Cluster synchronization control for the complex network with dynamic connection relations

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.

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

Complex networks are everywhere.Synchronization is a very important nonlinear phenomenon which universally exists in nature.Over the last decade,many researchers have further investigated the synchronization of complex dynamical networks,including identical synchronzation,clustering synchronization,partial synchronization,and so on.The characterization of synchronous speed of complex dynamical networks in most known res(?)lts is asymptotic.That is,complex networks can realize synchronization only when the time t tends to infinity.However,there are few results reported on how long complex networks can reach synchronization.Based on two kinds of typical complex dynamical networks with nonlinear coupling,this paper will further explore the finite-time synchronization of complex dynamical networks.In detail,under some suitable conditions,it is proved that the above complex dynamical networks can realize accurate synchronization within finite- time.Moreover,a typical numerical simulation is then given to validate the effectiveness of the proposed criteria for finitetime synchronization.It should be especially pointed out that the finite-time synchronization sucessfully overcomes the difficulty of infinite synchronous time.The above results have some important practical meaning for the real-world engineering application.

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.

This paper utilises the synthesised link observers to investigate the asymptotic outer synchronisation for two complex dynamical networks (CDNs) with time-varying links, in which one is called as the drive network (DN), another is the response network (RN). Meanwhile, each CDN is regarded to be composed of two mutually coupled subsystems: the node subsystem (NS) and the link subsystem (LS). In each CDN, the dynamics of NS is modelled as the vector differential equation with the unknown interaction caused by the nodes state of another network, and the LS is modelled as the matrix differential equation with link output. Two kinds of differential equations are regarded to be coupled with each other due to the coupling relation between nodes and their connections. By employing the above dynamical models of NSs and the outputs of LSs, the asymptotic state observers of two LSs are constructed, respectively, by which the adaptive controller of NS is synthesised for the DN under some mild conditions. The results in this paper show that the synthesised adaptive controller ensures not only the asymptotic outer synchronisation is achieved for DN and RN but also the corresponding states of LSs are asymptotically the same. Compared with the existing literature, the most prominent highlight of this paper is that two link observers are synthesised under the unavailable links to assist the adaptive controller of NS in RN to achieve the asymptotic outer synchronisation. Finally, the effectiveness of the theoretical results in this paper is proved by the numerical simulation.

In the existing research results of the complex dynamical networks controlled, the controllers are mainly used to guarantee the synchronization or stabilization of the nodes’ state, and the terms coupled with connection relationships may affect the behaviors of nodes, this obviously ignores the dynamic common behavior of the connection relationships between the nodes. In fact, from the point of view of large-scale system, a complex dynamical network can be regarded to be composed of two time-varying dynamic subsystems, which can be called the nodes subsystem and the connection relationships subsystem, respectively. Similar to the synchronization or stabilization of the nodes subsystem, some characteristic phenomena can be also emerged in the connection relationships subsystem. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. This paper focuses on the structural balance in dynamic complex networks. Generally speaking, the state of the connection relationships subsystem is difficult to be measured accurately in practical applications, and thus it is not easy to implant the controller directly into the connection relationships subsystem. It is noted that the nodes subsystem and the relationships subsystem are mutually coupled, which implies that the state of the connection relationships subsystem can be affected by the controllable state of nodes subsystem. Inspired by this observation, by using the structural balance theory of triad, the controller with the parameter adaptive law is proposed for the nodes subsystem in this paper, which may ensure the connection relationship matrix to approximate a given structural balance matrix in the sense of the uniformly ultimately bounded (UUB). That is, the structural balance may be obtained by employing the controlling state of the nodes subsystem. Finally, the simulations are used to show the validity of the method in this paper.

