- Email: [email protected]
Bubbling effect in the electro-optic delayed feedback oscillator coupled network
Optics Communications 387 (2017) 310–315
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Bubbling effect in the electro-optic delayed feedback oscillator coupled network
MARK
⁎
Lingfeng Liua, , Jun Lina, Suoxia Miaob a b
School of Software, Nanchang University, Nanchang 330031, People's Republic of China Faculty of Science, Nanchang Institute of Technology, Nanchang 330029, People's Republic of China
A R T I C L E I N F O
A BS T RAC T
Keywords: Electro-optic delayed feedback oscillator Synchronization Bubbling effects
Synchronization in the optical systems coupled network always suffers from bubbling events. In this paper, we numerically investigate the statistical properties of the synchronization characteristics and bubbling effects in the electro-optic delayed feedback oscillator coupled network with different coupling strength, delay time and gain coefficient. Furthermore, we compare our results with the synchronization properties of semiconductor laser (SL) coupled network, which indicates that the electro-optic delayed feedback oscillator can be better to suppress the bubbling effects in the synchronization of coupled network under the same conditions.
1. Introduction With the development of high-speed and large-capacity all-optical networks, the optical chaotic secure communication technology has been widely studied [1–3]. The nodes are described by optic chaotic systems, and the communications are based on the synchronization of these network nodes. Presently, two kinds of methods are commonly used to generate optical chaotic signal, one is based on the inner nonlinear effects of semiconductor lasers (SLs) [4,5], which can be described by the rate equations. The other is based on the nonlinear effects of some external nonlinear devices, such as Mach-Zehnder (MZ) modulator [6,7], which is modeled by Ikeda's delay differential equations. Comparatively speaking, the electro-optic chaotic system based on MZ modulator has great advantages for its high modulation speed, good stability, large Lyapunov dimension and good compatibility with the modern communication network [8]. Realizing stable synchronization is the key issue of network communication, which has been widely studied [9–15]. [9] provides that only generalized synchronization can be achieved in two mutual coupled identical SLs. Zero-lag synchronization can be observed for large networks with multiple nonlinear nodes [10]. [11] demonstrates multi-cluster synchronization of large networks of unidirectional coupled lasers with homogeneous and heterogeneous delays. [12] presents how two electro-optic delayed systems can be coupled together, and the conditions under which they can synchronize. Generally, most of the studies are focus on the synchronization performances of SLs described by the rate equations. For the electrooptic delayed feedback oscillator based on MZ modulator, its synchro-
⁎
nization characteristics have not been studied to the same extent. To our best knowledge, the network synchronization of nodes, which is described by electro-optic delayed feedback systems, has not been studied yet. However, even at well-synchronized systems, the de-synchronization phenomenon (bubbling) can be observed, which is induced by noise and/or parameter mismatch [16,17]. [18] investigates the behavior of bubbling effects in a SL coupled network with a star topology, and reveal the relationship between bubbling effects and couple strength, as well as delay times. Motivated by [18] and the introduction above, in this paper, we study on the bubbling effects in the electro-optic delayed feedback oscillator coupled network. The statistical properties of the synchronization characteristics and bubbling effects between different nodes with different coupling strengths, delay time and gain coefficients are numerically investigated. Furthermore, we compare our results with the synchronization properties of SL coupled network in [18], which demonstrates that the electro-optic delayed feedback oscillator can be better to suppress the bubbling effects in the synchronization of coupled network under the same conditions. The rest of this paper is organized as follows. In Section 2, the model of electro-optic delayed feedback oscillator coupled network with a star topology is introduced, and the bubbling effect is presented. In Section 3, the statistical properties of the synchronization characteristics and bubbling effects with different coupling strengths, delay time and gain coefficients are numerically investigated, and the results are compared with the synchronization properties of SL coupled network in [18]. Finally, Section 4 concludes the whole paper.
Corresponding author. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.optcom.2016.12.003 Received 9 November 2016; Received in revised form 30 November 2016; Accepted 1 December 2016 Available online 04 December 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
Optics Communications 387 (2017) 310–315
L. Liu et al.
t dxj ⎧ 1 2 ⎪ xj + τj dt + θj ∫t 0 xj (s ) ds = βj cos [kj1 xj (t − Tj ) + kH 2 xH (t − TH ) + Φj ] ⎪ ⎪ x + τ dxH + 1 t x (s ) ds ∫ H dt ⎨ H θH t 0 H ⎪ ⎪ = β cos2 ⎡k x (t − T ) + ∑10 k x (t − T ) + Φ ⎤ ⎢ H1 H H j H⎥ H j =1 j 2 j ⎪ ⎣ ⎦ ⎩
,
j = 1, 2, ... ,10.
