Mittag–Leffler synchronization of fractional-order uncertain chaotic systems

This paper deals with the synchronization of fractional-order chaotic systems with unknown parameters and unknown disturbances.An adaptive control scheme combined with fractional-order update laws is proposed.The asymptotic stability of the error system is proved in the sense of generalized Mittag–Leffler stability.The two fractional-order chaotic systems can be synchronized in the presence of model uncertainties and additive disturbances.Finally these new developments are illustrated in examples and numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.     

Coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions

In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation function, Filippov solution of the neural network is defined. Based on Brouwer's fixed point theorem and definition of Mittag–Leffler stability, sufficient criteria are established to ensure the existence of (2k + 3)~n (k ≥ 1) equilibrium points, among which (k + 2)~n equilibrium points are locally Mittag–Leffler stable. Compared with the existing results, the derived results cover local Mittag–Leffler stability of both fractional-order and integral-order recurrent neural networks. Meanwhile discontinuous networks might have higher storage capacity than the continuous ones. Two numerical examples are elaborated to substantiate the effective of the theoretical results.     

Finite-time modified combination synchronization of memristive FitzHugh–Nagumo circuit with unknown disturbances

The memristor-based FitzHugh-Nagumo neuron system shows special dynamical behaviors, which make it have a good application in the fields of image processing and signal transmission. The accurate dimensionality reduction model of the system is established, which facilitates the internal interpretation and physical control of the multistability that depends on the initial values. The dynamical behavior analyses of the dimensionality reduction memristive system with different original initial states are developed through bifurcation diagram and Lyapunov index spectrum. Multistability and antimonotonicity with periodic-chaotic bubbles are investigated. The sliding mode control method with a smooth function is designed to achieve the finite-time synchronization of the multistable memristive neuron systems, which make three different behaviors of systems synchronized. The sliding mode controller can enable the system to quickly and smoothly implement combination synchronization within finite time. Numerical simulations show the effectiveness of the sliding mode controller designed.     

Linear-control-based synchronization of coexisting attractor networks with time delays

This paper introduces the concept of linear-control-based synchronization of coexisting attractor networks with time delays.Within the new framework,closed loop control for each dynamic node is realized through linear state feedback around its own arena in a decentralized way,where the feedback matrix is determined through consideration of the coordination of the node dynamics,the inner connected matrix and the outer connected matrix.Unlike previously existing results,the feedback gain matrix here is decoupled from the inner matrix;this not only guarantees the flexible choice of the gain matrix,but also leaves much space for inner matrix configuration.Synchronization of coexisting attractor networks with time delays is made possible in virtue of local interaction,which works in a distributed way between individual neighbours,and the linear feedback control for each node.Provided that the network is connected and balanced,synchronization will come true naturally,where theoretical proof is given via a Lyapunov function.For completeness,several illustrative examples are presented to further elucidate the novelty and efficacy of the proposed scheme.     

H∞ synchronization of chaotic neural networks with time-varying delays

In this paper,we investigate the problem of H∞ synchronization for chaotic neural networks with time-varying delays.A new model of the networks with disturbances in both master and slave systems is presented.By constructing a suitable Lyapunov–Krasovskii functional and using a reciprocally convex approach,a novel H∞ synchronization criterion for the networks concerned is established in terms of linear matrix inequalities(LMIs)which can be easily solved by various effective optimization algorithms.Two numerical examples are given to illustrate the effectiveness of the proposed method.     

Novel delay dependent stability analysis of Takagi—Sugeno fuzzy uncertain neural networks with time varying delays

This paper presents the stability analysis for a class of neural networks with time varying delays that are represented by the Takagi-Sugeno (T-S) model. The main results given here focus on the stability criteria using a new Lyapunov functional. New relaxed conditions and new linear matrix inequality-based designs are proposed that outperform the previous results found in the literature. Numerical examples are provided to show that the achieved conditions are less conservative than the existing ones in the literature.     