<abstract><p>In this paper, cluster synchronization in finite/fixed time for semi-Markovian switching complex dynamical networks (CDNs) with discontinuous dynamic nodes is studied. Firstly, the global fixed-time convergence principle of nonlinear systems with semi-Markovian switching is developed. Secondly, the novel state-feedback controllers, which include discontinuous factors and integral terms, are designed to achieve the global stochastic finite/fixed-time cluster synchronization. In the framework of Filippov stochastic differential inclusion, the Lyapunov-Krasovskii functional approach, Takagi-Sugeno(T-S) fuzzy theory, stochastic analysis theory, and inequality analysis techniques are applied, and the global stochastic finite/fixed time synchronization conditions are proposed in the form of linear matrix inequalities (LMIs). Moreover, the upper bound of the stochastic settling time is explicitly proposed. In addition, the correlations among the obtained results are interpreted analytically. Finally, two numerical examples are given to illustrate the correctness of the theoretical results.</p></abstract>

Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach

The scope of complex dynamical networks (CDNs) with dynamic edges is very wide, as it is composed of a class of realistic networks including web-winding systems, communication networks, neural networks, etc. However, a classic research topic in CDNs, the synchronization control problem, has not been effectively solved for CDNs with dynamic edges. This paper will investigate the emergence mechanism of synchronization from the perspective of large-scale systems. Firstly, a CDN with dynamic edges is conceptualized as an interconnected coupled system composed of an edge subsystem (ES) and a node subsystem (NS). Then, based on the proposed improved directed matrix ES model and expanded matrix inequality, this paper overcomes the limitations of coupling term design in node models and the strong correlation of tracking targets between nodes and edges. Due to the effect of the synthesized node controller and the auxiliary effect of the ES, state synchronization can be realized in the CDN. Finally, through simulation examples, the validity and advantages of our work compared to existing methods are demonstrated.

SummaryThe issues of cluster synchronization and state estimation for a class of nonlinear complex dynamical networks are concurrently focused in this study. In precise, led by the design of complex dynamical networks, a proportional integral‐based observer is implemented for achieving robustness, by the virtue of which the states of examined networks are estimated. Thereupon, a proportional integral observer‐based hybrid triggered control scheme is proposed, wherein, the switch‐over between time‐ and event‐ triggered scheme is governed through a Bernoulli distributed random variable. Further, with the application of mixed and passivity performance technique, the external disturbances for the addressed complex dynamical networks are reduced. Based on the Lyapunov stability theory, the requisite sufficient conditions in respect of linear matrix inequalities are provided which ensures the discussed complex dynamical network's synchronization phenomenon. In a further, the presented criteria are exploited for the attainment of required controller, proportional and integral gain matrices. In conclusion, the applicability of proposed results and efficacy of the undertaken strategy are evidently proven with the illustration of two numerical examples with Chua's circuit and Goodwin oscillator model.

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.

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

This paper deals with the function projective synchronization of two complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences. Based on Lyapunov stability theory and inequality transformation method, the robust adaptive synchronization controller is designed, by which the drive and response systems can achieve synchronization according to the function scaling factor. Different from some existing studies on nonlinear system with sector nonlinear input, this paper studies the synchronization of two complex dynamic networks when the boundary of sector nonlinear input is unknown. The controller does not include the boundary value of the sector nonlinear input and the time delay term, so it is more practical and relatively easy to implement. The corresponding simulation examples demonstrate the effectiveness of the proposed scheme.

The first synchronization clustering (SynC) algorithm based on an extensive Kuromato model was presented in 2010. In 2017, an effective synchronization clustering (ESynC) algorithm, inspired by SynC algorithm and a linear version of Vicsek model, was proposed. When facing complex data distributions, ESynC algorithm may regard an irregular and whole cluster as some micro-clusters. To conquer this shortcoming, a combined clustering algorithm based on ESynC algorithm and a merging judgement process of micro-clusters (CESynC) is presented. CESynC algorithm first uses ESynC algorithm to detect clusters or micro-clusters, then merges those conjoint micro-clusters by using a merging judgement process. For some datasets that ESynC and SynC cannot detect correct clusters, CESynC can capture natural clusters. From the simulation experiments, we observe that CESynC can get better (or the same) clustering results than (or as) that of ESynC in many cases. We also observe that the clustering results of CESynC and ESynC are often better than that of SynC. Therefore, we can say CESynC can often obtain better clustering quality than ESynC and SynC in some kinds of datasets. Further comparison experiments with some classical clustering algorithms demonstrate the clustering effect of CESynC.

Cluster synchronization of heterogeneous nonlinear multi-agent systems with actuator faults and IQCs through adaptive fault-tolerant pinning control