Exploring synchronizability of complex dynamical networks from edges perspective

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.

Complex dynamical networks consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made complex networks such as social networks, food networks, neural networks, the World Wide Web, computer networks, electrical power grid and so on, can be effectively modelled by complex dynamical networks. Synchronization of a group of dynamical nodes in a complex network topology is one of the most interesting and significant phenomenon in complex dynamical networks. In this section, we shall discuss impulsive synchronization of the discrete delay networks. By employing the impulsive control method and the Razumikhin-type results obtained in previous chapter, two criteria for exponential synchronization will be derived.

Network topology and node dynamics play a key role in forming synchronization of complex networks. Unfortunately there is no effective synchronization criterion for pinning synchronization of complex dynamical networks with switching topology. In this paper, pinning synchronization of complex dynamical networks with switching topology is studied. Two basic problems are considered: one is pinning synchronization of switched complex networks under arbitrary switching; the other is pinning synchronization of switched complex networks by design of switching when synchronization cannot achieved by using any individual connection topology alone. For the two problems, common Lyapunov function method and single Lyapunov function method are used respectively, some global synchronization criteria are proposed and the designed switching law is given. Finally, simulation results verify the validity of the results.

This paper investigates the synchronisation of nonlinear coupled complex dynamical networks with different nonlinear nodes and different orders by using decentralised dynamical compensation controllers. We propose a dynamical network mathematical model with similar nonlinear nodes, whose dimensions are different. For this kind of network model, the decentralised dynamical compensation controllers are designed for the state synchronisation of the coupled nodes. In addition, the synchronisation manifold is defined as an invariant manifold, which is regarded as the generalised case of dynamical networks with the same nodes’ dynamics. Furthermore, some stability criteria for the synchronisation are derived by means of rigorous theoretical analysis. Finally, numerical examples are presented to verify the effectiveness of the obtained theoretical results.

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.

Although function projective synchronization in complex dynamical networks has been extensively studied in the literature, few papers deal with the problem between two different complex networks with correlated random disturbances. In this paper, we present some novel techniques to analyze the problem of synchronization. A probability approach is introduced to obtain an almost sure synchronization criterion. We also present some efficient approaches to analyze the problem of exponential synchronization. For the problem of synchronization in some complex networks, our approaches not only can replace the LaSalle-type theorem but also allow improvements of existing results in the literature. Finally, some numerical examples are provided to demonstrate the effectiveness of the proposed approaches.

In this chapter, we provide a basic introduction to complex and adaptive dynamical networks. We introduce graph theoretical preliminaries along with different classes of networks studied in the subsequent chapters. We discuss different types of models and coupling schemes. Further, we show how adaptivity is modeled in the context of synaptic plasticity and outline the reduction to phase oscillator models with phase difference-dependent plasticity.

This manuscript explores the analysis of passivity and synchronization criteria for a complex fractional-order dynamical network model with derivative couplings and time-varying delays. The passivity problem of the proposed network model is deduced using various inequality methods and presented as a linear matrix inequality. To ensure complete synchronization for a fractional-order complex dynamical network with derivative couplings (CDNMDC), we derive suitable criteria using an adaptive feedback control method. Additionally, we investigate the synchronization criterion of these complex networks while accounting for parameter uncertainties. Finally, we provide an example to demonstrate the effectiveness of the proposed solutions.

This paper is devoted to the study of synchronization problems in uncertain dynamical networks with time-delayed nodes and coupling delays. First, a complex dynamical network model with time-delayed nodes and coupling delays is given. Second, for a complex dynamical network with known or unknown but bounded nonlinear couplings, an adaptive controller is designed, which can ensure that the state of a dynamical network asymptotically synchronizes at the individual node state locally or globally in an arbitrary specified network. Then, the Lyapunov-Krasovskii stability theory is employed to estimate the network coupling parameters. The main results provide sufficient conditions for synchronization under local or global circumstances, respectively. Finally, two typical examples are given, using the M-G system as the nodes of the ring dynamical network and second-order nodes in the dynamical network with time-varying communication delays and switching communication topologies, which illustrate the effectiveness of the proposed controller design methods.

Complex networks are ubiquitous even in the form of defense networks in many fields such as ecological immune systems, biological immune systems, brain networks, air defense systems, national security, cyber security, peace and conflict studies, and so on. In the real world, complex networks exist in both static and dynamic forms. By understanding probabilistic failures and their possible defense mechanisms in static and dynamic complex networks, scientists and researchers can make decisions in developing and implementing methodologies of many fields as we can say, from the fields of biology to fields of technology. In this research, static and dynamic scale-free and small world networks have been generated as complex networks with network generation algorithms. Besides that, probabilistic failures and their possible defense mechanisms have also been generated and designed for previously generated complex networks. On these generated probabilistic failures and their possible defense mechanism, several mechanistic and statistical analysis methods have been applied. To generate these static and dynamic complex networks and their probabilistic failures and possible defense mechanism, two modified network generation algorithms along with several network measurements algorithms have been implemented where parameters have been chosen to produce possible types of static and dynamic scale-free and small world networks, and their corresponding probabilistic failures and possible defense mechanisms. Besides analyzing probabilistic failures and possible defense mechanisms for generated complex networks; the robustness, stability and sustainability of those networks have been studied and analyzed for probabilistic failures and their possible defense mechanisms.

Adaptive synchronization of the complex dynamical network with non-derivative and derivative coupling

In this paper, we study the adaptive pinning synchronization of complex dynamical networks based on intelligent materials with coupling delays and delays in the dynamical nodes. Via the theory of Lyapunov stability and adaptive pinning control technique, we design adaptive pinning feedback synchronization controller. The numerical example of the adaptive pinning synchronization problem has been also provided to demonstrate the effectiveness of the theory.

Finite-time synchronization of complex dynamical networks under delayed impulsive effects

In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically 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.