Chaos synchronization based on cluster fusion in asymmetric coupling semiconductor lasers networks

Shiqin Liu; Ning Jiang; Yiqun Zhang; Anke Zhao; Jiafa Peng; Kun Qiu; Qianwu Zhang

doi:10.1364/OE.426751

1. Introduction

In the last decades, laser chaos and its synchronization have attracted great attention due to the potential applications in physical-layer chaotic communication [1–5], secure key distribution [6,7], random bit generator [8,9], and machine learning [10–12]. In 1990, the first demonstration of chaos synchronization between two chaotic systems was reported [13]. Since then, many efforts have been made to study the synchronization properties of coupled chaotic lasers in the past years [14–19]. However, most of the previously reported works mainly focused on the synchronization of two lasers and did not consider the multipoint-to-multipoint scenarios which are more common in practice.

Until most recent years, a few studies on network-type chaos synchronization were reported. The cluster chaos synchronization in mutually-coupled ring-type laser networks was experimentally investigated in [20], and the laser networks with complex topologies were also numerically discussed in [21–23], where synchronization can be achieved between lasers within same cluster while lasers of different clusters could not be synchronized. Besides, the synchronization mechanism in hierarchical tree-type networks composed of mutually coupled semiconductor lasers (SLs) was numerically studied in [24], and the results show that only structurally symmetrical SLs can be synchronized. Additionally, global chaos synchronization was experimentally observed in a highly-symmetric global laser network consisting of 16 completely-coupled SLs [25], that is, all the SLs are identically synchronized with each other. Nevertheless, in most previous studies on network chaos synchronization, synchronous states are only observed among lasers that are structurally symmetric, while asymmetric coupling lasers could not be synchronized. The investigations about the chaos synchronization between asymmetric coupling nodes in laser networks are still sorely lacked. Moreover, to satisfy the various demands in real applications, it is valuable to study the chaos synchronization among asymmetric coupling lasers in SLs networks.

In this paper, by employing a novel cluster fusion method, we numerically demonstrate the chaos synchronization in hybrid clusters composed of ACSLs. With the introduction of cluster fusion, the ACSLs from different clusters can be combined into a newly generated hybrid cluster, as such, the original ACSLs can be symmetrically coupled in new SLs clusters. Here, three types of SLs hybrid clusters are systematically discussed, namely, trivial-hybrid cluster, trivial-nontrivial-hybrid cluster, and nontrivial-hybrid cluster that are generated by implementing cluster fusion to distinct trivial clusters, distinct trivial and non-trivial clusters, and distinct nontrivial clusters, respectively. It is demonstrated that compared with the original exemplary 7-SLs network, the intra-cluster SLs of the trivial-hybrid cluster, trivial-nontrivial-hybrid cluster, and nontrivial-hybrid cluster can be synchronized over a wide dynamic operation range, and the chaos synchronization is robust to the mismatch of ACSLs’ intrinsic parameters and injection strength mismatch between cluster fusion-introduced chaotic injections and previous ones to some extent. Meanwhile, the partial cluster fusion-based chaos synchronization in typical star-type SLs networks is also systematically proved, which indicates that the proposed cluster fusion is appropriate for the ACSLs’ synchronization in general SLs networks.

