Mathematical problems for complex networks

Copyright @ 2012 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is made available through the Brunel Open Access Publishing Fund.Complex networks do exist in our lives. The brain is a neural network. The global economy is a network of national economies. Computer viruses routinely spread through the Internet. Food-webs, ecosystems, and metabolic pathways can be represented by networks. Energy is distributed through transportation networks in living organisms, man-made infrastructures, and other physical systems. Dynamic behaviors of complex networks, such as stability, periodic oscillation, bifurcation, or even chaos, are ubiquitous in the real world and often reconfigurable. Networks have been studied in the context of dynamical systems in a range of disciplines. However, until recently there has been relatively little work that treats dynamics as a function of network structure, where the states of both the nodes and the edges can change, and the topology of the network itself often evolves in time. Some major problems have not been fully investigated, such as the behavior of stability, synchronization and chaos control for complex networks, as well as their applications in, for example, communication and bioinformatics

Fixed-Time Synchronization for Hybrid Coupled Dynamical Networks with Multilinks and Time-Varying Delays

This paper concerns the problem of fixed/finite-time synchronization of hybrid coupled dynamical networks. The considered dynamical networks with multilinks contain only one transmittal time-varying delay for each subnetwork, which makes us get hold of more interesting and practical points. Two kinds of delay-dependent feedback controllers with multilinks as well as appropriate Lyapunov functions are defined to achieve the goal of fixed-time synchronization and finite-time synchronization for the networks. Some novel and effective criteria of hybrid coupled networks are derived based on fixed-time and finite-time stability analysis. Finally, two numerical simulation examples are given to show the effectiveness of the results proposed in our paper

Impulsive mean square exponential synchronization of stochastic dynamical networks with hybrid time-varying delays

This paper investigates the mean square exponential synchronization problem for complex dynamical networks with stochastic disturbances and hybrid time-varying delays, both internal delay and coupling delay are considered in the model. At the same time, the coupled time-delay is also probabilistic in two time interval. Impulsive control method is applied to force all nodes synchronize to a chaotic orbit, and impulsive input delay is also taken into account. Based on the theory of stochastic differential equation, an impulsive differential inequality and some analysis techniques, several simple and useful criteria are derived to ensure mean square exponential synchronization of the stochastic dynamical networks. Furthermore, pinning impulsive strategy is studied. An effective method is introduced to select the controlled nodes at each impulsive constants. Numerical simulations are exploited to demonstrate the effectiveness of the theory results in this paper

Finite-time synchronization of multi-layer nonlinear coupled complex networks via intermittent feedback control

This paper addresses the problem of finite-time synchronization for a class of multi-layer nonlinear coupled complex networks via intermittent feedback control. Firstly, based on finite-time stability theory, some novel criteria are given to guarantee that the error system of drive-response systems is still finite-time stable under an inherently discontinuous controller. Then, by proposing two kinds of intermittent feedback control laws, sufficient conditions of finite-time synchronization of two kinds of multi-layer complex networks are derived, respectively. The time delay between different layers is also taken into consideration. Finally, a numerical example is provided to verify the effectiveness of the proposed methods.http://www.elsevier.com/locate/neucom2018-02-28hb2017Electrical, Electronic and Computer Engineerin

Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations

This article, we explore the asymptotic stability and asymptotic synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous neuron activation functions (FCGNNDDs). First, under the framework of Filippov theory and differ- ential inclusion theoretical analysis, the global existence of Filippov solution for FCGNNDDs is studied by means of the given growth condition. Second, by virtue of suitable Lyapunov functional, Young inequality and comparison theorem for fractional order delayed linear system, some global asymptotic stability conditions for such system is derived by limiting discontinuous neuron activations. Third, the global asymptotic synchronization condition for FCGNNDDs is obtained based on the pinning control. At last, two numerical simula- tions are given to verify the theoretical findings.N/

Adaptive Synchronization of Complex Dynamical Multilinks Networks with Similar Nodes

