Multiple ψ -Type Stability and Its Robustness for Recurrent Neural Networks With Time-Varying Delays

Multistability and Robustness of Competitive Neural Networks With Time-Varying Delays

This article is devoted to analyzing the multistability and robustness of competitive neural networks (NNs) with time-varying delays. Based on the geometrical structure of activation functions, some sufficient conditions are proposed to ascertain the coexistence of equilibrium points, of them are locally exponentially stable, where represents a dimension of system and is the parameter related to activation functions. The derived stability results not only involve exponential stability but also include power stability and logarithmical stability. In addition, the robustness of stable equilibrium points is discussed in the presence of perturbations. Compared with previous papers, the conclusions proposed in this article are easy to verify and enrich the existing stability theories of competitive NNs. Finally, numerical examples are provided to support theoretical results.

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.

An Overview of the Stability Analysis of Recurrent Neural Networks With Multiple Equilibria

The stability analysis of recurrent neural networks (RNNs) with multiple equilibria has received extensive interest since it is a prerequisite for successful applications of RNNs. With the increasing theoretical results on this topic, it is desirable to review the results for a systematical understanding of the state of the art. This article provides an overview of the stability results of RNNs with multiple equilibria including complete stability and multistability. First, preliminaries on the complete stability and multistability analysis of RNNs are introduced. Second, the complete stability results of RNNs are summarized. Third, the multistability results of various RNNs are reviewed in detail. Finally, future directions in these interesting topics are suggested.

Analysis and Design of Multivalued High-Capacity Associative Memories Based on Delayed Recurrent Neural Networks

This article aims at analyzing and designing the multivalued high-capacity-associative memories based on recurrent neural networks with both asynchronous and distributed delays. In order to increase storage capacities, multivalued activation functions are introduced into associative memories. The stored patterns are retrieved by external input vectors instead of initial conditions, which can guarantee accurate associative memories by avoiding spurious equilibrium points. Some sufficient conditions are proposed to ensure the existence, uniqueness, and global exponential stability of the equilibrium point of neural networks with mixed delays. For neural networks with n neurons, m -dimensional input vectors, and 2k -valued activation functions, the autoassociative memories have (2k)ⁿ storage capacities and heteroassociative memories have min {(2k)ⁿ,(2k)^m} storage capacities. That is, the storage capacities of designed associative memories in this article are obviously higher than the 2ⁿ and min {2ⁿ,2^m} storage capacities of the conventional ones. Three examples are given to support the theoretical results.

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

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.

New Criteria on Stability of Dynamic Memristor Delayed Cellular Neural Networks

Dynamic memristor (DM)-cellular neural networks (CNNs), which replace a linear resistor with flux-controlled memristor in the architecture of each cell of traditional CNNs, have attracted researchers' attention. Compared with common neural networks, the DM-CNNs have an outstanding merit: when a steady state is reached, all voltages, currents, and power consumption of DM-CNNs disappeared, in the meantime, the memristor can store the computation results by serving as nonvolatile memories. The previous study on stability of DM-CNNs rarely considered time delay, while delay is quite common and highly impacts the stability of the system. Thus, taking the time delay effect into consideration, we extend the original system to DM-D(delay)CNNs model. By using the Lyapunov method and the matrix theory, some new sufficient conditions for the global asymptotic stability and global exponential stability with a known convergence rate of DM-DCNNs are obtained. These criteria generalized some known conclusions and are easily verified. Moreover, we find DM-DCNNs have 3ⁿ equilibrium points (EPs) and 2ⁿ of them are locally asymptotically stable. These results are obtained via a given constitutive relation of memristor and the appropriate division of state space. Combine with these theoretical results, the applications of DM-DCNNs can be extended to other fields, such as associative memory, and its advantage can be used in a better way. Finally, numerical simulations are offered to illustrate the effectiveness of our theoretical results.

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

This article mainly investigates the multiple Mittag-Leffler stability of delayed fractional-order Cohen-Grossberg neural networks with time-varying delays. By using mixed monotone operator pair, the conditions of the coexistence of multiple equilibrium points are obtained for fractional-order Cohen-Grossberg neural networks, and these conditions are eventually transformed into algebraic inequalities based on the vertex of the divided region. In particular, when the symbols of these inequalities are determined by the dominant term, several verifiable corollaries are given. And then, the sufficient conditions of the Mittag-Leffler stability are derived for fractional-order Cohen-Grossberg neural networks with time-varying delays. In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results.