2. Network topologies and theoretical model

A 7 mutually-coupled SLs network whose topology is shown in Fig. 1(a) is taken for instance to systematically discuss the cluster fusion and synchronization performance in the following. As demonstrated in Refs. [21,23], the SLs that are structurally symmetric can be divided into same clusters, thus, the SLs of original network are classified into five clusters referred as: C(1, 2), C(3), C(4), C(5, 7), and C(6). Among them, C(1, 2) and C(5, 7) that include more than one SL are nontrivial clusters, and the other clusters are trivial clusters which only contain one SL. Under such a scenario, chaos synchronization can be only achieved between intra-cluster SLs, as such, the chaotic synchronization-based applications in the SLs networks would be limited to some extent. To overcome this limitation, we propose a novel method referred as cluster fusion and apply it to the original 7-SLs network. The cluster fusion can be implemented among two or more asymmetric coupling SLs groups. Here, taking the cluster fusion of two different clusters for instance, the cluster fusion is performed by firstly finding the adjacent SLs sets S₁ and S₂ of two distinct clusters (cluster 1 and cluster 2), and the common adjacent SLs set is the intersection of two sets, which is mathematically expressed as: S₁∩ S₂. Then, new injections from adjacent SLs set S₂-S₁∩S₂ are introduced to each SL in cluster 1, and similarly, the outputs of adjacent SLs set S₁-S₁∩S₂ are symmetrically injected to each SL in cluster 2. Accordingly, the inter-cluster SLs of two clusters have identical adjacent SLs. That is, the SLs originally belonging to two different clusters are symmetric to each other, and consequently a hybrid cluster composed of two original clusters is produced after using cluster fusion. For instance, as shown in Fig. 1(b), by introducing additional injections from the adjacent SLs (SL3, SL5, SL7) of SL4 to SL6, two trivial clusters C(4) and C(6) in Fig. 1(a) become symmetrical and merge into a trivial-hybrid cluster referred as C(4, 6). Similarly, Figs. 1(c) and 1(d) respectively present the other two available scenarios: cluster fusion of one trivial cluster C(6) and one non-trivial cluster C(5, 7) as well as cluster fusion of two non-trivial clusters C(1, 2) and C(5, 7), where a trivial-nontrivial-hybrid cluster C(5, 6, 7) and a nontrivial-hybrid cluster C(1, 2, 5, 7) are obtained. It is shown that the original ACSLs can be symmetrically coupled in structure with the contribution of cluster fusion, and based on this, chaos synchronization can be achievable. Note that, the cluster fusion among multiple asymmetric coupling groups is similar to that between two clusters, and for the sake of simplicity, the cluster fusion process between multiple SLs groups is not given here.

Fig. 1. The topologies of (a) exemplary original SLs network, and SLs networks under cluster fusions of (b) two trivial clusters, (c) one trivial cluster and one non-trivial cluster, and (d) two non-trivial clusters. Here, the SLs with same color belong to same cluster. Especially, the generated hybrid clusters are cycled by red dotted lines and the red solid lines in (b), (c), and (d) denote the directed injections introduced by cluster fusion.

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For the purpose of simulation, the Lang-Kobayashi rate equations modified by taking the chaotic coupling terms into consideration are regarded as the mathematical model for the SLs in the original and cluster-fusion SLs networks, and are described as [26–29]:

(1)$$\begin{aligned} \frac{{d{E_m}^{(i)}{(}t{)}}}{{dt}} &= \frac{{(1 + i\alpha )}}{2}({G_m}^{(i)}\textrm{(}t\textrm{)} - \frac{1}{{{\tau _p}}}){E_m}^{(i)}{(}t{)}\\ & + {\sigma ^{(i)}}\sum\limits_{l = 1}^n {{A_{ml}}^{(i)}} {E_l}^{(i)}(t - \tau )\exp ( - i\omega \tau ) + \sqrt {2\beta {N_m}^{(i)}(t)} {\chi _m}^{(i)}(t), \end{aligned}$$

(2)$$\frac{{d{N_m}^{(i)}(t)}}{{dt}} = \frac{{\mu {I_{th}}}}{q} - \frac{{{N_m}^{(i)}(t)}}{{{\tau _e}}} - {G_m}^{(i)}(t){||{{E_m}} ^{(i)}}(t{ ) ||^2},$$

(3)$${G_m}^{(i)}(t) = \frac{{g({N_m}^{(i)}(t) - {N_\textrm{0}})}}{{1 + s||{{E_m}^{(i)}} (t{{ ) ||}^2}}},$$

(4)$${A^{(1)}} = \left( {\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&1&0&0&0&0\\ 1&1&0&1&1&0&1\\ 0&0&1&0&1&1&1\\ 0&0&1&1&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&1&1&0&0&0 \end{array}} \right),\quad {A^{(2)}} = \left( {\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&1&0&0&0&0\\ 1&1&0&1&1&0&1\\ 0&0&1&0&1&1&1\\ 0&0&1&1&0&0&0\\ 0&0&1&1&1&0&1\\ 0&0&1&1&0&0&0 \end{array}} \right),$$