Robust adaptive synchronization of chaotic neural networks by slide technique

In this paper, we focus on the robust adaptive synchronization between two coupled chaotic neural networks with all the parameters unknown and time-varying delay. In order to increase the robustness of the two coupled neural networks, the key idea is that a sliding-mode-type controller is employed. Moreover, without the estimate values of the network unknown parameters taken as an updating object, a new updating object is introduced in the constructing of controller. Using the proposed controller, without any requirements for the boundedness, monotonicity and differentiability of activation functions, and symmetry of connections, the two coupled chaotic neural networks can achieve global robust synchronization no matter what their initial states are. Finally, the numerical simulation validates the effectiveness and feasibility of the proposed technique.     

Error and attack tolerance of synchronization in Hindmarsh–Rose neural networks with community structure

Synchronization is one of the most important features observed in large-scale complex networks of interacting dynamical systems. As is well known, there is a close relation between the network topology and the network synchronizability. Using the coupled Hindmarsh–Rose neurons with community structure as a model network, in this paper we explore how failures of the nodes due to random errors or intentional attacks affect the synchronizability of community networks. The intentional attacks are realized by removing a fraction of the nodes with high values in some centrality measure such as the centralities of degree, eigenvector, betweenness and closeness. According to the master stability function method, we employ the algebraic connectivity of the considered community network as an indicator to examine the network synchronizability. Numerical evidences show that the node failure strategy based on the betweenness centrality has the most influence on the synchronizability of community networks. With this node failure strategy for a given network with a fixed number of communities, we find that the larger the degree of communities, the worse the network synchronizability; however, for a given network with a fixed degree of communities, we observe that the more the number of communities, the better the network synchronizability.     

H_∞ synchronization of chaotic Hopfield networks with time-varying delay: a resilient DOF control approach

This paper focuses on the issue of resilient dynamic output-feedback(DOF) control for H_∞ synchronization of chaotic Hopfield networks with time-varying delay. The aim is to determine a DOF controller with gain perturbations ensuring that the H_∞ norm from the external disturbances to the synchronization error is less than or equal to a prescribed bound. A delaydependent criterion for the H_∞ synchronization is derived by employing the Lyapunov functional method together with some recent inequalities. Then, with the help of some decoupling techniques, sufficient conditions on the existence of the resilient DOF controller are developed for both the time-varying and constant time-delay cases. Lastly, an example is used to illustrate the applicability of the results obtained.     

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen–Grossberg neural networks

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order CohenGrossberg neural networks(FOCGNNs) with time delays.Based on Brouwer's fixed point theorem,sufficient conditions are established to ensure the existence of Π_(i=1)~n(2 K_i+1) equilibrium points for FOCGNNs.Through the use of Hardy inequality,fractional Halanay inequality,and Lyapunov theory,some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of Π_(i=1)~n(K_i+1) equilibrium points for FOCGNNs·The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases.The activation functions are nonlinear and nonmonotonic.There could be many corner points in this general class of activation functions.The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points.Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories.Finally,two numerical examples are provided to illustrate the effectiveness of the obtained results.     

Order parameters and synchronization of FitzHugh—Nagumo small-world networks

This paper numerically investigates the order parameter and synchronisation in the small world connected FitzHugh--Nagumo excitable systems. The simulations show that the order parameter continuously decreases with increasing d, the quality of the synchronisation worsens for large noise intensity. As the coupling intensity goes up, the quality of the synchronisation worsens, and it finds that the larger rewiring probability becomes the larger order parameter. It obtains the complete phase diagram for a wide range of values of noise intensity d and control parameter g.     

More relaxed condition for dynamics of discrete time delayed Hopfield neural networks

The dynamics of discrete time delayed Hopfield neural networks is investigated. By using a difference inequality combining with the linear matrix inequality, a sufficient condition ensuring global exponential stability of the unique equilibrium point of the networks is found. The result obtained holds not only for constant delay but also for time-varying delays.     

