ f þ u v; ¼ g þ u v; dt dt
ð14Þ
with u and v defined above (9), for F 6¼ f and G 6¼ g on the synchronization manifold x1 ðtÞ ¼ y1 ðtÞ, x2 ðtÞ ¼ y2 ðtÞ one can easily arrive at the dual synchronization condition (13). Next we consider dual-cross synchronization, when one synchronizes the slave lasers to their complementary master lasers: EM1 ðtÞ ¼ ES2 ðtÞ and EM2 ðtÞ ¼ ES1 ðtÞ. Following the procedure above
leading to Eq. (10) and to condition (13), we find the existence condition for dual-cross synchronization: 1 ¼ 2 ¼ 1=2:
ð15Þ
Notice that condition (13) also holds for the dual-cross synchronization case. Thus we conclude that both dual and dual-cross synchronizations are possible if the coupling parameters (or contribution weights) to the joint error signal from the oscillators to be synchronized become equal. In other words symmetry in contributions to the coupling plays a crucial role in the synchronization phenomenon studied here. This conclusion is also valid for larger number of constituent parts of the transmitter and receiver. Indeed if we have three master ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ and three slave ðy1 ðtÞ; y2 ðtÞ; y3 ðtÞÞ oscillators in the transmitter and receiver, respectively, it is possible to establish that the triple synchronization x1 ðtÞ ¼ y1 ðtÞ, x2 ðtÞ ¼ y2 ðtÞ, x3 ðtÞ ¼ y3 ðtÞ existence condition is 1 þ 2 þ 3 ¼ 1. Triple-cross synchronization, e.g. x1 ðtÞ ¼ y3 ðtÞ; x2 ðtÞ ¼ y1 ðtÞ; x3 ðtÞ ¼ y2 ðtÞ, takes place if 1 ¼ 2 ¼ 3 ¼ 1=3. In the case of three and more oscillators we also obtain the case of mixed synchronization, when some of the oscillators are coupled in crossed configuration, others in the uncrossed configuration. For example, in the case of three oscillators in both the transmitter and receiver we can define the following synchronization manifold as a mixed synchronization manifold: x1 ðtÞ ¼ y2 ðtÞ, x2 ðtÞ ¼ y1 ðtÞ, x3 ðtÞ ¼ y3 ðtÞ. Again by use of error dynamics approach one obtains that the mixed synchronizationÕs existence condition is: 2 þ 3 ¼ 1 with 1 ¼ 2 ¼ . In other words, we find that all types of synchronizations exist if the contribution weights to the joint error signal from the oscillators to be synchronized are equal. Generalization to the case of N master and N slave oscillators is straightforward. To summarize, we have investigated dual synchronization and dual-cross synchronization between a transmitter and receiver each consisting of two master and slave systems. By use of error dynamics we have derived the existence conditions for dual and dual-cross synchronizations. For the
E.M. Shahverdiev et al. / Optics Communications 216 (2003) 179–183
first time we have established that in all the cases studied synchronization occurs, when the sum of the contribution weights from individual master (or slave) systems in the transmitter (or receiver) is equal to unity; an arbitrarily chosen master system from the transmitter synchronizes with the slave system from the receiver which has an equal contribution weight to the joint error signal. The synchronization conditions identified here are those which must be satisfied to obtain exact synchronization. For practical applications in chaotic communications it is not essential to meet such a stringent condition. Experimental demonstrations of chaos synchronization using chaotic external cavity laser diodes [3,10] are remarkably robust with synchronization having been maintained for long periods of time using commercially available laser diodes where parameter mismatches are the expectations rather than exceptions. Moreover, these experiments have demonstrated that exact synchronization is not required for successful gigahertz message transmission and extraction [11]. With this experimental experience we consider that practical demonstration of the phenomena examined here is eminently feasible. As such we believe that the present findings will be of interest in the context of multichannel chaotic communication schemes where message decoding is dependent on the synchronization of several chaotic transmitters and receivers.
183
Acknowledgements This work is supported by the UK Engineering and Physical Sciences Research Council Grant GR/R22568/01 and GR/N63093/01. References [1] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821; E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [2] G. Chen, X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, 1998; H.G. Schuster (Ed.), Handbook of Chaos Control, WileyVCH, Weinheim, 1999. [3] R. Roy, K.S. Thornburg Jr., Phys. Rev. Lett. 72 (1994) 2009; S. Sivaprakasam, K.A. Shore, Opt. Lett. 24 (1999) 466; S. Sivaprakasam, K.A. Shore, Opt. Lett. 24 (1999) 1200; S. Sivaprakasam, K.A. Shore, IEEE J. Quantum Electron. 36 (2000) 35. [4] S. Sundar, A.A. Minai, Phys. Rev. Lett. 85 (2000) 5456. [5] Y. Liu, P. Davis, Phys. Rev. E 61 (2000) R2176. [6] R. Lang, K. Kobayashi, IEEE J. Quantum Electron. 16 (1980) 347. [7] S. Sivaprakasam, E.M. Shahverdiev, K.A. Shore, Phys. Rev. E 62 (2000) 7505. [8] E.M. Shahverdiev, S. Sivaprakasam, K.A. Shore, Phys. Lett. A 292 (2002) 320. [9] E.M. Shahverdiev, Phys. Rev. E 60 (1999) 3905. [10] S. Sivaprakasam, E.M. Shahverdiev, P.S. Spencer, K.A. Shore, Phys. Rev. Lett. 87 (2001) 154101. [11] J. Paul, S. Sivaprakasam, P.S. Spencer, P. Rees, K.A. Shore, Electron. Lett. 38 (2002) 28.