Robust Stability of Recurrent Neural Networks With Time-Varying Delays and Input Perturbation

This paper addresses the robust stability of recurrent neural networks (RNNs) with time-varying delays and input perturbation, where the time-varying delays include discrete and distributed delays. By employing the new ψ -type integral inequality, several sufficient conditions are derived for the robust stability of RNNs with discrete and distributed delays. Meanwhile, the robust boundedness of neural networks is explored by the bounded input perturbation and L¹ -norm constraint. Moreover, RNNs have a strong anti-jamming ability to input perturbation, and the robustness of RNNs is suitable for associative memory. Specifically, when input perturbation belongs to the specified and well-characterized space, the results cover both monostability and multistability as special cases. It is revealed that there is a relationship between the stability of neural networks and input perturbation. Compared with the existing results, these conditions proposed in this paper improve and extend the existing stability in some literature. Finally, the numerical examples are given to substantiate the effectiveness of the theoretical results.

Multistability of delayed fractional-order competitive neural networks

This paper is concerned with the multistability of fractional-order competitive neural networks (FCNNs) with time-varying delays. Based on the division of state space, the equilibrium points (EPs) of FCNNs are given. Several sufficient conditions and criteria are proposed to ascertain the multiple O(t^-α)-stability of delayed FCNNs. The O(t^-α)-stability is the extension of Mittag-Leffler stability of fractional-order neural networks, which contains monostability and multistability. Moreover, the attraction basins of the stable EPs of FCNNs are estimated, which shows the attraction basins of the stable EPs can be larger than the divided subsets. These conditions and criteria supplement and improve the previous results. Finally, the results are illustrated by the simulation examples.

Multiple and Complete Stability of Recurrent Neural Networks With Sinusoidal Activation Function

This article presents new theoretical results on multistability and complete stability of recurrent neural networks with a sinusoidal activation function. Sufficient criteria are provided for ascertaining the stability of recurrent neural networks with various numbers of equilibria, such as a unique equilibrium, finite, and countably infinite numbers of equilibria. Multiple exponential stability criteria of equilibria are derived, and the attraction basins of equilibria are estimated. Furthermore, criteria for complete stability and instability of equilibria are derived for recurrent neural networks without time delay. In contrast to the existing stability results with a finite number of equilibria, the new criteria, herein, are applicable for both finite and countably infinite numbers of equilibria. Two illustrative examples with finite and countably infinite numbers of equilibria are elaborated to substantiate the results.

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

This article addresses the multistability and attraction of fractional-order neural networks (FONNs) with unbounded time-varying delays. Several sufficient conditions are given to ensure the coexistence of equilibrium points (EPs) of FONNs with concave-convex activation functions. Moreover, by exploiting the analytical method and the property of the Mittag-Leffler function, it is shown that the multiple Mittag-Leffler stability of delayed FONNs is derived and the obtained criteria do not depend on differentiable time-varying delays. In particular, the criterion of the Mittag-Leffler stability can be simplified to M-matrix. In addition, the estimation of attraction basin of delayed FONNs is studied, which implies that the extension of attraction basin is independent of the magnitude of delays. Finally, three numerical examples are given to show the validity of the theoretical results.

Lagrange Stability and Finite-Time Stabilization of Fuzzy Memristive Neural Networks With Hybrid Time-Varying Delays

This paper focuses on Lagrange exponential stability and finite-time stabilization of Takagi-Sugeno (T-S) fuzzy memristive neural networks with discrete and distributed time-varying delays (DFMNNs). By resorting to theories of differential inclusions and the comparison strategy, an algebraic condition is developed to confirm Lagrange exponential stability of the underlying DFMNNs in Filippov's sense, and the exponentially attractive set is estimated. When external input is not considered, global exponential stability of DFMNNs is derived directly, which includes some existing ones as special cases. Furthermore, finite-time stabilization of the addressed DFMNNs is analyzed by exploiting inequality techniques and the comparison approach via designing a nonlinear state feedback controller. The boundedness assumption of activation functions is removed herein. Finally, two simulations are presented to demonstrate the validness of the outcomes, and an application is performed in pseudorandom number generation.