(5)$${A^{(3)}} = \left( {\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&1&0&0&0&0\\ 1&1&0&1&1&0&1\\ 0&0&1&0&1&1&1\\ 0&0&1&1&0&0&0\\ 0&0&1&1&0&0&0\\ 0&0&1&1&0&0&0 \end{array}} \right),\quad {A^{(4)}} = \left( {\begin{array}{ccccccc} 0&0&1&1&0&0&0\\ 0&0&1&1&0&0&0\\ 1&1&0&1&1&0&1\\ 0&0&1&0&1&1&1\\ 0&0&1&1&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&1&1&0&0&0 \end{array}} \right),$$

In these equations, the subscript i indicates the i-th SLs network shown in Fig. 1, (m), (l)=1, 2, …, n, denote SLm and SLl, and the size of SLs networks is n. E(t) denotes the slowly varying complex electric field, N(t) and G(t) represent the carrier density and the optical gain, respectively. The second term in the right hand of Eq. (1) stands for mutual couplings in which A_ml=1 means the output of SLl is injected to SLm, whereas A_ml=0 indicates there is no injection from SLl to SLm. The adjacent matrices A⁽¹⁾, A⁽²⁾, A⁽³⁾, and A⁽⁴⁾ of the exemplary original SLs network and those with cluster fusion are given in Eqs. (4) and (5). Additionally, in these SLs, the spontaneous emission noise is modeled by the Gaussian noise χ(t) with unity variance and zero mean. The internal parameters of SLs in the networks shown in Fig. 1 are set as the typical values reported in [30,31]: the linewidth-enhancement factor α=5, the photon lifetime τ_p=2ps, the current factor µ=1.5, the threshold current I_th=14.7 mA, the electric charge q=1.6×10⁻¹⁹C, the carrier lifetime τ_e=2 ns, the spontaneous emission rate β=1.5×10⁻⁶ns^-1, the differential gain coefficient g=1.5×10⁴s^-1, the transparency carrier density N₀=1.5×10⁸, the gain saturation coefficient s=5×10⁻⁷, the wavelength of SLs λ₀=1550 nm, the frequency ω=1.216×10¹⁵rad/s, the coupling time delay τ=5 ns. Unless otherwise stated, the coupling strength σ of SLs networks in Figs. 1(a)–1(d) are 41ns^-1, 40ns^-1, 35ns^-1, 35ns^-1, respectively. In our simulations, the fourth-order Runge–Kutta algorithm is used to solve the rate equations mentioned above.

To mathematically and quantitatively describe the quality of chaos synchronization, the cross-correlation function (CCF) that is usually adopted to quantify the synchronization quality is introduced. And the CCF for SLm and SLl is written as [32,33]:

(6)$${C_{m,l}}^{(i)}(\Delta t) = \frac{{\left\langle {({P_m}^{(i)}(t + \Delta t) - \left\langle {{P_m}^{(i)}(t + \Delta t)} \right\rangle ) \cdot ({P_l}^{(i)}(t) - \left\langle {{P_l}^{(i)}(t)} \right\rangle )} \right\rangle }}{{\sqrt {\left\langle {{{({P_m}^{(i)}(t + \Delta t) - \left\langle {{P_m}^{(i)}(t + \Delta t)} \right\rangle )}^2}} \right\rangle \cdot \left\langle {{{({P_l}^{(i)}(t) - \left\langle {{P_l}^{(i)}(t)} \right\rangle )}^2}} \right\rangle } }},$$

where Δt denotes the lag time, P(t)=|E(t)²| is the time series of chaos outputted by SLs, <•> means time average. The cross-correlation coefficient (CC) is defined as the maximum value of CCF within the interval of Δt${\in}$[-10 ns,10 ns], mathematically, C_m_,l⁽ⁱ⁾=max|C_m_,l⁽ⁱ⁾(Δt)|. SLm and SLl of i-th SLs network are regarded as synchronized only when C_m_,l⁽ⁱ⁾>0.95. In regard to the CC calculation of more than two SLs, the averaged CC is employed to evaluate the corresponding synchronization quality and defined as [24]:

(7)$${C_{5,6,7}}^{(i)} = \frac{{{C_{5,6}}^{(i)} + {C_{6,7}}^{(i)} + {C_{5,7}}^{(i)}}}{3},$$

(8)$${C_{1,2,5,7}}^{(i)} = \frac{{{C_{1,2}}^{(i)} + {C_{1,5}}^{(i)} + {C_{1,7}}^{(i)} + {C_{2,5}}^{(i)} + {C_{2,7}}^{(i)} + {C_{5,7}}^{(i)}}}{6},$$

3. Numerical results and discussion

The temporal waveforms of the nontrivial clusters in original SLs network, nontrivial clusters and new hybrid clusters in cluster-fusion SLs networks are presented in Fig. 2, and the corresponding pairwise CC plots of the SLs networks are illustrated in Fig. 3. As shown in Fig. 2(a), the waveforms of SLs in two original nontrivial clusters are isochronously consistent, indicating that satisfactory chaos synchronization is achieved. Correspondingly, the CC in Fig. 3(a) demonstrates that only the SLs belonging to same clusters are synchronized with C_1,2⁽¹⁾>0.95, C_5,7⁽¹⁾>0.95, while the ACSLs of different clusters are asynchronous due to their asymmetric couplings. Nevertheless, in the SLs networks with cluster fusion, hybrid clusters consisting of two trivial clusters (C(4), C(6)), one trivial cluster (C(6)) and one nontrivial cluster (C(5,7)), and two nontrivial clusters (C(1,2), C(5,7)) are respectively obtained. As shown in Figs. 2(b)–2(d), hybrid clusters C(4, 6), C(5, 6, 7) and C(1, 2, 5, 7) are generated with similar fluctuations within each hybrid cluster. For instance, in the cluster fusion scenario of two trivial clusters, it is demonstrated in Figs. 2(b) and 3(b) that the hybrid cluster C(4, 6) composed of two ACSLs is generated and well synchronized by using cluster fusion. Besides, the results in Figs. 3(b)–3(d) also show that high-quality chaos synchronization can be achieved in another two types of hybrid clusters, which verifies the feasibility of the proposed cluster fusion method.

Fig. 2. Temporal waveforms of (a) nontrivial clusters in the original SLs network, (b)-(d) nontrivial clusters and hybrid clusters in SLs networks with cluster fusion.

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Fig. 3. The pairwise CC of (a) the original SLs network, and (b)-(d) the SLs networks with cluster fusion as shown in Figs. 1(b)–1(d), respectively.

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To investigate the influence of coupling strength on the quality of chaos synchronization, Fig. 4 presents the CC of SLs in original network and those with cluster fusion versus coupling strength. In the original SLs network, it can be seen that the CC development trends are sensitive to the coupling strength. Obviously, for C(1,2) and C(5,7) in original SLs network, chaos synchronization can be achieved with a properly large coupling strength, while low correlations (C_4,6⁽¹⁾, C_5,6,7⁽¹⁾ and C_1,2,5,7⁽¹⁾) are observed among chaotic ACSLs belonging to different clusters. It is noticed that although large values of CC might be obtained in the weak coupling regions, the SLs are not chaotically synchronous due to their non-chaotic states here. In contrast with the desynchronized chaotic ACSLs in original network, as shown in Figs. 4(b)–4(d), satisfactory chaos synchronization can be achieved in the hybrid clusters C(4,6), C(5,6,7) and C(1,2,5,7) generated with cluster fusion by adjusting the coupling strength to the chaos synchronization range. Besides, regarding the previously remaining nontrivial clusters presented in Figs. 4(b) and 4(c), high-quality chaos synchronization can also be realized with development trends of CC similar to those in original SLs network. Therefore, it is indicated that compared with the original SLs network, along with the introduction of cluster fusion, new synchronization modes are found between ACSLs that belong to different clusters, and chaos synchronization can be still achieved in the previously remaining nontrivial clusters. That is, our proposed cluster fusion method is very effective for realizing the chaos synchronization of ACSLs in different scenarios.

