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

Finite-time Anti-synchronization of Memristive Stochastic BAM Neural Networks with Probabilistic Time-varying Delays

This paper investigates the drive-response finite-time anti-synchronization for memristive bidirectional associative memory neural networks (MBAMNNs). Firstly, a class of MBAMNNs with mixed probabilistic time-varying delays and stochastic perturbations is first formulated and analyzed in this paper. Secondly, an nonlinear control law is constructed and utilized to guarantee drive-response finite-time anti-synchronization of the neural networks. Thirdly, by employing some inequality technique and constructing an appropriate Lyapunov function, some anti-synchronization criteria are derived. Finally, a number simulation is provided to demonstrate the effectiveness of the proposed mechanism

Event-triggered communication for passivity and synchronisation of multi-weighted coupled neural networks with and without parameter uncertainties

A multi-weighted coupled neural networks (MWCNNs) model with event-triggered communication is studied here. On the one hand, the passivity of the presented network model is studied by utilising Lyapunov stability theory and some inequality techniques, and a synchronisation criterion based on the obtained output-strict passivity condition of MWCNNs with eventtriggered communication is derived. On the other hand, some robust passivity and robust synchronisation criteria based on output-strict passivity of the proposed network with uncertain parameters are presented. At last, two numerical examples are provided to testify the effectiveness of the output-strict passivity and robust synchronisation results

Fixed-time control of delayed neural networks with impulsive perturbations

This paper is concerned with the fixed-time stability of delayed neural networks with impulsive perturbations. By means of inequality analysis technique and Lyapunov function method, some novel fixed-time stability criteria for the addressed neural networks are derived in terms of linear matrix inequalities (LMIs). The settling time can be estimated without depending on any initial conditions but only on the designed controllers. In addition, two different controllers are designed for the impulsive delayed neural networks. Moreover, each controller involves three parts, in which each part has different role in the stabilization of the addressed neural networks. Finally, two numerical examples are provided to illustrate the effectiveness of the theoretical analysis

Synchronization of inertial memristive neural networks with time-varying delays via static or dynamic event-triggered control

Funding Information: This work was supported in part by the National Natural Science Foundation of China under Grant 61971185, the Major Research Project of the National Natural Science Foundation of China under Grant 91964108 and the Open Fund Project of Key Laboratory in Hunan Universities under Grant 18K010. Publisher Copyright: © 2020 Elsevier B.V.This paper investigates the synchronization problem of inertial memristive neural networks (IMNNs) with time-varying delays via event-triggered control (ETC) scheme and state feedback controller for the first time. First, two types of state feedback controllers are designed; the first type of controller is added to the transformational first-order system, and the second type of controller is added to the original second-order system. Next, based on each feedback controller, static event-triggered control (SETC) condition and dynamic event-triggered control (DETC) condition are presented to significantly reduce the update times of controller and decrease the computing cost. Then, some sufficient conditions are given such that synchronization of IMNNs with time-varying delays can be achieved under ETC schemes. Finally, a numerical simulation and some data analyses are given to verify the validity of the proposed results.Peer reviewe

General decay lag anti-synchronization of multi-weighted delayed coupled neural networks with reaction–diffusion terms

We propose a new anti-synchronization concept, called general decay lag anti-synchronization, by combining the definitions of decay synchronization and lag synchronization. Novel criteria for the decay lag anti-synchronization of multi-weighted delayed coupled reaction–diffusion neural networks (MWDCRDNNs) with and without bounded distributed delays are derived by constructing an appropriate nonlinear controller and using the Lyapunov functional method. Moreover, the robust decay lag anti-synchronization of MWDCRDNNs with and without bounded distributed delays is considered. Finally, two numerical simulations are performed to validate the obtained results

Razumikhin and Krasovskii stability of impulsive stochastic delay systems via uniformly stable function method

This paper generalizes Razumikhin-type theorem and Krasovskii stability theorem of impulsive stochastic delay systems. By proposing uniformly stable function (USF) in the form of impulse as a new tool, some properties about USF and some novel pth moment decay theorems are derived. Based on these new theorems, the stability theorems of impulsive stochastic linear delay system are acquired via the Razumikhin method and the Krasovskii method. The obtained results enhance the elasticity of the impulsive gain by comparing the previous results. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results

Pinning Cluster Synchronization in Linear Hybrid Coupled Delayed Dynamical Networks

The problem on cluster synchronization will be investigated for a class of delayed dynamical networks based on pinning control strategy. Through utilizing the combined convex technique and Kronecker product, two sufficient conditions can be derived to ensure the desired synchronization when the designed feedback controller is employed to each cluster. Moreover, the inner coupling matrices are unnecessarily restricted to be diagonal and the controller design can be converted into solving a series of linear matrix inequalities (LMIs), which greatly improve the present methods. Finally, two numerical examples are provided to demonstrate the effectiveness and reduced conservatism