(1)
where kH1=kj1=1−k, kH1 and kj1 denote the self-feedback strength of central and star node, respectively. Then, the sum of self-feedback and cross-coupled feedback of all nodes (including central and star nodes) will be the same. xj and xH are the state variables, βj and βHare the gain coefficients, Tj and TH are the delay time of star and central nodes, respectively. Furthermore, τj,H=1/2πfj,H, θj,H=1/2πFj,H, Фj,H=πVj,H/2Wj,H, where fj and fH are the cutoff frequencies of high-pass filters, Fj and FH are the cutoff frequencies of low-pass filters, Vj and VH are the biased voltages, Wj and WH are the dc halfwave voltages of star and central nodes, respectively. A more detailed description can be found in [6]. In this paper, we always assume βj=βH=β, τj=τH=τ, θj=θH=θ, Фj=ФH=Ф and Tj=TH=T. In the numerical simulations of Eq. (1), the following values of parameters are employed unless otherwise stated: Ф=-π/4, β=5, τ=25 ps, θ=5 μs, T=30 ns, k=0.5. The Eq. (1) are numerically solved on the Matlab 2014a by using the common fourth-order Runge-Kutta method with a time step of 0.6 ps. The synchronization of this star network is shown in Fig. 2. From Fig. 2, we can conclude that all the nodes can realize synchronization in a very short time (in the ps order). However, with time evolution, the bubbling effects will appear. Fig. 3 shows the synchronization errors between node H2 and H7 (We randomly choose a pair of nodes to present the bubbling effect. Factually, the bubbling effects also exist in other nodes, which are omitted here to avoid redundancy). From Fig. 3 we can find that the de-synchronization event can be observed intermittently in the process of synchronization.
Fig. 1. The star topology network composed of electro-optic delayed feedback oscillators.
Fig. 2. Synchronization in the star network.
2. Bubbling effect in the star network In this paper, the following star coupled network with 11 nodes is considered, see Fig. 1. In the network, the nodes are described by the identical electro-optic delayed feedback oscillators. H denotes the central node in the network, and H1, H2, …, H10 denote the star nodes. The central node and star nodes are mutual coupling by using a coupling strength of kH2=k, from the central node towards each star node, and a coupling strength of kj2=0.1k, from each star node towards the central node. The mathematical model of each node can be described by Ikeda's delay differential equations [6], and the whole network can be modeled by
3. Analysis of bubbling effects In this section, we will analyze the statistical properties of the synchronization characteristics and bubbling effects between different nodes with different network parameters. Here, the synchronization of the star nodes in the network is evaluated by calculating the mean zerolag cross-correlation Cm and mean zero-lag synchronization error Em between the star nodes, which can be calculated as [18]
Cij =
(xi (t ) − xi (t ) )⋅(xj (t ) − xj (t ) ) (xi (t ) − xi (t ) )2 ⋅(xj (t ) − xj (t ) )2
Fig. 3. (a) Synchronization error between node H2 and H7; (b) Enlargement of (a).
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Fig. 4. (a) Mean cross-correlation for all oscillator pairs; (b) Enlargement of (a).
Fig. 5. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus time delay. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus time delay. (c) Mean duration of bubbling effect versus time delay. (d) Number of bubbles versus time delay. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Cm = Cij , i ≠ j
Eij =
sequentially by a time period of Tm=25 ps to assure a high resolution tracing in the Bubbling events. By averaging the sliding window crosscorrelation performances for all delayed oscillators, we can get the mean sliding window cross-correlation, see Fig. 4. From Fig. 4 we can find that the cross-correlation value will decrease rapidly when the bubbling event appears. In order to qualify to performances of the synchronization of network, including bubbling duration and number of bubbles, we set a threshold value of mean cross-correlation Cthr=0.99 in this paper.
