Fractional-order Neural Networks Research Articles

Fixed-deviation stabilization and synchronization for delayed fractional-order complex-valued neural networks.

In this paper, we study fixed-deviation stabilization and synchronization for fractional-order complex-valued neural networks with delays. By applying fractional calculus and fixed-deviation stability theory, sufficient conditions are given to ensure the fixed-deviation stabilization and synchronization for fractional-order complex-valued neural networks under the linear discontinuous controller. Finally, two simulation examples are presented to show the validity of theoretical results.

Asymptotic synchronization analysis of fractional-order octonion-valued neural networks with impulsive effects

<abstract><p>This paper deals with a class of fractional-order octonion-valued neural networks (FOOVNNs) with impulsive effects. Firstly, although the multiplication of octonion numbers does not satisfy the commutativity and associativity, we don't need to separate an octonion-valued system into eight real-valued systems. Secondly, by applying the appropriate Lyapunov function, and inequality techniques, we obtain the global asymptotical synchronization of FOOVNNs. Finally, we give two illustrative examples to illustrate the feasibility of the proposed method.</p></abstract>

Global Mittag-Leffler stability of Caputo fractional-order fuzzy inertial neural networks with delay

<abstract><p>This paper deals with the global Mittag-Leffler stability (GMLS) of Caputo fractional-order fuzzy inertial neural networks with time delay (CFOFINND). Based on Lyapunov stability theory and global fractional Halanay inequalities, the existence of unique equilibrium point and GMLS of CFOFINND have been established. A numerical example is given to illustrate the effectiveness of our results.</p></abstract>

Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control.

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in $ (\tau_{1}, \tau_{2}) $ plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system.

The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays

<abstract><p>The dynamic signal transmission process can be regarded as an anti-periodic process, and fractional-order inertial neural networks are widely used in signal processing and other fields, so anti-periodicity is also regarded as an important dynamic feature of inertial neural networks. This paper mainly studies the existence and Mittag-Leffler stability of anti-periodic solutions for a class of fractional-order inertial BAM neural networks with time-delays. By introducing variable substitution, the model with two different fractional-order derivatives is transformed into a model with only one fractional-order derivative of the same order. Using the properties of fractional-order calculus, the relationship between the fractional-order integral of the state function with and without time-delays is given. Firstly, the sufficient conditions for the boundedness and the Mittag-Leffler stability of the solutions for the system are derived. Secondly, by constructing the sequence solution of the function for the system and applying Ascoli-Arzela theorem, the sufficient conditions for the existence and Mittag-Leffler stability of the anti-periodic solution are given. Finally, the correctness of the conclusion is verified by a numerical example.</p></abstract>

Dissipative and dissipativity analysis for quaternion-valued fractional-order discrete-time memristive neural networks

Dissipative and dissipativity analysis for quaternion-valued fractional-order discrete-time memristive neural networks

Chaotic resonance in discrete fractional-order LIF neural network motifs

Chaotic resonance in discrete fractional-order LIF neural network motifs

Multistability of State-Dependent Switched Fractional-Order Hopfield Neural Networks With Mexican-Hat Activation Function and Its Application in Associative Memories.

The multistability and its application in associative memories are investigated in this article for state-dependent switched fractional-order Hopfield neural networks (FOHNNs) with Mexican-hat activation function (AF). Based on the Brouwer's fixed point theorem, the contraction mapping principle and the theory of fractional-order differential equations, some sufficient conditions are established to ensure the existence, exact existence and local stability of multiple equilibrium points (EPs) in the sense of Filippov, in which the positively invariant sets are also estimated. In particular, the analysis concerning the existence and stability of EPs is quite different from those in the literature because the considered system involves both fractional-order derivative and state-dependent switching. It should be pointed out that, compared with the results in the literature, the total number of EPs and stable EPs increases from 5l1 3l2 and 3l1 2l2 to 7l1 5l2 and 4l1 3l2 , respectively, where 0 ≤ l1 + l2 ≤ n with n being the system dimension. Besides, a new method is designed to realize associative memories for grayscale and color images by introducing a deviation vector, which, in comparison with the existing works, not only improves the utilization efficiency of EPs, but also reduces the system dimension and computational burden. Finally, the effectiveness of the theoretical results is illustrated by four numerical simulations.

Synchronization analysis of coupled fractional-order neural networks with time-varying delays.

In this paper, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems.

Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays

<abstract> <p>Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays are investigated in the article. We derive the relationship between the fractional-order integrals of the state function with and without delays through the division of time interval, using the properties of fractional calculus, and initial conditions. Moreover, by constructing the sequence solution of the system function which converges to a continuous function uniformly with the Arzela-Asoli theorem, a sufficient condition is obtained to ensure the existence of an anti-periodic solution and Mittag-Leffler stabilization of the system. In the final, we verify the correctness of the conclusion by numerical simulation.</p> </abstract>

PROJECTION SYNCHRONIZATION OF FUNCTIONAL FRACTIONAL-ORDER NEURAL NETWORKS WITH VARIABLE COEFFICIENTS

In this paper, the projection synchronization problem of functional fractional-order neural networks with variable coefficients and Caputo derivatives is studied. Firstly, a simple global projection synchronization scheme is designed according to the open-loop and adaptive feedback control. Secondly, by constructing a suitable Lyapunov function and utilizing the properties of delayed fractional-order differential inequalities, some criteria for the global projective synchronization of the variable coefficient functional neural networks with Caputo derivatives are obtained. Finally, a numerical example with many numerical simulations is employed to demonstrate the correctness and validity of the proposed method in this paper.

Neutral-Type and Mixed Delays in Fractional-Order Neural Networks: Asymptotic Stability Analysis

The lack of a conventional Lyapunov theory for fractional-order (FO) systems makes it difficult to study the dynamics of fractional-order neural networks (FONNs). Instead, the existing literature derives necessary conditions for various dynamic properties of FONNs using Halanay-type lemmas. However, when these lemmas are used, the results are frequently more conservative than those produced for integer-order neural networks (NNs). In order to provide sufficient criteria that are less conservative than those found in other research, a novel application of the Halanay-type lemma is made within this study. Thus, for extremely general FONNs containing neutral-type, time-varying, and distributed delays, sufficient conditions presented by way of linear matrix inequalities (LMIs) and algebraic inequalities are achieved. For the FO scenario, a model this broad and including so many different kinds of delays is developed for the first time. Additionally, a novel form of Lyapunov-like function is built, which results in less stringent algebraic inequalities. One of the first times in the setting of FONNs, the free-weighting matrix method is also used to further lower the conservativeness of the obtained conditions. Based on different Lyapunov-type functions, three theorems are developed regarding the asymptotic stability of the proposed networks. Three numerical simulations are used to demonstrate the theoretical developments.

Switching-Jumps-Dependent Quasi-Synchronization Criteria for Fractional-Order Memrisive Neural Networks

This paper investigates the switching-jumps-dependent quasi-synchronization issue for fractional-order memristive neural networks (FMNNs). First, a simplied linear feedback controller is applied. Then, in terms of several fractional order differential inequalities and two kinds of Lyapunov functions, two quasi-synchronization criteria expressed by linear matrix inequality (LMI)-based form and algebraic form are established, respectively. Meanwhile, the co-designed scheme for error bound and control gain is established. Compared with the previous quasi-synchronization results, a strong assumption that the system states must be bounded is removed. Finally, some simulation examples are carried out to display the feasibility and validity of the proposed analysis methods.

Bifurcation Mechanism for Fractional-Order Three-Triangle Multi-delayed Neural Networks

Bifurcation Mechanism for Fractional-Order Three-Triangle Multi-delayed Neural Networks

Stability and Synchronization of Fractional-Order Complex-Valued Inertial Neural Networks: A Direct Approach

This paper is dedicated to the asymptotic stability and synchronization for a type of fractional complex-valued inertial neural network by developing a direct analysis method. First, a new fractional differential inequality is presented for nonnegative functions, which provides an effective tool for the convergence analysis of fractional-order systems. Moreover, instead of the previous separation analysis for complex-valued neural networks, a class of Lyapunov functions composed of the complex-valued states and their fractional derivatives is constructed, and some compact stability criteria are derived. In synchronization analysis, unlike the existing control schemes for reduced-order subsystems, some feedback and adaptive control schemes, formed by the linear part and the fractional derivative part, are directly designed for the response fractional inertial neural networks, and some synchronization conditions are derived using the established fractional inequality. Finally, the theoretical analysis is supported via two numerical examples.