Fig. 4. The CC of SLs in (a) the original network, and (b)-(d) the SLs networks with cluster fusion as the function of coupling strength.

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In order to quantitatively explore the parameter regions for chaos synchronization in original SLs clusters and hybrid clusters, we study the two dimensional maps of CC in original SLs network and SLs networks with cluster fusion, in the parameter spaces of coupling strength and current factor. Figure 5 displays the CC maps of corresponding SLs clusters, where the white dotted lines denote the boundaries (CC=0.95) between synchronization and asynchronization regions. As shown in Figs. 5(a1) and 5(a2), although the parameter ranges for chaos synchronization of two clusters are distinct from each other due to the asymmetric couplings between different clusters, both two original clusters C(1,2) and C(5,7) can be synchronized over wide parameter regions. However, in the original SLs network, ACSLs from different clusters are structurally asymmetrically coupled, so that chaotic synchronization is difficult to be observed among inter-cluster ACSLs as shown in Figs. 5(a3)–5(a5). Regarding the SLs networks with cluster fusion, on the one hand, new hybrid clusters C(4, 6), C(5, 6, 7) and C(1, 2, 5, 7) are generated, and compared with the CC maps of ACSLs in original network, wide operation regions for stable chaos synchronization in the parameter spaces of coupling strength and current factor are obtained as presented in Figs. 5(b2), 5(c2), and 5(d). On the other hand, as seen in Figs. 5(b1), 5(b3), and 5(c1), some changes of synchronization regions are observed in C(1, 2) and C(5, 7) of the cluster-fusion SLs networks in contrast with those in original SLs network. This is because with the introduction of cluster fusion, some new chaotic injections which would influence the dynamic behaviors of the SLs are introduced, and consequently the synchronization within nontrivial clusters would be affected to some extent. Nevertheless, wide dynamic operation range is still found in the previously remaining SLs clusters. Thus, it can be concluded that in the ACSLs network with cluster fusion, not only the newly-emerging hybrid clusters but also the previously-remaining clusters can be synchronized in wide parameter spaces of coupling strength and current factor.

Fig. 5. Two dimensional maps of CC (a) C_1,2⁽¹⁾, C_5,7⁽¹⁾, C_4,6⁽¹⁾, C_5,6,7⁽¹⁾ and C_1,2,5,7⁽¹⁾ in the original 7-SLs network, (b) C_1,2⁽²⁾, C_5,7⁽²⁾ and C_4,6⁽²⁾ of the remaining clusters C(1, 2) and C(5, 7) and new trivial-hybrid cluster C(4, 6), (c) C_1,2⁽³⁾ and C_5,6,7⁽³⁾ of remaining cluster C(1, 2) and trivial-nontrivial-hybrid cluster C(5, 6, 7), and (d) C_1,2,5,7⁽⁴⁾ of nontrivial-hybrid cluster C(1, 2, 5, 7) in the SLs networks with cluster fusion, as the functions of coupling strength and current factor.

