Multiple Mittag-Leffler Stability of Delayed Fractional-Order Cohen-Grossberg Neural Networks via Mixed Monotone Operator Pair

On Ordered Weighted Averaging Operator and Monotone Takagi-Sugeno-Kang Fuzzy Inference Systems

The necessary and/or sufficient conditions for a Takagi-Sugeno-Kang Fuzzy Inference System (TSK-FIS) to be monotone has been a key research direction in the last two decades. In this article, we first define fuzzy membership functions (FMFs) with single and continuous support; and consider TSK-FIS with a grid partition strategy for computing its firing strengths with product T-norm (here after denoted as TSK-FIS-product). We also define a more general joint necessary condition, whereby each constituent itself is a necessary condition for the TSK-FIS-product model. The first necessary condition indicates that the normalized firing strength must not be indeterminate (i.e., 0/0), i.e., susceptible to the tomato classification problem. The second necessary condition indicates that all restricted consequents of fuzzy if-then rules must be defined. Based on the principle of the ordered weighted averaging (OWA) operator as well as the concept of increasing orness in OWA and hyperboxes, a general joint sufficient condition for a TSK-FIS-product model to be monotone is derived. Three case studies of the developed methods for undertaking failure mode and effect analysis (FMEA) and image processing tasks are presented. The results are compared, analyzed, and discussed, demonstrating the usefulness of our developed methods.

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 and to and , respectively, where with 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.

Mittag-Leffler stability and application of delayed fractional-order competitive neural networks

In the article, the Mittag-Leffler stability and application of delayed fractional-order competitive neural networks (FOCNNs) are developed. By virtue of the operator pair, the conditions of the coexistence of equilibrium points (EPs) are discussed and analyzed for delayed FOCNNs, in which the derived conditions of coexistence improve the existing results. In particular, these conditions are simplified in FOCNNs with stepped activations. Furthermore, the Mittag-Leffler stability of delayed FOCNNs is established by using the principle of comparison, which enriches the methodologies of fractional-order neural networks. The results on the obtained stability can be used to design the horizontal line detection of images, which improves the practicability of image detection results. Two simulations are displayed to validate the superiority of the obtained results.

On convergence properties of the brain-state-in-a-convex-domain

Convergence in the presence of multiple equilibrium points is one of the most fundamental dynamical properties of a neural network (NN). Goal of the paper is to investigate convergence for the classic Brain-State-in-a-Box (BSB) NN model and some of its relevant generalizations named Brain-State-in-a-Convex-Body (BSCB). In particular, BSCB is a class of discrete-time NNs obtained by projecting a linear system onto a convex body of Rn. The main result in the paper is that the BSCB is convergent when the matrix of the linear system is symmetric and positive semidefinite or, otherwise, it is symmetric and the step size does not exceed a given bound depending only on the minimum eigenvalue of the matrix. This result generalizes previous results in the literature for BSB and BSCB and it gives a solid foundation for the use of BSCB as a content addressable memory (CAM). The result is proved via Lyapunov method and LaSalle's Invariance Principle for discrete-time systems and by using some fundamental inequalities enjoyed by the projection operator onto convex sets as Bourbaki-Cheney-Goldstein inequality.

Detections of bifurcation in a fractional-order Cohen-Grossberg neural network with multiple delays

The dynamics of integer-order Cohen-Grossberg neural networks with time delays has lately drawn tremendous attention. It reveals that fractional calculus plays a crucial role on influencing the dynamical behaviors of neural networks (NNs). This paper deals with the problem of the stability and bifurcation of fractional-order Cohen-Grossberg neural networks (FOCGNNs) with two different leakage delay and communication delay. The bifurcation results with regard to leakage delay are firstly gained. Then, communication delay is viewed as a bifurcation parameter to detect the critical values of bifurcations for the addressed FOCGNN, and the communication delay induced-bifurcation conditions are procured. We further discover that fractional orders can enlarge (reduce) stability regions of the addressed FOCGNN. Furthermore, we discover that, for the same system parameters, the convergence time to the equilibrium point of FONN is shorter (longer) than that of integer-order NNs. In this paper, the present methodology to handle the characteristic equation with triple transcendental terms in delayed FOCGNNs is concise, neoteric and flexible in contrast with the prior mechanisms owing to skillfully keeping away from the intricate classified discussions. Eventually, the developed analytic results are nicely showcased by the simulation examples.

Lyapunov approach to manifolds stability for impulsive Cohen-Grossberg-type conformable neural network models

In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen-Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.

Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays

This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m₁+m₂=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.

Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control

This paper studies the sliding mode control method for coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks on a directed non-strongly connected topology. A novel fractional integral sliding mode surface and the corresponding control law are designed to realize global Mittag-Leffler synchronization. The sufficient conditions for synchronization and reachability of the sliding mode surface are derived via the hierarchical method and the Lyapunov method. Finally, simulations are provided to verify our theoretical findings.

Multimode function multistability for Cohen-Grossberg neural networks with mixed time delays

In this paper, we are concerned with the multimode function multistability for Cohen-Grossberg neural networks (CGNNs) with mixed time delays. It is introduced the multimode function multistability as well as its specific mathematical expression, which is a generalization of multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and asymptotic stability. Also, according to the neural network (NN) model and the maximum and minimum values of activation functions, n pairs of upper and lower boundary functions are obtained. Via the locations of the zeros of the n pairs of upper and lower boundary functions, the state space is divided into ∏_i=1ⁿ(2H_i+1) parts correspondingly. By virtue of the reduction to absurdity, continuity of function, Brouwer's fixed point theorem and Lyapunov stability theorem, the criteria for multimode function multistability are acquired. Multiple types of multistability, including multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and multiple asymptotic stability, can be achieved by selecting different types of function P(t). Two numerical examples are offered to substantiate the generality of the obtained criteria over the existing results.

Multistability and Stabilization of Fractional-Order Competitive Neural Networks With Unbounded Time-Varying Delays

This article investigates the multistability and stabilization of fractional-order competitive neural networks (FOCNNs) with unbounded time-varying delays. By utilizing the monotone operator, several sufficient conditions of the coexistence of equilibrium points (EPs) are obtained for FOCNNs with concave-convex activation functions. And then, the multiple μ -stability of delayed FOCNNs is derived by the analytical method. Meanwhile, several comparisons with existing work are shown, which implies that the derived results cover the inverse-power stability and Mittag-Leffler stability as special cases. Moreover, the criteria on the stabilization of FOCNNs with uncertainty are established by designing a controller. Compared with the results of fractional-order neural networks, the obtained results in this article enrich and improve the previous results. Finally, three numerical examples are provided to show the effectiveness of the presented results.