(3)
|xi (t ) − xj (t )| 0.5⋅(xi (t ) + xj (t ))
Em = Eij ,
i≠j
(4) (5)
where〈 • 〉indicates a time average, xi is the optical power of the ith optical system. Cm and Em stand for averaging over all oscillator pairs (i, j). In our analysis, the mean sliding window cross-correlation introduced in [7] is used. Set the window Tw=0.125 ns and shift it
1) The synchronization performances with different delay times
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Fig. 6. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus coupling strength. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus coupling strength. (c) Mean duration of bubbling effect versus coupling strength. (d) Number of bubbles versus coupling strength. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with different coupling strengths are shown in Fig. 6. As Fig. 6(a) shown, the mean cross-correlation curve is similar to a parabola. When the coupling strength is near 0.5, the cross-correlation will be large, which indicates a good synchronization characteristics. Otherwise, the cross-correlation will drop below Cthr. This conclusion can also be proved by estimating the synchronization error, see Fig. 6(b). Fig. 6(c) and (d) performs the bubble duration and number of bubbles with different coupling strengths. As Fig. 6(c) and (d) shown, both bubble duration and number of bubbles will firstly decrease with the coupling strength increased from 0.1 to 0.5, and then gradually increase with the coupling strength increased from 0.5 to 1, which present a “U” like graph. At the bottom (k=0.5–0.7), the bubble duration and number of bubbles are stabilized at about 33 ps and 3, respectively. Therefore, in order to avoid strong bubble effects, the coupling strength should be set near 0.5.
In this simulations, we set coupling strength k=0.45 and gain coefficient β=5 for all the delayed oscillators. The synchronization performances, including mean zero-lag cross-correlation Cm, mean zero-lag synchronization error Em, bubble duration and number of bubbles, with different delay time are shown in Fig. 5. Fig. 5(a) depicts the mean (squares) and minimum (asterisks) cross-correlation among all oscillator pairs in the star network. From Fig. 5(a) we can find that the cross-correlation has oscillations with a small delay time, and the Cm may drop below Cthr. When the delay time is larger than 16 ns, the cross-correlation will be stable. Alternatively, a similar result can be observed by estimating the synchronization error for the corresponding cases, see Fig. 5(b). Fig. 5(c) and (d) indicates that the bubble effect will approach to be stable with the delay time increased. When the delay time grows to about 20 ns, the bubble duration and number of bubbles will be 33 ps and 9, respectively, and will not change with the delay time increased. Furthermore, the bubble duration and number of bubbles will reach their minimum value in such delays, which implies a weaker bubble effect, and a better synchronization performances. Thus, according to the simulation results, in an electro-optic delayed oscillators coupled network, we should try to make the delay time larger than 20 ns to avoid strong bubble effects, and make the synchronization as stable as possible. 2) The synchronization strengths
performances
with
different
3) The synchronization performances with different gain coefficients In this simulations, the delay time T and coupling strength k are fixed to be 15 ns and 0.45, respectively. For different gain coefficients, the synchronization performances are presented in Fig. 7. From Fig. 7(a) we can find that the mean cross-correlation is larger than Cthr with a low gain coefficient. Once the gain coefficient is larger than 6, the cross-correlation will drop rapidly, which implies a bad synchronization performances. Fig. 7(b) depicts the synchronization error among all oscillator pairs, which can verify an equivalent performance as Fig. 7(a). Fig. 7(c) and (d) performs the bubble duration and number of bubbles with different gain coefficients. From these two figures we can
coupling
In this simulations, we set delay time T=15 ns and gain coefficient β=5 for all the delayed oscillators. The synchronization performances
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Fig. 7. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus gain coefficient. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus gain coefficient. (c) Mean duration of bubbling effect versus gain coefficient. (d) Number of bubbles versus gain coefficient. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
find that the bubble duration will gradually increase with the gain coefficients grows from 3 to 15, while the number of bubbles will first increase with the gain coefficients grows from 3 to 8, and then keep stable from 8 to 15. Although the results are not completely the same, we can still get the conclusion that the lower the gain coefficient is, the weaker the bubble effect performs. It is noted that, the dynamical complexity of the electro-optic delayed feedback oscillator will increase with the gain coefficient grow [19]. Therefore, we have to choose a compromise scheme between complexity and synchronization stability. Previous studies show that when the gain coefficient β≥3, the delayed oscillator will be chaotic, which is with a quite complex dynamics [6]. Thus, choosing β=3 may satisfy these two requirements.
less than the mean duration of bubble events of SL coupled network in [18] (The mean duration of bubble events of SL coupled network in [18] is about 1 ns). Furthermore, the number of bubbles of our synchronized network is still less than the SL coupled network. Therefore, the synchronization of electro-optic delayed feedback oscillator coupled network can be better to suppress the bubbling effects in the synchronization, which is better for application in the optical secure communication in this sense. 4. Conclusions Synchronization in the optical systems coupled network always suffers from bubbling events (de-synchronization phenomenon). In this paper, the bubbling effects in the electro-optic delayed feedback oscillator coupled star network is studied. In the numerical experiments, the mean zero-lag cross-correlation, zero-lag synchronization error, bubble duration and number of bubbles are used to evaluate the synchronization performances between different nodes with different coupling strength, delay time and gain coefficient. The results show that.
