Cluster synchronization in networks of coupled nonidentical dynamical systems

Cluster synchronization in networks of partial-state coupled linear systems via pinning control technique

The neural networks of the human brain exhibit dual structural characteristics of smallworld and scale-free networks, and their electrical activity is readily modulated by electromagnetic fields. Constructing neural network models that mimic biological structures is crucial for elucidating the brain's information processing mechanisms and the pathological basis of neurological disorders. This paper constructs a Small-World Scale-Free Neural Network (SWSFNN) model under electromagnetic effects using discrete memristors. By optimizing network topology via graph theory, we systematically investigate how memristor initial values and electromagnetic induction intensity influence the network dynamics. Numerical simulations reveal that memristor initial values affect neuronal firing patterns and regulate network synchronization. We also discovered a spontaneous “synchronization – cluster synchronization – synchronization” transition under constant parameters. This demonstrates that even without parameter changes, the inherent nonlinear interactions within the neural network system can drive spontaneous state transitions, yielding rich dynamical behaviors. Furthermore, increasing electromagnetic induction intensity similarly enhances network synchronization. This study provides theoretical foundations for understanding the nonlinear dynamical mechanisms and synchronization control of neural networks in electromagnetic environments, offering insights for neural computation and information processing.

The synchronization phenomenon is one of the crucial collective behaviors of the brain network, with its abnormal patterns in the brain networks of individuals with mental disorders like attention deficit hyperactivity disorder (ADHD) capturing the interest of researchers. ADHD is a prevalent disorder among children, which is linked to adverse outcomes such as impaired recognition of others' emotional facial expressions, leading to awkward social interactions. Despite numerous studies focusing on the synchronization analysis of brain networks in ADHD children while observing emotional facial expressions, the path to synchronization in these networks has been overlooked. For this reason, this research aims to investigate the differences in the synchronization transition of brain networks between healthy and ADHD children while processing emotional facial expressions based on their electroencephalogram signals. The analysis of synchronization cluster sequences in children's brain networks reveals that the formation and progression of synchronization clusters toward complete synchronization follow a more organized and goal-directed pattern in the healthy group than in the ADHD group. Furthermore, examining synchronization clusters with significant differences in stability (P-values < 0.05) during the clustering sequence between the two groups highlights abnormal synchronization in the frontal brain region during the early stages of neural processing in the ADHD group. This finding may contribute to deficits in attentional control and emotional regulation observed in ADHD children. Moreover, the classification results indicate the high potential of synchronization transition-based features to distinguish between ADHD and healthy children, with maximum accuracy, precision, and sensitivity of 82%, 77%, and 90%, respectively.

This paper concentrates on the cluster synchronization such that each community achieves linear-generalized synchronization in complex networks. Here two control schemes are introduced to realize cluster synchronization. One is to adjust the coupling strength of edges which connect different groups automatically, and to use the pinning control simultaneously. The other scheme is to adjust the coupling strength of edges in each group and to select only one node in each group to control. Finally, we present some simulations to demonstrate the correction of our conclusion.

Comment on “Cluster synchronization in the adaptive complex dynamical networks via a novel approach” [Phys. Lett. A 375 (2011) 1559

In this paper, cluster synchronization in community network with nonidentical nodes is investigated. By combining intermittency with a pinning control scheme, some effective controllers are designed. In the control scheme, only one node in each community is controlled and coupling weights of a spanning tree in each community are enhanced. Based on the Lyapunov function method and mathematical analysis technique, two results for achieving cluster synchronization are obtained. Noticeably, by introducing an adaptive strategy, some universal adaptive intermittent pinning controllers are designed for different networks. Finally, two numerical simulations are performed to verify the correctness of the derived results.