H_∞ synchronization of the coronary artery system with input time-varying delay

This paper investigates the H_∞ synchronization of the coronary artery system with input delay and disturbance.We focus on reducing the conservatism of existing synchronization strategies.Base on the triple integral forms of the Lyapunov–Krasovskii functional(LKF),we utilize single and double integral forms of Wirtinger-based inequality to guarantee that the synchronization feedback controller has good performance against time-varying delay and external disturbance.The effectiveness of our strategy can be exhibited by simulations under the different time-varying delays and different disturbances.     

Exponential synchronization of chaotic Lur’e systems with time-varying delay via sampled-data control

In this paper, we study the exponential synchronization of chaotic Lur＇e systems with time-varying delays via sampled-data control by using sector nonlinearties. In order to make full use of information about sampling intervals and interval time-varying delays, new Lyapunov-Krasovskii functionals with triple integral terms are introduced. Based on the convex combination technique, two kinds of synchronization criteria are derived in terms of linear matrix inequal- ities, which can be efficiently solved via standard numerical software. Finally, three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.     

Adaptive coupled synchronization of non-autonomous systems in ring networks

The adaptive coupled synchronization method for non-autonomous systems is proposed. This method can avoid estimating the value of coupling coefficient. Under the uniform Lipschitz assumption, we derive the asymptotical synchronization for a general coupling ring network with N identical non-autonomous systems~ even when N is large enough. Strict theoretical proofs are given. Numerical simulations illustrate the effectiveness of the present method.     

Stochastic finite-time stability of reaction-diffusion Cohen–Grossberg neural networks with time-varying delays

This paper investigates the problem of the stochastic finite-time stability of reaction-diffusion Cohen-Grossberg neural networks with time varying delays using the Dirichlet boundary condition. The concept of finite-time stability for the system is first derived by using the stochastic conditions. By constructing a new Lyapunov–Krasovskii functional and utilizing Jensen’s inequality, Wirtinger’s type inequality technique, the Gronwall inequality and the linear matrix inequality (LMI) frame work, conditions are obtained which guarantee the stochastically finite-time stability of Cohen–Grossberg neural networks. Finally, two numerical examples are given to show the effectiveness of the proposed results.     

Global exponential stability of reaction diffusion neural networks with discrete and distributed time-varying delays

This paper investigates the global exponential stability of reaction-diffusion neural networks with discrete and distributed time-varying delays.By constructing a more general type of Lyapunov-Krasovskii functional combined with a free-weighting matrix approach and analysis techniques,delay-dependent exponential stability criteria are derived in the form of linear matrix inequalities.The obtained results are dependent on the size of the time-varying delays and the measure of the space,which are usually less conservative than delay-independent and space-independent ones.These results are easy to check,and improve upon the existing stability results.Some remarks are given to show the advantages of the obtained results over the previous results.A numerical example has been presented to show the usefulness of the derived linear matrix inequality(LMI)-based stability conditions.     

Application of adaptive filter in time delay estimation——generalized quadratic interpolation method

The time delay estimation(TDE)of two different received sig-nals from the same source has attracted many interests of researchers inthe field of signal processing.A method described for precisely estimatingtime delay in this paper is based on the assumption that little prioriknowledge on statistical characteristics is available for the received signals.The variance of the estimate is derived.The basic architecture of thismethod is to use the adaptive noise canceller,in the steady state,andto interpolate the weight coefficients by using a generalized quadratic inter-polation matrix.The formula of the time delay estimation is presented.The method proposed by F.A.Reed is a special case of this method.The hardware implementation is much easier than that of theconventional time delay estimation method.The results of the systemsimulation and the experimental results at sea show a good agreementwith the theoretical analysis.     

Nonlinear H∞ control of structured uncertain stochastic neural networks with discrete and distributed time varying delays

This paper is concerned with the problem of robust H∞ control for structured uncertain stochastic neural networks with both discrete and distributed time varying delays. A sufficient condition is presented for the existence of H∞ control based on the Lyapunov stability theory. The stability criterion is described in terms of linear matrix inequalities （LMIs）, which can be easily checked in practice. An example is provided to demonstrate the effectiveness of the proposed result.