4) Synchronization performances comparison with SL coupled network The synchronization performances of SL coupled star network is analyzed in [18]. In this paper, the synchronization performances of electro-optic delayed feedback oscillator coupled network with the same topology is studied. Comparatively, on the whole, the synchronization performances of SL coupled network is more stable, as the cross-correlation and synchronization error do not have a big change versus the network parameters. However, the synchronization performances of electrooptic delayed feedback oscillator coupled network in this paper are greatly influenced by the network and system parameters. As the network parameters can be artificially adjusted, therefore, we will always choose those parameters which make the network under the best working condition. Under the best case, the mean duration of bubble events of our synchronized network is about 33 ps, which is far
1) The bubbling effect will gradually tends to be stable when the delay time be larger than 20 ns. 2) The bubbling effect will be weak when the coupling strength be about 0.5. 3) The lower the gain coefficient is, the weaker the bubble effect performs. These three results are very important and can be used as the guidance in the synchronization of electro-optic delayed feedback 314
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electro-optical systems, Phys. Rev. Lett. 95 (2005) 203903. [7] Y.C. Kouomou, L. Larger, T. Hervé, B. Ryad, R. Enrico, P. Colet, Dynamic instabilities of microwaves generated with optoelectronic oscillators, Opt. Lett. 32 (17) (2007) 2571–2573. [8] H.P. Hu, W. Su, L.F. Liu, Z.L. Yu, Electro-optic intensity chaotic system with varying parameters, Phys. Lett. A 378 (3) (2014) 184–190. [9] J. Mulet, C.R. Mirasso, T. Heil, I. Fischer, Synchronization scenario of two distant mutually coupled semiconductor lasers, J. Opt. B Quantum Semiclass Opt. 6 (2004) 97–105. [10] M.B. ourmpos, A. Argyris, D. Syvridis, Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers, J. Lightwave Technol. 30 (16) (2012) 2618–2624. [11] M. Nixon, M. Fridman, E. Ronen, A.A. Fruesem, N. Davidson, I. Kanter, Controlling synchronization in large laser networks, Phys. Rev. Lett. 108 (2012) 214101. [12] T.E. Murphy, A.B. Cohen, B. Ravoori, et al., Complex dynamics and synchronization of delayed-feedback nonlinear oscillators, Philos. Trans. R. Soc. A 368 (2010) 343–366. [13] E.A.R. Dakin, J.G. Ojalvo, D.J. Deshazer, R. Roy, Synchronization and symmetry breaking in mutually coupled fiber lasers, Phys. Rev. E 73 (2006) 045201. [14] A. Uchida, F. Rogister, J.G. Ojalvo, R. Roy, Synchronization and communication with chaotic laser systems, in: Progress in Optics, Elsevier, Amsterdam, The Netherlands, 2005, 48, pp. 203–341. [15] E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, I. Kanter, Stable isochronal synchronization of mutually coupled chaotic lasers, Phys. Rev. E 73 (2006) 066214. [16] J. Tiana-Alsina, et al., Zero-lag synchronization and bubbling in delay-coupled lasers, Phys. Rev. E 85 (2) (2012) 206209. [17] K. Hicke, O. D’Huys, V. Flunkert, E. Scholl, J. Danckaert, I. Fisher, Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled laser, Phys. Rev. E 83 (5) (2011) 056211. [18] M. Bourmpos, A. Argyris, D. Syvridis, Analysis of the bubbling effect in synchronized networks with semiconductor lasers, IEEE Photonics Technol. Lett. 25 (9) (2013) 817–820. [19] D. Rontani, A. Locquet, M. Sciamanna, et al., Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback, Opt. Lett. 32 (20) (2007) 2960–2962.
