Mittag-Leffler Synchronization Research Articles

Global Mittag-Leffler Synchronization for Fractional-Order BAM Neural Networks with Impulses and Multiple Variable Delays via Delayed-Feedback Control Strategy

This paper is concerned with the global Mittag-Leffler synchronization schemes for the Caputo type fractional-order BAM neural networks with multiple time-varying delays and impulsive effects. Based on the delayed-feedback control strategy and Lyapunov functional approach, the sufficient conditions are established to ensure the global Mittag-Leffler synchronization, which are described as the algebraic inequalities associated with the network parameters. The control gain constants can be searched in a wider range following the proposed synchronization conditions. The obtained results are more general and less conservative. A numerical example is also presented to illustrate the feasibility and effectiveness of the theoretical results based on the modified predictor–corrector algorithm.

Formula omitted]-synchronization and Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations

This paper investigates O(t−α)-synchronization and adaptive Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations. Firstly, based on the framework of Filippov solution and differential inclusion theory, using a Razumikhin-type method, some sufficient conditions ensuring the global O(t−α)-synchronization of considered networks are established via a linear-type discontinuous control. Next, a new fractional differential inequality is established and two new discontinuous adaptive controller is designed to achieve Mittag-Leffler synchronization between the drive system and the response systems using this inequality. Finally, two numerical simulations are given to show the effectiveness of the theoretical results. Our approach and theoretical results have a leading significance in the design of synchronized fractional-order memristive neural networks circuits involving discontinuous activations and time-varying delays.

Robust Mittag-Leffler Synchronization for Uncertain Fractional-Order Discontinuous Neural Networks via Non-fragile Control Strategy

This paper deals with the global robust non-fragile Mittag-Leffler synchronization issue for uncertain fractional-order neural networks with discontinuous activations. Firstly, a new inequality, which is concerned with the fractional derivative of the variable upper limit integral for the non-smooth integrable functional, is developed, and to be applied in the main results analysis. Then, the appropriate non-fragile controller with two types of gain perturbations is designed, and the global asymptotical stability is discussed for the synchronization error dynamical system by applying Lyapunov functional approach, non-smooth analysis theory and inequality analysis technique. In addition, the robust non-fragile Mittag-Leffler synchronization conditions are addressed in terms of linear matrix inequalities. Finally, two numerical examples are given to demonstrate the feasibility of the proposed non-fragile controller and the validity of the theoretical results.

Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction–diffusion terms using impulsive and linear controllers

In this paper, we propose a fractional-order neural network system with time-varying delays and reaction–diffusion terms. We first develop a new Mittag-Leffler synchronization strategy for the controlled nodes via impulsive controllers. Using the fractional Lyapunov method sufficient conditions are given. We also study the global Mittag-Leffler synchronization of two identical fractional impulsive reaction–diffusion neural networks using linear controllers, which was an open problem even for integer-order models. Since the Mittag-Leffler stability notion is a generalization of the exponential stability concept for fractional-order systems, our results extend and improve the exponential impulsive control theory of neural network system with time-varying delays and reaction–diffusion terms to the fractional-order case. The fractional-order derivatives allow us to model the long-term memory in the neural networks, and thus the present research provides with a conceptually straightforward mathematical representation of rather complex processes. Illustrative examples are presented to show the validity of the obtained results. We show that by means of appropriate impulsive controllers we can realize the stability goal and to control the qualitative behavior of the states. An image encryption scheme is extended using fractional derivatives.

Mittag-Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes.

This paper is concerned with the drive-response synchronization for a class of fractional-order bidirectional associative memory neural networks with time delays, as well as in the presence of discontinuous activation functions. The global existence of solution under the framework of Filippov for such networks is firstly obtained based on the fixed-point theorem for condensing map. Then the state feedback and impulsive controllers are, respectively, designed to ensure the Mittag-Leffler synchronization of these neural networks and two new synchronization criteria are obtained, which are expressed in terms of a fractional comparison principle and Razumikhin techniques. Numerical simulations are presented to validate the proposed methodologies.

Global synchronization in finite time for fractional-order neural networks with discontinuous activations and time delays

This paper is concerned with the global Mittag-Leffler synchronization and the synchronization in finite time for fractional-order neural networks (FNNs) with discontinuous activations and time delays. Firstly, the properties with respect to Mittag-Leffler convergence and convergence in finite time, which play a critical role in the investigation of the global synchronization of FNNs, are developed, respectively. Secondly, the novel state-feedback controller, which includes time delays and discontinuous factors, is designed to realize the synchronization goal. By applying the fractional differential inclusion theory, inequality analysis technique and the proposed convergence properties, the sufficient conditions to achieve the global Mittag-Leffler synchronization and the synchronization in finite time are addressed in terms of linear matrix inequalities (LMIs). In addition, the upper bound of the setting time of the global synchronization in finite time is explicitly evaluated. Finally, two examples are given to demonstrate the validity of the proposed design method and theoretical results.

Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays

This paper investigates the finite-time synchronization of delayed fractional-order memristive bidirectional associative memory neural networks (FMBAMNNs). Based on Lyapunov theory, fractional-order differential inequalities, norm properties and linear feedback controller, the separated criteria are obtained to ensure the finite-time synchronization of studied FMBAMNNs with different fractional order α, respectively. Moreover, the derived criteria are easily checked and they contribute to the reduction of amount of calculation. Finally, two numerical examples are given to demonstrate the effectiveness of the theoretical results.

Global Mittag-Leffler synchronization of fractional-order neural networks with discontinuous activations

This paper is concerned with the global Mittag-Leffler synchronization for a class of fractional-order neural networks with discontinuous activations (FNNDAs). We give the concept of Filippov solution for FNNDAs in the sense of Caputo’s fractional derivation. By using a singular Gronwall inequality and the properties of fractional calculus, the existence of global solution under the framework of Filippov for FNNDAs is proved. Based on the nonsmooth analysis and control theory, some sufficient criteria for the global Mittag-Leffler synchronization of FNNDAs are derived by designing a suitable controller. The proposed results enrich and enhance the previous reports. Finally, one numerical example is given to demonstrate the effectiveness of the theoretical results.

Mittag–Leffler synchronization of fractional-order uncertain chaotic systems**Project supported by the National Natural Science Foundation of China (Grant No. 61171034) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. R1110443).

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.

Robust synchronization for a class of fractional-order dynamical system via linear state variable

This paper reports the issue of robust synchronization for a class of fractional-order dynamical system with parameter mismatch and stochastic disturbance. Based on the fractional Lyapunov stability theorem, a linear controller is obtained to guarantee robust Mittag–Leffler synchronization of the fractional-order dynamical system. The gain matrix has been derived by solving the linear matrix inequality. As illustrative examples, the synchronization of fractional-order Newton–Leipnik chaotic system and fractional-order Chen hyperchaotic system are executed to verify the effectiveness of the proposed scheme.