Synchronization in small networks of time-delay coupled chaotic diode lasers

Regulation of cluster synchronization in multilayer networks of delay coupled semiconductor lasers with the use of disjoint layer symmetry

In real-world complex systems, heterogeneous components often interact in complex connection patterns and could be schematized by a formalism of multilayer network. In this work, the synchronization characteristics of multilayer network composed of semiconductor lasers (SLs) are investigated systematically. It is demonstrated that the interplay between different layers plays an important role on the synchronization patterns. We elucidate that the performance of cluster synchronization could be facilitated effectively with the introduction of disjoint layer symmetry into network topology. Intertwined stability of clusters from different layers could be decoupled into independent, and the parameter spaces for stable synchronization are extended significantly. The robustness of our proposed regulation scheme on operation parameters is numerically evaluated. Furthermore, the generality of presented theoretical results is validated in networks with more complex topology and multiple layers.

Matrix Measure-Based Projective Synchronization on Coupled Neural Networks With Clustering Trees

This article mainly studies the projective quasisynchronization for an array of nonlinear heterogeneous-coupled neural networks with mixed time-varying delays and a cluster-tree topology structure. For the sake of the mismatched parameters and the mutual influence among distinct clusters, the exponential and global quasisynchronization within a prescribed error bound instead of complete synchronization for the coupled neural networks with clustering trees is investigated. A kind of pinning impulsive controllers is designed, which will be imposed on the selected neural networks with some largest norms of error states at each impulsive instant in different clusters. By employing the concept of the average impulsive interval, the matrix measure method, and the Lyapunov stability theorem, sufficient conditions for the realization of the cluster projective quasisynchronization are derived. Meanwhile, in terms of the formula of variation of parameters and the comparison principle for the impulsive systems with mixed time-varying delays, the convergence rate and the synchronization error bound are precisely estimated. Furthermore, the synchronization error bound is efficiently optimized based on different functions of the impulsive effects. Finally, a numerical experiment is given to prove the results of theoretical analysis.

Hierarchical-dependent cluster synchronization in directed networks with semiconductor lasers

Cluster synchronization in complex networks with mutually coupled semiconductor lasers (SLs) has recently been extensively studied. However, most of the previous works on cluster synchronization patterns have concentrated on undirected networks. Here, we numerically study the complete cluster synchronization patterns in directed networks composed of SLs, and demonstrate that the values of the SLs parameter and network parameter play a prominent role on the formation and stability of cluster synchronization patterns. Moreover, it is shown that there is a hierarchical dependency between the synchronization stability of different clusters in directed networks. The stability of one cluster can be affected by another cluster, but not vice versa. Without loss of generality, the results are validated in another SLs network with more complex topology.

Strong cluster synchronization in complex semiconductor laser networks with time delay signature suppression

Cluster synchronization is a state where clusters of nodes inside the network exhibit isochronous synchronization. Here, we present a mechanism to realize the strong cluster synchronization in semiconductor laser (SL) networks with complex topology, where stable cluster synchronization is achieved with decreased correlation between dynamics of different clusters and time delay signature concealment. We elucidate that, with the removal of intra-coupling within clusters, the stability of cluster synchronization could be enhanced effectively, while the statistical correlation among dynamics of each cluster decreases. Moreover, it is demonstrated that the correlation between clusters can be further reduced with the introduction of dual-path injection and frequency detuning. The robustness of strong cluster synchronization on operation parameters is discussed systematically. Time delay signature in chaotic outputs of SL network is concealed simultaneously with heterogeneous inter-coupling among different clusters. Our results suggest a new approach to control the cluster synchronization in complex SL networks and may potentially lead to new network solutions for communication schemes and encryption key distribution.

Chaotic synchronization of a distant star-type laser network with multiple optical injections

A novel multi-injection module (MIM) is introduced into a typical distant star-type laser network, which is composed of a hub semiconductor laser node (H-SLN), star semiconductor laser nodes (S-SLNs) and tens of kilometers of fiber links. The chaotic synchronization of this distant network is investigated both experimentally and theoretically. As a result of using the MIM, a significantly low correlation (about 0.2) is successfully achieved between the H-SLN and S-SLNs in different clusters. This correlation is much lower than in previously reported results. Even when the fiber length is extended to 80 kilometers a low correlation (about 0.18) between the H-SLN and S-SLNs in different clusters is also obtained. Moreover, the dependence of chaotic synchronization on the operating conditions, such as the injection power, frequency detuning, and frequency mismatch between arbitrary nodes are examined. Lastly, using a theoretical model, we discuss the broad conditions for achieving chaotic synchronization among S-SLNs in the same cluster, and analyze the effect of the MIM branch number on chaotic synchronization.

Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression

The isochronous cluster synchronization with time delay (TD) signature suppression in delay-coupled vertical-cavity surface-emitting laser (VCSEL) networks subject to variable-polarization optical injection (VPOI) is theoretically and numerically studied. Based on the inherent symmetries of network topology, parameter spaces for stable cluster synchronization are presented, and zero-lag synchronization are achieved for VCSELs in same clusters. Additionally, the TD signature reduction for the dynamics of VCSELs in the stable clusters are systematically discussed. It is shown that both moderate polarizer angle and frequency detuning between different clusters have strengthen the effect of TD signature suppression. Moreover, the isochronous cluster synchronization with TD signature concealment is also verified in another VPOI-VCSEL network with different topology, indicating the generality of proposed results. Our results shed a new light on the research of chaos synchronization and chaos-based secure communications in VCSEL networks.

Mutual coupling and synchronization of optically coupled quantum-dot micropillar lasers at ultra-low light levels

Synchronization of coupled oscillators at the transition between classical physics and quantum physics has become an emerging research topic at the crossroads of nonlinear dynamics and nanophotonics. We study this unexplored field by using quantum dot microlasers as optical oscillators. Operating in the regime of cavity quantum electrodynamics (cQED) with an intracavity photon number on the order of 10 and output powers in the 100 nW range, these devices have high β-factors associated with enhanced spontaneous emission noise. We identify synchronization of mutually coupled microlasers via frequency locking associated with a sub-gigahertz locking range. A theoretical analysis of the coupling behavior reveals striking differences from optical synchronization in the classical domain with negligible spontaneous emission noise. Beyond that, additional self-feedback leads to zero-lag synchronization of coupled microlasers at ultra-low light levels. Our work has high potential to pave the way for future experiments in the quantum regime of synchronization.

Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback

The cluster synchronization of mutually coupled vertical-cavity surface-emitting lasers (VCSELs) networks subject to variable-polarization optical feedback (VPOF) with symmetric structure is theoretically investigated. Zero-lag synchronization could be achieved between different VCSELs within same cluster in such networks, which is solely derived from the intrinsic symmetry of network topology. The influences of significant parameters of VCSELs networks on the stability of cluster synchronization are further discussed. Moreover, it is shown that the polarizer angle of optical feedback in VCSELs plays a particularly important role on the formation of cluster.

Synchronization of fluctuating delay-coupled chaotic networks

We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. Focusing on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone, we compare the synchronization properties of static and fluctuating networks in the regime of large delays. We find that random network switching may enhance the stability of synchronized states. Synchronization appears to be maximally stable when fluctuations are much faster than the time-delay, whereas it disappears for very slow fluctuations. For fluctuation time scales of the order of the time-delay, we report a resynchronizing effect in finite-size networks. Moreover, we observe characteristic oscillations in all regimes, with a periodicity related to the time-delay, as the system approaches or drifts away from the synchronized state.

Stationary patterns in star networks of bistable units: Theory and application to chemical reactions

We present theoretical and experimental studies on pattern formation with bistable dynamical units coupled in a star network configuration. By applying a localized perturbation to the central or the peripheral elements, we demonstrate the subsequent spreading, pinning, or retraction of the activations; such analysis enables the characterization of the formation of stationary patterns of localized activity. The results are interpreted with a theoretical analysis of a simplified bistable reaction-diffusion model. Weak coupling results in trivial pinned states where the activation cannot propagate. At strong coupling, a uniform state is expected with active or inactive elements at small or large degree networks, respectively. A nontrivial stationary spatial pattern, corresponding to an activation pinning, is predicted to occur at an intermediate number of peripheral elements and at intermediate coupling strengths, where the central activation of the network is pinned, but the peripheral activation propagates toward the center. The results are confirmed in experiments with star networks of bistable electrochemical reactions. The experiments confirm the existence of the stationary spatial patterns and the dependence of coupling strength on the number of peripheral elements for transitions between pinned and retreating or spreading fronts in forced network configurations (where the central or periphery elements are forced to maintain their states).

