Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays

The synchronization problem for impulsive fractional-order neural networks with both time-varying bounded and distributed delays is studied. We study the case when the neural networks and the fractional derivatives of all neurons depend significantly on the moments of impulses and we consider both the cases of state coupling controllers and output coupling controllers. The fractional generalization of the Razumikhin method and Lyapunov functions is applied. Initially, a brief overview of the basic fractional derivatives of Lyapunov functions used in the literature is given. Some sufficient conditions are derived to realize the global Mittag–Leffler synchronization of impulsive fractional-order neural networks. Our results are illustrated with examples.

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Global Mittag–Leffler Synchronization for Impulsive Fractional-Order Neural Networks with Delays

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2017-0179 ◽

2018 ◽

Vol 19 (2) ◽

pp. 205-213 ◽

Cited By ~ 2

Author(s):

Ramziya Rifhat ◽

Ahmadjan Muhammadhaji ◽

Zhidong Teng

Keyword(s):

Neural Networks ◽

Fractional Order ◽

Lyapunov Method ◽

Sufficient Conditions ◽

Distributed Delays ◽

Time Varying ◽

Synchronization Problem

AbstractIn this paper, we investigate the synchronization problem of impulsive fractional-order neural networks with both time-varying and distributed delays. By using the fractional Lyapunov method and Mittag–Leffler function, some sufficient conditions are derived to realize the global Mittag–Leffler synchronization of impulsive fractional-order neural networks and one illustrative example is given to demonstrate the effectiveness of the obtained results.

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Impulsive Cohen-Grossberg neural networks with s-type distributed delays

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-011-0001-9 ◽

2011 ◽

Vol 48 (1) ◽

pp. 1-13

Author(s):

Haydar Akça ◽

Valéry Covachev

Keyword(s):

Neural Networks ◽

Exponential Stability ◽

Equilibrium Point ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Distributed Delays ◽

Unique Equilibrium ◽

Analysis Techniques

Abstract We study impulsive Cohen-Grossberg neural networks with S-type distributed delays. This type of delays in the presence of impulses is more general than the usual types of delays studied in the literature. Using analysis techniques we prove the existence of a unique equilibrium point. By means of simple and efficient Lyapunov functions we present some sufficient conditions for the exponential stability of the equilibrium.

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Synchronization of bidirectional N-coupled fractional-order chaotic systems with ring connection based on antisymmetric structure

Advances in Difference Equations ◽

10.1186/s13662-019-2380-1 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 17

Author(s):

Cuimei Jiang ◽

Akbar Zada ◽

M. Tamer Şenel ◽

Tongxing Li

Keyword(s):

Fractional Order ◽

Lyapunov Functions ◽

Chaotic Systems ◽

Design Method ◽

Sufficient Conditions ◽

Order Error ◽

Synchronization Problem ◽

Direct Design ◽

Error System ◽

Theoretical Results

Abstract This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results.

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Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0066 ◽

2018 ◽

Vol 21 (5) ◽

pp. 1238-1261 ◽

Cited By ~ 8

Author(s):

Mikhail I. Gomoyunov

Keyword(s):

Fractional Order ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Original System ◽

Control Problems ◽

Fractional Order Systems ◽

Control Procedures ◽

Fractional Differential ◽

And Control ◽

Derivatives Of

Abstract The paper is devoted to the development of control procedures with a guide for fractional order dynamical systems controlled under conditions of disturbances, uncertainties or counteractions. We consider a dynamical system which motion is described by ordinary fractional differential equations with the Caputo derivative of an order α ∈ (0, 1). For the case when the guide is, in a certain sense, a copy of the system, we propose a mutual aiming procedure between the original system and guide. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an essentially bounded measurable function. This estimate can be considered as a generalization of the known estimates of such type. We give an example that illustrates the workability of the proposed control procedures with a guide.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽

10.2298/fil1716217a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5217-5239 ◽

Cited By ~ 5

Author(s):

Ravi Agarwal ◽

Snehana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Intervals ◽

Dini Derivative ◽

Comparison Results ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

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Complex projection synchronization of fractional order uncertain complex-valued networks with time-varying coupling

Modern Physics Letters B ◽

10.1142/s0217984919503512 ◽

2019 ◽

Vol 33 (29) ◽

pp. 1950351 ◽

Cited By ~ 1

Author(s):

Dawei Ding ◽

Xiaolei Yao ◽

Hongwei Zhang

Keyword(s):

Fractional Order ◽

Coupling Strength ◽

Dynamic Networks ◽

Sufficient Conditions ◽

Feedback Gain ◽

Time Varying ◽

Order Complex ◽

Unknown Parameters ◽

Synchronization Problem ◽

Complex Valued

In this paper, the complex projection synchronization problem of fractional complex-valued dynamic networks is investigated. Considering the time-varying coupling and unknown parameters of the fractional order complex network, several decentralized adaptive strategies are designed to adjust the coupling strength and controller feedback gain in order to investigate the complex projection synchronization problem of the system. Moreover, based on the designed identification law, the uncertain parameters in the network can be estimated. Using adaptive law which balances the time-varying coupling strength and the feedback gain of the controller, some sufficient conditions are obtained for the complex projection synchronization of complex networks. Finally, numerical simulation examples are provided to illustrate the efficiency of the complex projection synchronization strategies of the fractional order complex dynamic networks.

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Impulsive Disturbances on the Dynamical Behavior of Complex-Valued Cohen-Grossberg Neural Networks with Both Time-Varying Delays and Continuously Distributed Delays

Complexity ◽

10.1155/2017/3826729 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12 ◽

Cited By ~ 5

Author(s):

Xiaohui Xu ◽

Jiye Zhang ◽

Quan Xu ◽

Zilong Chen ◽

Weifan Zheng

Keyword(s):

Neural Networks ◽

Exponential Stability ◽

Equilibrium Point ◽

Global Exponential Stability ◽

Dynamical Behavior ◽

Sufficient Conditions ◽

Distributed Delays ◽

Time Varying ◽

Homeomorphism Mapping ◽

Complex Valued

This paper studies the global exponential stability for a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays. Firstly, the existence and uniqueness of the equilibrium point of the system are analyzed by using the corresponding property of M-matrix and the theorem of homeomorphism mapping. Secondly, the global exponential stability of the equilibrium point of the system is studied by applying the vector Lyapunov function method and the mathematical induction method. The established sufficient conditions show the effects of both delays and impulsive strength on the exponential convergence rate. The obtained results in this paper are with a lower level of conservatism in comparison with some existing ones. Finally, three numerical examples with simulation results are given to illustrate the correctness of the proposed results.

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Existence and Globally Asymptotic Stability of Equilibrium Solution for Fractional-Order Hybrid BAM Neural Networks with Distributed Delays and Impulses

Complexity ◽

10.1155/2017/6875874 ◽

2017 ◽

Vol 2017 ◽

pp. 1-13 ◽

Cited By ~ 9

Author(s):

Hai Zhang ◽

Renyu Ye ◽

Jinde Cao ◽

Ahmed Alsaedi

Keyword(s):

Neural Networks ◽

Asymptotic Stability ◽

Fractional Order ◽

Lyapunov Functional ◽

Equilibrium Solution ◽

Distributed Delays ◽

Bam Neural Networks ◽

Network Systems ◽

Globally Asymptotic Stability ◽

Stability Of Equilibrium

This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the U-layer and V-layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.

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