State estimation problem for fractional-order neural networks using event-triggered state observers

This article considers the state estimation problem for fractional-order neural networks subject to external disturbances using event-triggered state observers. For the first time, a novel method based on a discrete-time event-triggered mechanism and a nonlinear fractional-order state observer is obtained. A new discrete-time event-triggered mechanism is first proposed. Then a new nonlinear fractional-order state observer based on the newly discrete-time event-triggered mechanism is designed to estimate state vectors of the fractional-order neural networks robustly. The discrete-time event-triggered mechanism is Zeno-free, and it helps reduce unnecessary continuous signal transmission. A fractional-order-dependent condition to guarantee the existence of the discrete-time event-triggered nonlinear fractional-order state observer is established. This condition is translated into a convex optimization problem, which can be solved easily by MATLAB. The solution to this problem minimizes the attenuation levels and thus reduces the error in estimating the state vector of the fractional-order neural networks. The effectiveness of the proposed method is illustrated by two numerical examples and simulation results.

Delay-dependent and order-dependent conditions for stability and stabilization of fractional-order memristive neural networks with time-varying delays

The stability and stabilization problems of fractional-order memristive neural networks with time-varying delays are addressed. Based on Jensen-based integral inequalities, novel delay-dependent and order-dependent stability conditions for fractional-order memristive neural networks with time-varying delays are established. The proposed stability criterion is in terms of linear matrix inequalities and is easy to be verified and applied. Then, a linear feedback control law that stabilizes fractional-order memristive neural networks with time-varying delays is designed by utilizing the obtained stability conditions. Two numerical examples are used to illustrate the effectiveness and less conservativeness of the proposed results.

New insight into bifurcation of fractional-order 4D neural networks incorporating two different time delays

Delay has a vital influence on the dynamics of neural networks. Exploring the effect of time delay on the dynamics of neural networks has become a hot issue in mathematics and engineering fields. In this current manuscript, on the basis of the earlier publications, we put forward a new fractional-order 4D neural networks incorporating two different time delays. First of all, the existence and uniqueness, boundedness of the solution of the fractional-order 4D neural networks incorporating two different time delays. are analyzed by applying contraction mapping principle, construct of an adaptive function, respectively. Next, the stability and the emergence of Hopf bifurcation are explored by making use of the stability and bifurcation theory of fractional-order dynamical system. A series of novel stability criteria and bifurcation conditions guaranteeing the stability and the emergence of Hopf bifurcation of the considered fractional-order 4D neural networks under the different delay cases are built. What,s more, the impact of delay on stabilizing neural networks and controlling the emergence of Hopf bifurcation of neural networks is adequately uncovered. At last, Matlab simulation figures are presented to confirm scientificness of the derived prime conclusions. The derived prime conclusions of this manuscript are perfectly innovative and own momentous theoretical reference value in the control issue and design aspect of neural networks.

Settling-Time Estimation for Finite-Time Stabilization of Fractional-Order Quaternion-Valued Fuzzy NNs

This article solves the problems of finite-time control and settling-time estimation for fractional-order quaternion-valued fuzzy neural networks (FQFNNs) with time delays. A novel fractional differential inequality is established to estimate the settling time of the addressed system, which is more general and less conservative than the existing results. In addition, owing to the noncommutativity of multiplication of quaternions, the decomposition method is usually adopted to discuss the finite-time stabilization (FTS) of quaternion-valued neural networks, which inevitably doubles the dimensionality of the system and brings a great computational burden. To avoid the aforementioned issues, some new properties of the quaternion-valued signum function are presented for exploring the FTS of FQFNNs by the direct quaternion method without any decomposition. Then, 1-norm and 2-norm control strategies are designed to stabilize the addressed system in finite time, and several sufficient criteria are derived to ensure the FTS of FQFNNs. Finally, the validity of the obtained theoretical results and the superiority of the proposed estimation method are illustrated by numerical simulations.

Stability of the Caputo fractional-order inertial neural network with delay-dependent impulses

This article investigates the stability of the Caputo fractional-order inertial neural network (CFOINN) with destabilizing and stabilizing delay-dependent impulses, separately. Firstly, based on average impulsive delay (AID), stability conditions for the Caputo fractional-order (CFO) system with destabilizing and stabilizing delay-dependent impulses are separately given by utilizing properties of the CFO derivative. Then, by constructing the Lyapunov function for the CFOINN with delayed impulses, some stability criteria are obtained. Finally, numerical examples are presented to verify the effectiveness of the obtained results.