oscillator coupled network. Furthermore, we compare our results with the synchronization properties of SL coupled network in [18]. The results show that the electro-optic delayed feedback oscillators coupled network can be better to suppress the bubbling effects in the synchronization, and is better for application in the optical secure communication in this sense. Acknowledgements This work is supported by the National Natural Science Foundation of China (61601215, 61505061 and 61603175), and Science & Technology Research Project of Education Department of Jiangxi Province (GJJ150104). References [1] Z.X. Kang, J. Sun, L. Ma, Multimode synchronization of chaotic semiconductor ring laser and its potential in chaos communication, IEEE J. Quantum Electron. 50 (3) (2014) 148–157. [2] S.Y. Xiang, W. Pan, L.S. Yan, Impact of unpredictability on chaos synchronization of vertical-cavity surface-emitting lasers with variable-polarization optical feedback, Opt. Lett. 36 (17) (2011) 3497–3499. [3] A. Argyris, D. Syvridis, L. Larger, L.V. Annovazzi, P. Colet, I. Fischer, J.G. Ojalvo, C.R. Mirasso, L. Pesquera, K.A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature 438 (2005) 343–346. [4] Y.Y. Liu, W. Pan, N. Jiang, S.Y. Xiang, Y.D. Lin, Isochronal chaos synchronization of a chain mutually coupled semiconductor lasers, Acta Phys. Sin. 62 (2) (2013) 024208. [5] W. Deng, G.Q. Xia, Z.M. Wu, Dual-channel chaos synchronization and communication based on a vertical-cavity surface emitting laser with double optical feedback, Acta Phys. Sin. 62 (16) (2013) 164209. [6] Y.C. Kouomou, P. Colet, L. Larger, N. Gastaud, Chaotic breathers in delayed
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Bubbling effect in the electro-optic delayed feedback oscillator coupled network
MARK
⁎
Lingfeng Liua, , Jun Lina, Suoxia Miaob a b
School of Software, Nanchang University, Nanchang 330031, People's Republic of China Faculty of Science, Nanchang Institute of Technology, Nanchang 330029, People's Republic of China
A R T I C L E I N F O
A BS T RAC T
Keywords: Electro-optic delayed feedback oscillator Synchronization Bubbling effects
Synchronization in the optical systems coupled network always suffers from bubbling events. In this paper, we numerically investigate the statistical properties of the synchronization characteristics and bubbling effects in the electro-optic delayed feedback oscillator coupled network with different coupling strength, delay time and gain coefficient. Furthermore, we compare our results with the synchronization properties of semiconductor laser (SL) coupled network, which indicates that the electro-optic delayed feedback oscillator can be better to suppress the bubbling effects in the synchronization of coupled network under the same conditions.
1. Introduction With the development of high-speed and large-capacity all-optical networks, the optical chaotic secure communication technology has been widely studied [1–3]. The nodes are described by optic chaotic systems, and the communications are based on the synchronization of these network nodes. Presently, two kinds of methods are commonly used to generate optical chaotic signal, one is based on the inner nonlinear effects of semiconductor lasers (SLs) [4,5], which can be described by the rate equations. The other is based on the nonlinear effects of some external nonlinear devices, such as Mach-Zehnder (MZ) modulator [6,7], which is modeled by Ikeda's delay differential equations. Comparatively speaking, the electro-optic chaotic system based on MZ modulator has great advantages for its high modulation speed, good stability, large Lyapunov dimension and good compatibility with the modern communication network [8]. Realizing stable synchronization is the key issue of network communication, which has been widely studied [9–15]. [9] provides that only generalized synchronization can be achieved in two mutual coupled identical SLs. Zero-lag synchronization can be observed for large networks with multiple nonlinear nodes [10]. [11] demonstrates multi-cluster synchronization of large networks of unidirectional coupled lasers with homogeneous and heterogeneous delays. [12] presents how two electro-optic delayed systems can be coupled together, and the conditions under which they can synchronize. Generally, most of the studies are focus on the synchronization performances of SLs described by the rate equations. For the electrooptic delayed feedback oscillator based on MZ modulator, its synchro-
⁎
nization characteristics have not been studied to the same extent. To our best knowledge, the network synchronization of nodes, which is described by electro-optic delayed feedback systems, has not been studied yet. However, even at well-synchronized systems, the de-synchronization phenomenon (bubbling) can be observed, which is induced by noise and/or parameter mismatch [16,17]. [18] investigates the behavior of bubbling effects in a SL coupled network with a star topology, and reveal the relationship between bubbling effects and couple strength, as well as delay times. Motivated by [18] and the introduction above, in this paper, we study on the bubbling effects in the electro-optic delayed feedback oscillator coupled network. The statistical properties of the synchronization characteristics and bubbling effects between different nodes with different coupling strengths, delay time and gain coefficients are numerically investigated. Furthermore, we compare our results with the synchronization properties of SL coupled network in [18], which demonstrates that the electro-optic delayed feedback oscillator can be better to suppress the bubbling effects in the synchronization of coupled network under the same conditions. The rest of this paper is organized as follows. In Section 2, the model of electro-optic delayed feedback oscillator coupled network with a star topology is introduced, and the bubbling effect is presented. In Section 3, the statistical properties of the synchronization characteristics and bubbling effects with different coupling strengths, delay time and gain coefficients are numerically investigated, and the results are compared with the synchronization properties of SL coupled network in [18]. Finally, Section 4 concludes the whole paper.