Experimental synchrony of semiconductor lasers in coupled networks

The emission and synchronization of mutually-coupled semiconductor lasers with short cavities has been already recorded, with transversely unstable solutions existing within the chaotic synchronization manifold. Noise and laser-mismatch induced instabilities cause short de-synchronization events within the overall generalized synchronization, influencing the pragmatism of using these signals in secure data exchange applications. However, such operation can be functional for user authentication and sensing applications by assessing a time-averaged performance of synchrony. Until now, this has not been examined either in large-scale laser network configurations or in large transmission coupling paths, as real network implementations oblige. Here we present the first implementation of a fully-coupled fiber network with up to 16 semiconductor lasers, independently controlled and coupled through long interacting cavities. High level of consistent global or cluster synchrony via chaotic signals is demonstrated among all devices of the same origin and under appropriate operation. Devices that are not identical fail to synchronize at any condition, when coupled to the network. Under multiplexed operation, groups of lasers that emit at spectral distances as low as 50pm are shown to preserve intra-cluster synchronization when transmitted in the same fiber-optic channel, despite their large bandwidth of emitted signals.

Clustering in delay-coupled smooth and relaxational chemical oscillators

We investigate cluster synchronization in networks of nonlinear systems with time-delayed coupling. Using a generic model for a system close to the Hopf bifurcation, we predict the order of appearance of different cluster states and their corresponding common frequencies depending upon coupling delay. We may tune the delay time in order to ensure the existence and stability of a specific cluster state. We qualitatively and quantitatively confirm these results in experiments with chemical oscillators. The experiments also exhibit strongly nonlinear relaxation oscillations as we increase the voltage, i.e., go further away from the Hopf bifurcation. In this regime, we find secondary cluster states with delay-dependent phase lags. These cluster states appear in addition to primary states with delay-independent phase lags observed near the Hopf bifurcation. Extending the theory on Hopf normal-form oscillators, we are able to account for realistic interaction functions, yielding good agreement with experimental findings.

Coupled lasers: phase versus chaos synchronization

The synchronization of chaotic lasers and the optical phase synchronization of light originating in multiple coupled lasers have both been extensively studied. However, the interplay between these two phenomena, especially at the network level, is unexplored. Here, we experimentally compare these phenomena by controlling the heterogeneity of the coupling delay times of two lasers. While chaotic lasers exhibit deterioration in synchronization as the time delay heterogeneity increases, phase synchronization is found to be independent of heterogeneity. The experimental results are found to be in agreement with numerical simulations for semiconductor lasers.

Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators

We experimentally demonstrate group synchrony in a network of four nonlinear optoelectronic oscillators with time-delayed coupling. We divide the nodes into two groups of two each, by giving each group different parameters and by enabling only intergroup coupling. When coupled in this fashion, the two groups display different dynamics, with no isochronal synchrony between them, but the nodes in a single group are isochronally synchronized, even though there is no intragroup coupling. We compare experimental behavior with theoretical and numerical results.

Synchronization by elastic neuronal latencies

Psychological and physiological considerations entail that formation and functionality of neuronal cell assemblies depend upon synchronized repeated activation such as zero-lag synchronization. Several mechanisms for the emergence of this phenomenon have been suggested, including the global network quantity, the greatest common divisor of neuronal circuit delay loops. However, they require strict biological prerequisites such as precisely matched delays and connectivity, and synchronization is represented as a stationary mode of activity instead of a transient phenomenon. Here we show that the unavoidable increase in neuronal response latency to ongoing stimulation serves as a nonuniform gradual stretching of neuronal circuit delay loops. This apparent nuisance is revealed to be an essential mechanism in various types of neuronal time controllers, where synchronization emerges as a transient phenomenon and without predefined precisely matched synaptic delays. These findings are described in an experimental procedure where conditioned stimulations were enforced on a circuit of neurons embedded within a large-scale network of cortical cells in vitro, and are corroborated and extended by simulations of circuits composed of Hodgkin-Huxley neurons with time-dependent latencies. These findings announce a cortical time scale for time controllers based on tens of microseconds stretching of neuronal circuit delay loops per spike. They call for a reexamination of the role of the temporal periodic mode in brain functionality using advanced in vitro and in vivo experiments.

Cluster and group synchronization in delay-coupled networks

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.