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In this work, the proposed cluster fusion method is implemented by introducing new chaotic injections to improve the symmetry between ACSLs. Hence, it is valuable to discuss the influences of intrinsic parameter mismatches of ACSLs and injection strength mismatch between newly introduced and previous chaotic injections on the quality of chaos synchronization in three types of SLs hybrid clusters. Here, the intrinsic parameter mismatches that are defined as [7]: α’=α(1+u), g’=g(1+u), τ_p’=τ_p(1+u), τ_e’=τ_e(1-u), N₀’=N₀(1-u), s’=s(1-u), are respectively implemented between SL4 and SL6; SL6 and SL5, SL7; SL1, SL2 and SL5, SL7. Besides, we introduce the mismatched injection strength σ’=σ(1+u_σ) into one of the new chaotic injections in each hybrid cluster. Figure 6 presents the CC of each hybrid cluster as the functions of mismatches in intrinsic parameters and injection strength. As illustrated in Fig. 6(a), when u is set around 0, the CC of three clusters is almost equal to 1. As the intrinsic parameters mismatch deviates from zero, the values of CC decrease slowly. Nevertheless, high-quality chaos synchronization of three hybrid clusters can be obtained over the interval of u = [-6%, 6%], which means that the ACSLs’ chaos synchronization based on cluster fusion method has no strict requirement for a good match of intrinsic parameters of ACSLs. With respect to mismatch in injection strength, it is obviously observed that the chaos synchronization in C(4, 6) is more robust to the mismatch of injection strength than synchronization in C(5, 6, 7) and C(1, 2, 5, 7). This is because the proportion of mismatched chaotic injection to all injections from other SLs to C(4, 6) is lower than those of other two hybrid clusters. Therefore, it can be concluded that the chaos synchronization in trivial-hybrid cluster, trivial-nontrivial-hybrid cluster and nontrivial-hybrid cluster is all robust to the intrinsic parameters mismatch between ACSLs and injection strength mismatch between newly-introduced chaotic injections and previous chaotic injections to some extent.

Fig. 6. CC of three types of hybrid clusters C(4, 6), C(5, 6, 7) and C(1, 2, 5, 7) versus (a) mismatch of intrinsic parameters between ACSLs and (b) mismatch of injection strength between newly introduced injection and previous chaotic injections.

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In the above, based on an exemplary 7-SLs network, we systematically discuss the synchronization properties and the parameters spaces for chaos synchronization in original ACSLs network and newly-emerging hybrid clusters generated with cluster fusion for comparison. The results show that by using cluster fusion, new hybrid clusters consisting of original ACSLs are generated, and satisfactory chaos synchronization in trivial-hybrid cluster, trivial-nontrivial-hybrid cluster and nontrivial-hybrid cluster can be respectively achieved over a wide dynamic operation range. To demonstrate the universality of the proposed cluster fusion method, on the one hand, the partial cluster fusion deserves exploration. On the other hand, the demonstration of feasibility of cluster fusion in other types of SLs networks is also required. Therefore, in the typical star-type SLs networks [34,35], the partial cluster-fusion-based chaos synchronization in two star-type SLs networks is further investigated in the following.

(9)$${A_1} = \left( {\begin{array}{ccccccccc} 0&1&1&1&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&1&1&1&1\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0&0 \end{array}} \right){A_2} = \left( {\begin{array}{ccccccccc} 0&1&1&1&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 1&0&0&0&1&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&1&1&1&1\\ 1&0&0&0&1&0&0&0&0\\ 1&0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0&0 \end{array}} \right),$$

The topologies of two star-type SLs networks and their corresponding partial-cluster-fusion SLs network are shown in Fig. 7. Here, the parameters of SLs in the star-type SLs networks and network with cluster fusion are identical to those in 7-SLs networks. Besides, the adjacent matrixes A₁ of two star-type SLs networks and A₂ of SLs network with cluster fusion are presented in Eq. (9). In each star-type SLs network, the SLs can be classified into two SLs clusters, that is, one nontrivial cluster composed of side SLs and one trivial cluster including only a hub SL. Thus, the clusters shown in Figs. 7(a) can be referred as: C(1), C(2, 3, 4), C(5), C(6, 7, 8, 9). In this part, the partial cluster fusion is considered by applying cluster fusion to partial ACSLs of two nontrivial clusters C(2, 3, 4) and C(6, 7, 8, 9). Without loss of generality, one SL of C(2, 3, 4) and two SLs of C(6, 7, 8, 9) are taken for instance to verify the feasibility of partial cluster fusion in star-type SLs networks. It is worth mentioning that the cluster fusion between ACSLs of two independent star-type SLs networks is similar to that within a SLs network shown in Fig. 1(a). Namely, as shown in Fig. 7(b), the cluster fusion between SL3, SL6, and SL7 is implemented by introducing additional injections from the adjacent SL1 of SL3 to SL6 and SL7, and the output of SL5 that adjacent to SL6 and SL7 is injected to SL3. Then, SL3, SL6 and SL7 form a new hybrid cluster C(3, 6, 7).