Corresponding author. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.optcom.2016.12.003 Received 9 November 2016; Received in revised form 30 November 2016; Accepted 1 December 2016 Available online 04 December 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
Optics Communications 387 (2017) 310–315
L. Liu et al.
t dxj ⎧ 1 2 ⎪ xj + τj dt + θj ∫t 0 xj (s ) ds = βj cos [kj1 xj (t − Tj ) + kH 2 xH (t − TH ) + Φj ] ⎪ ⎪ x + τ dxH + 1 t x (s ) ds ∫ H dt ⎨ H θH t 0 H ⎪ ⎪ = β cos2 ⎡k x (t − T ) + ∑10 k x (t − T ) + Φ ⎤ ⎢ H1 H H j H⎥ H j =1 j 2 j ⎪ ⎣ ⎦ ⎩
,
j = 1, 2, ... ,10.
(1)
where kH1=kj1=1−k, kH1 and kj1 denote the self-feedback strength of central and star node, respectively. Then, the sum of self-feedback and cross-coupled feedback of all nodes (including central and star nodes) will be the same. xj and xH are the state variables, βj and βHare the gain coefficients, Tj and TH are the delay time of star and central nodes, respectively. Furthermore, τj,H=1/2πfj,H, θj,H=1/2πFj,H, Фj,H=πVj,H/2Wj,H, where fj and fH are the cutoff frequencies of high-pass filters, Fj and FH are the cutoff frequencies of low-pass filters, Vj and VH are the biased voltages, Wj and WH are the dc halfwave voltages of star and central nodes, respectively. A more detailed description can be found in [6]. In this paper, we always assume βj=βH=β, τj=τH=τ, θj=θH=θ, Фj=ФH=Ф and Tj=TH=T. In the numerical simulations of Eq. (1), the following values of parameters are employed unless otherwise stated: Ф=-π/4, β=5, τ=25 ps, θ=5 μs, T=30 ns, k=0.5. The Eq. (1) are numerically solved on the Matlab 2014a by using the common fourth-order Runge-Kutta method with a time step of 0.6 ps. The synchronization of this star network is shown in Fig. 2. From Fig. 2, we can conclude that all the nodes can realize synchronization in a very short time (in the ps order). However, with time evolution, the bubbling effects will appear. Fig. 3 shows the synchronization errors between node H2 and H7 (We randomly choose a pair of nodes to present the bubbling effect. Factually, the bubbling effects also exist in other nodes, which are omitted here to avoid redundancy). From Fig. 3 we can find that the de-synchronization event can be observed intermittently in the process of synchronization.
Fig. 1. The star topology network composed of electro-optic delayed feedback oscillators.
Fig. 2. Synchronization in the star network.
2. Bubbling effect in the star network In this paper, the following star coupled network with 11 nodes is considered, see Fig. 1. In the network, the nodes are described by the identical electro-optic delayed feedback oscillators. H denotes the central node in the network, and H1, H2, …, H10 denote the star nodes. The central node and star nodes are mutual coupling by using a coupling strength of kH2=k, from the central node towards each star node, and a coupling strength of kj2=0.1k, from each star node towards the central node. The mathematical model of each node can be described by Ikeda's delay differential equations [6], and the whole network can be modeled by
3. Analysis of bubbling effects In this section, we will analyze the statistical properties of the synchronization characteristics and bubbling effects between different nodes with different network parameters. Here, the synchronization of the star nodes in the network is evaluated by calculating the mean zerolag cross-correlation Cm and mean zero-lag synchronization error Em between the star nodes, which can be calculated as [18]
Cij =
(xi (t ) − xi (t ) )⋅(xj (t ) − xj (t ) ) (xi (t ) − xi (t ) )2 ⋅(xj (t ) − xj (t ) )2
Fig. 3. (a) Synchronization error between node H2 and H7; (b) Enlargement of (a).