This paper studies the synchronization of complex dynamical networks with multilinks and similar nodes. The dynamics of all the nodes in the networks are impossible to be completely identical due to the differences of parameters or the existence of perturbations. Networks with similar nodes are universal in the real world. In order to depict the similarity of the similar nodes, we give the definition of the minimal similarity of the nodes in the network for the first time. We find the threshold of the minimal similarity of the nodes in the network. If the minimal similarity of the nodes is bigger than the threshold, then the similar nodes can achieve synchronization without controllers. Otherwise, adaptive synchronization method is adopted to synchronize similar nodes in the network. Some new synchronization criteria are proposed based on the Lyapunov stability theory. Finally, numerical simulations are given to illustrate the feasibility and the effectiveness of the proposed theoretical results

Finite-time synchronisation of neural networks with discrete and distributed delays via periodically intermittent memory feedback control

In this paper, finite-time synchronization between two chaotic systems with discrete and distributed delays is investigated by using periodically intermittent memory feedback control. Based on finite-time stability theory, some novel and effective synchronization criteria of intermit- tent control are derived by means of linear matrix inequalities (LMIs) and differential inequality techniques. Furthermore, a necessary condition of finite-time synchronization of intermittent con- trol is given for neural networks with discrete and distributed delays. A numerical example on two chaotic neural networks shows the effectiveness and correctness of the derived theoretical results. In addition, a secure communication synchronization problem is presented to demonstrate practical effectiveness of the proposed method.National Natural Science Foundation of China (Grant No. 61273183, No. 61374028 and No. 61374085).http://www.ietdl.orgIET-CTAhb2016Electrical, Electronic and Computer Engineerin

Synchronization of Complex-Valued Dynamical Networks

Dynamical networks (DNs) have been broadly applied to describe natural and human systems consisting of a large number of interactive individuals. Common examples include Internet, food webs, social networks, neural networks, etc. One of the crucial and significant collective behaviors of DNs is known as synchronization. In reality, synchronization phenomena may occur either inside a network or between two or more networks, which are called “inner synchronization” and “outer synchronization”, respectively. On the other hand, many real systems are more suitably characterized by complex-valued dynamical systems, such as quantum systems, complex Lorenz system, and complex-valued neural networks. The main focus of this thesis is on synchronization of complex-valued dynamical networks (CVDNs). In this thesis, we firstly design a delay-dependent pinning impulsive controller to study synchronization of time-delay CVDNs. By taking advantage of the Lyapunov function in the complex field, some delay-independent synchronization criteria of CVDNs are established, which generalizes some existing synchronization results. Then, by employing the Lyapunov functional in the complex field, several delay-dependent sufficient conditions on synchronization of CVDNs with various sizes of delays are constructed. Moreover, we study synchronization of CVDNs with time-varying delays under distributed impulsive controllers. By taking advantage of time-varying Lyapunov function/ functional in the complex domain, several synchronization criteria for CVDNs with time-varying delays are derived in terms of complex-valued linear matrix inequalities (LMIs). Then, we propose a memory-based event-triggered impulsive control (ETIC) scheme with three levels of events in the complex field to investigate the synchronization problem of CVDNs with both discrete and distributed time delays, and we further consider an event-triggered pinning impulsive control (ETPIC) scheme combining the proposed ETIC and a pinning algorithm to study synchronization of time-delay CVDNs. Results show that the proposed ETIC scheme and ETPIC scheme can effectively synchronize CVDNs with the desired trajectory. Secondly, we study generalized outer synchronization of drive-response time-delayed CVDNs via hybrid control. A hybrid controller is proposed in the complex domain to construct response complex-valued networks. Some generalized outer synchronization criteria for drive-response CVDNs are established, which extend the existing generalized outer synchronization results to the complex field. Thirdly, we study the average-consensus problem of potential complex-valued multi-agent systems. A complex-variable hybrid consensus protocol is proposed, and time delays are taken into account in both the continuous-time protocol and the discrete-time protocol. Delay-dependent sufficient conditions are established to guarantee the proposed complex-variable hybrid consensus protocol can solve the average-consensus problem. Lastly, as a practical application for complex-valued networked systems, the synchronization problem of master-slave complex-valued neural networks (CVNNs) is studied via hybrid control and delayed ETPIC, respectively. We also investigate the state estimation problem of CVNNs by designing the adaptive impulsive observer in the complex field

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...