Fig. 7. Topologies of the (a) star-type SLs networks with three side SLs and four side SLs, respectively, and (b) the SLs network with partial cluster fusion of two side-SLs clusters.

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Figure 8 presents the pairwise CC of the original star-type SLs networks and the SLs network with partial cluster fusion of SL3 from C(2, 3, 4) and SL6 and SL7 from C(6, 7, 8, 9). It is shown in Fig. 8(a) that without cluster fusion, chaos synchronization can only be achieved between intra-cluster SLs of side-SLs clusters C(2, 3, 4) and C(6, 7, 8, 9), respectively, while undoubtedly there is no synchronization between SLs from two independent star-type SLs networks. However, with partial cluster fusion of two independent side-SLs clusters, a new hybrid cluster consisting of SL3, SL6, and SL7 are generated, and the other SLs that are not involved in cluster fusion remain unchanged. Consequently, there are two remaining nontrivial clusters C(2, 4), C(8, 9) and one newly-generated hybrid cluster C(3, 6, 7) in the SLs network with partial cluster fusion as shown in Fig. 7(b). By appropriately setting the coupling strength of the cluster-fusion star-type network, it is observed in Fig. 8(b) that not only the intra-cluster SLs of two remaining nontrivial clusters but also the SLs inside hybrid cluster C(3, 6, 7) are well synchronized. That is, in comparison with the low correlations of SL3, SL6, and SL7 observed in original star-type networks, satisfactory chaos synchronization is obtained among SL3, SL6, and SL7 with the contribution of partial cluster fusion. Thus, we can conclude that the cluster fusion is effective for the achievement of chaos synchronization between partial ACSLs of two clusters in star-type SLs networks, so as to be able to satisfy common synchronization demands. Moreover, the results indicate that our proposed cluster fusion method is also applicable to other SLs networks, which further demonstrates its universality in network-scenario ACSLs synchronization and its applications.

Fig. 8. Pairwise CC of (a) original star-type networks and (b) SLs network with partial cluster fusion, with coupling strength equaling to 55ns^-1 and 50ns^-1, respectively.

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4. Conclusion

We systematically and numerically investigate the chaos synchronization between ACSLs of different clusters by utilizing a novel cluster fusion method. The cluster fusion-based chaos synchronization among ACSLs of different trivial clusters, different trivial cluster and non-trivial cluster, and different non-trivial clusters in an exemplary 7-SLs network are thoroughly discussed. To validate the universality of the proposed cluster fusion, the synchronization properties of partial ACSLs of different clusters in typical star-type SLs networks are further explored. The results show that the ACSLs from different clusters can be symmetrically coupled and form new hybrid clusters with the contribution of cluster fusion. Based on this, high-quality chaos synchronization can be respectively achieved in the trivial-hybrid cluster, trivial-nontrivial-hybrid cluster and nontrivial-hybrid cluster over wide parameter regions. Besides, the chaos synchronization in the generated hybrid clusters is robust to the intrinsic parameter mismatch between ACSLs and the mismatch of injection strength between introduced chaotic injections and original ones to some extent. Moreover, satisfactory chaos synchronization between ACSLs belonging to two independent star-type SLs networks is also available, which further verifies the universality of the cluster fusion method. Our results offer a novel way to realize chaos synchronization among ACSLs in general SLs networks, which have potential applications in the network-scenario key distribution and communication.

Funding

National Natural Science Foundation of China (61671119, 61805031); Fundamental Research Funds for the Central Universities (ZYGX2019J003); Sichuan Province Science and Technology Support Program (2021JDJQ0023); Science and Technology Commission of Shanghai Municipality (SKLSFO2020-05).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

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Chaos synchronization based on cluster fusion in asymmetric coupling semiconductor lasers networks

Abstract

1. Introduction

2. Network topologies and theoretical model

3. Numerical results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (9)

Optics Express