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Fig. 4. (a) Mean cross-correlation for all oscillator pairs; (b) Enlargement of (a).
Fig. 5. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus time delay. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus time delay. (c) Mean duration of bubbling effect versus time delay. (d) Number of bubbles versus time delay. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Cm = Cij , i ≠ j
Eij =
sequentially by a time period of Tm=25 ps to assure a high resolution tracing in the Bubbling events. By averaging the sliding window crosscorrelation performances for all delayed oscillators, we can get the mean sliding window cross-correlation, see Fig. 4. From Fig. 4 we can find that the cross-correlation value will decrease rapidly when the bubbling event appears. In order to qualify to performances of the synchronization of network, including bubbling duration and number of bubbles, we set a threshold value of mean cross-correlation Cthr=0.99 in this paper.
(3)
|xi (t ) − xj (t )| 0.5⋅(xi (t ) + xj (t ))
Em = Eij ,
i≠j
(4) (5)
where〈 • 〉indicates a time average, xi is the optical power of the ith optical system. Cm and Em stand for averaging over all oscillator pairs (i, j). In our analysis, the mean sliding window cross-correlation introduced in [7] is used. Set the window Tw=0.125 ns and shift it
1) The synchronization performances with different delay times
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Fig. 6. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus coupling strength. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus coupling strength. (c) Mean duration of bubbling effect versus coupling strength. (d) Number of bubbles versus coupling strength. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with different coupling strengths are shown in Fig. 6. As Fig. 6(a) shown, the mean cross-correlation curve is similar to a parabola. When the coupling strength is near 0.5, the cross-correlation will be large, which indicates a good synchronization characteristics. Otherwise, the cross-correlation will drop below Cthr. This conclusion can also be proved by estimating the synchronization error, see Fig. 6(b). Fig. 6(c) and (d) performs the bubble duration and number of bubbles with different coupling strengths. As Fig. 6(c) and (d) shown, both bubble duration and number of bubbles will firstly decrease with the coupling strength increased from 0.1 to 0.5, and then gradually increase with the coupling strength increased from 0.5 to 1, which present a “U” like graph. At the bottom (k=0.5–0.7), the bubble duration and number of bubbles are stabilized at about 33 ps and 3, respectively. Therefore, in order to avoid strong bubble effects, the coupling strength should be set near 0.5.
In this simulations, we set coupling strength k=0.45 and gain coefficient β=5 for all the delayed oscillators. The synchronization performances, including mean zero-lag cross-correlation Cm, mean zero-lag synchronization error Em, bubble duration and number of bubbles, with different delay time are shown in Fig. 5. Fig. 5(a) depicts the mean (squares) and minimum (asterisks) cross-correlation among all oscillator pairs in the star network. From Fig. 5(a) we can find that the cross-correlation has oscillations with a small delay time, and the Cm may drop below Cthr. When the delay time is larger than 16 ns, the cross-correlation will be stable. Alternatively, a similar result can be observed by estimating the synchronization error for the corresponding cases, see Fig. 5(b). Fig. 5(c) and (d) indicates that the bubble effect will approach to be stable with the delay time increased. When the delay time grows to about 20 ns, the bubble duration and number of bubbles will be 33 ps and 9, respectively, and will not change with the delay time increased. Furthermore, the bubble duration and number of bubbles will reach their minimum value in such delays, which implies a weaker bubble effect, and a better synchronization performances. Thus, according to the simulation results, in an electro-optic delayed oscillators coupled network, we should try to make the delay time larger than 20 ns to avoid strong bubble effects, and make the synchronization as stable as possible. 2) The synchronization strengths
performances
with
different
3) The synchronization performances with different gain coefficients In this simulations, the delay time T and coupling strength k are fixed to be 15 ns and 0.45, respectively. For different gain coefficients, the synchronization performances are presented in Fig. 7. From Fig. 7(a) we can find that the mean cross-correlation is larger than Cthr with a low gain coefficient. Once the gain coefficient is larger than 6, the cross-correlation will drop rapidly, which implies a bad synchronization performances. Fig. 7(b) depicts the synchronization error among all oscillator pairs, which can verify an equivalent performance as Fig. 7(a). Fig. 7(c) and (d) performs the bubble duration and number of bubbles with different gain coefficients. From these two figures we can
coupling
In this simulations, we set delay time T=15 ns and gain coefficient β=5 for all the delayed oscillators. The synchronization performances
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Fig. 7. (a) Mean (squares, red) and minimum (asterisks, blue) cross-correlation versus gain coefficient. (b) Mean (squares, red) and maximum (asterisks, blue) synchronization error versus gain coefficient. (c) Mean duration of bubbling effect versus gain coefficient. (d) Number of bubbles versus gain coefficient. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
find that the bubble duration will gradually increase with the gain coefficients grows from 3 to 15, while the number of bubbles will first increase with the gain coefficients grows from 3 to 8, and then keep stable from 8 to 15. Although the results are not completely the same, we can still get the conclusion that the lower the gain coefficient is, the weaker the bubble effect performs. It is noted that, the dynamical complexity of the electro-optic delayed feedback oscillator will increase with the gain coefficient grow [19]. Therefore, we have to choose a compromise scheme between complexity and synchronization stability. Previous studies show that when the gain coefficient β≥3, the delayed oscillator will be chaotic, which is with a quite complex dynamics [6]. Thus, choosing β=3 may satisfy these two requirements.