Cluster synchronization of complex dynamic networks under pinning control via a limited capacity communication channel

Pinning control of cluster synchronization in a globally connected network of chaotic oscillators is studied. It is found in simulations that when the pinning strength exceeds a critical value, the oscillators are synchronized into two different clusters, one formed by the pinned oscillators and the other one formed by the unpinned oscillators. The numerical results are analyzed by the generalized method of master stability function (MSF), in which it is shown that whereas the method is able to predict the synchronization behaviors of the pinned oscillators, it fails to predict the synchronization behaviors of the unpinned oscillators. By checking the trajectories of the oscillators in the phase space, it is found that the failure is attributed to the deformed synchronization manifold of the unpinned oscillators, which is clearly deviated from that of isolated oscillator under strong pinnings. A similar phenomenon is also observed in the pinning control of cluster synchronization in a complex network of symmetric structures and in the self-organized cluster synchronization of networked neural oscillators. The findings are important complements to the generalized MSF method and provide an alternative approach to the manipulation of synchronization behaviors in complex network systems.

formula omitted] cluster synchronization and state estimation for complex dynamical networks with mixed time delays

With the rapid growth of information technology, controlling the synchronization of large-scale networks has become more challenging and costly. To address this, we propose a scalable regulation scheme of cluster synchronization in a directed complex semiconductor lasers (SLs) network. The stability of chaotic synchronization for the entire network could be scalably regulated by only modifying the parameters of the driver cluster based on the hierarchical dependency of cluster synchronization in directed network, thus improving the control efficiency and reducing costs. The influence of parameters associated with the driver cluster is discussed systematically, and we numerically and experimentally demonstrate that a complex SLs network could achieve ideal cluster synchronization with the introduction of the proposed scalable regulation scheme. At last, encrypting and decrypting images is realized by leveraging the controlled chaotic cluster synchronization in complex, SLs networks.

The issue of cluster synchronization and disturbance rejection of multi‐weighted complex dynamical networks subject to nonlinearities and external disturbances are studied in this article. In particular, a hybrid‐triggering control scheme is implemented to mitigate the burden of the network transmission according to the prespecified triggering condition, under which each node will transmit its signal independently to the controller. To be more specific, a Bernoulli distributed stochastic variable is introduced to describe the switching law of the communication scheme. Also, the disturbance rejection performance is carried out with the aid of an improved equivalent input disturbance estimator approach. The sufficient criterion in the form of linear matrix inequalities is derived by constructing a suitable Lyapunov–Krasovskii functional and employing stochastic theory. Based on these settings, the feedback gain matrices and the trigger parameters affirming the desired cluster synchronization performance of the considered networks are established. Eventually, two illustrative examples including Lur'e networks with simulations are presented, which evidently proves the effectiveness of the obtained results.

We study the evolution of coupled chaotic dynamics on networks and investigate the role of degree-degree correlation in the networks' cluster synchronizability. We find that an increase in the disassortativity can lead to an increase or a decrease in the cluster synchronizability depending on the degree distribution and average connectivity of the network. Networks with heterogeneous degree distribution exhibit significant changes in cluster synchronizability as well as in the phenomena behind cluster synchronization as compared to those of homogeneous networks. Interestingly, cluster synchronizability of a network may be very different from global synchronizability due to the presence of the driven phenomenon behind the cluster formation. Furthermore, we show how degeneracy at the zero eigenvalues provides an understanding of the occurrence of the driven phenomenon behind the synchronization in disassortative networks. The results demonstrate the importance of degree-degree correlations in determining cluster synchronization behavior of complex networks and hence have potential applications in understanding and predicting dynamical behavior of complex systems ranging from brain to social systems.

This article investigates a bounded cluster synchronization problem of dynamical systems, which can be of the generic linear type or Lipschitz nonlinear type, over directed switching network. Each cluster is equipped with a virtual leader which produces the desired trajectory for the agents to track. It is required only a fraction of systems is influenced by the leader. The interaction topology, which describes the information exchange among the dynamical systems as well as the virtual leaders, is allowed to be time varying with a well-defined average over an infinite horizon. That is, each augmented cluster, consisting of the agents as well as the corresponding virtual leader, in the time-average network topology is required to have a directed spanning tree. We then transform the cluster synchronization problem into a stability problem via the averaging method. It is proved that the convergence property for both types of dynamical systems is exclusively determined by the averaging system if the network topology switches sufficiently fast compared to original systems. Finally, it is concluded that if the intracluster coupling strength of the time-average topology is stronger than a threshold, then bounded cluster synchronization can be realized for a fast switching linear or nonlinear systems. Two examples are provided to verify our results.

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.

Projective cluster synchronization of dynamical networks is considered via pinning impulsive control. The projective cluster synchronization, which is try to make nodes of response network realize cluster synchronization with the control about the driven network. A sufficient condition is given to solve the projective cluster synchronization problem based on Lyapunov theory. Finally, the effectiveness of control mechanism is demonstrated by a numerical example.