less than the mean duration of bubble events of SL coupled network in [18] (The mean duration of bubble events of SL coupled network in [18] is about 1 ns). Furthermore, the number of bubbles of our synchronized network is still less than the SL coupled network. Therefore, the synchronization of electro-optic delayed feedback oscillator coupled network can be better to suppress the bubbling effects in the synchronization, which is better for application in the optical secure communication in this sense. 4. Conclusions Synchronization in the optical systems coupled network always suffers from bubbling events (de-synchronization phenomenon). In this paper, the bubbling effects in the electro-optic delayed feedback oscillator coupled star network is studied. In the numerical experiments, the mean zero-lag cross-correlation, zero-lag synchronization error, bubble duration and number of bubbles are used to evaluate the synchronization performances between different nodes with different coupling strength, delay time and gain coefficient. The results show that.
4) Synchronization performances comparison with SL coupled network The synchronization performances of SL coupled star network is analyzed in [18]. In this paper, the synchronization performances of electro-optic delayed feedback oscillator coupled network with the same topology is studied. Comparatively, on the whole, the synchronization performances of SL coupled network is more stable, as the cross-correlation and synchronization error do not have a big change versus the network parameters. However, the synchronization performances of electrooptic delayed feedback oscillator coupled network in this paper are greatly influenced by the network and system parameters. As the network parameters can be artificially adjusted, therefore, we will always choose those parameters which make the network under the best working condition. Under the best case, the mean duration of bubble events of our synchronized network is about 33 ps, which is far
1) The bubbling effect will gradually tends to be stable when the delay time be larger than 20 ns. 2) The bubbling effect will be weak when the coupling strength be about 0.5. 3) The lower the gain coefficient is, the weaker the bubble effect performs. These three results are very important and can be used as the guidance in the synchronization of electro-optic delayed feedback 314
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oscillator coupled network. Furthermore, we compare our results with the synchronization properties of SL coupled network in [18]. The results show that the electro-optic delayed feedback oscillators coupled network can be better to suppress the bubbling effects in the synchronization, and is better for application in the optical secure communication in this sense. Acknowledgements This work is supported by the National Natural Science Foundation of China (61601215, 61505061 and 61603175), and Science & Technology Research Project of Education Department of Jiangxi Province (GJJ150104). References [1] Z.X. Kang, J. Sun, L. Ma, Multimode synchronization of chaotic semiconductor ring laser and its potential in chaos communication, IEEE J. Quantum Electron. 50 (3) (2014) 148–157. [2] S.Y. Xiang, W. Pan, L.S. Yan, Impact of unpredictability on chaos synchronization of vertical-cavity surface-emitting lasers with variable-polarization optical feedback, Opt. Lett. 36 (17) (2011) 3497–3499. [3] A. Argyris, D. Syvridis, L. Larger, L.V. Annovazzi, P. Colet, I. Fischer, J.G. Ojalvo, C.R. Mirasso, L. Pesquera, K.A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature 438 (2005) 343–346. [4] Y.Y. Liu, W. Pan, N. Jiang, S.Y. Xiang, Y.D. Lin, Isochronal chaos synchronization of a chain mutually coupled semiconductor lasers, Acta Phys. Sin. 62 (2) (2013) 024208. [5] W. Deng, G.Q. Xia, Z.M. Wu, Dual-channel chaos synchronization and communication based on a vertical-cavity surface emitting laser with double optical feedback, Acta Phys. Sin. 62 (16) (2013) 164209. [6] Y.C. Kouomou, P. Colet, L. Larger, N. Gastaud, Chaotic breathers in delayed
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