Multistability of Switched Neural Networks With Piecewise Linear Activation Functions Under State-Dependent Switching

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.

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive-response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Synchronization of coupled switched neural networks subject to hybrid stochastic disturbances

In this paper, the theoretical analysis on exponential synchronization of a class of coupled switched neural networks suffering from stochastic disturbances and impulses is presented. A control law is developed and two sets of sufficient conditions are derived for the synchronization of coupled switched neural networks. First, for desynchronizing stochastic impulses, the synchronization of coupled switched neural networks is analyzed by Lyapunov function method, the comparison principle and a impulsive delay differential inequality. Then, for general stochastic impulses, by partitioning impulse interval and using the convex combination technique, a set of sufficient condition on the basis of linear matrix inequalities (LMIs) is derived for the synchronization of coupled switched neural networks. Eventually, two numerical examples and a practical application are elaborated to illustrate the effectiveness of the theoretical 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.

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.

Multistability of Dynamic Memristor Delayed Cellular Neural Networks With Application to Associative Memories

Recently, dynamic memristor (DM)-cellular neural networks (CNNs) have received widespread attention due to their advantage of low power consumption. The previous works showed that DM-CNNs have at most 318 equilibrium points (EPs) with n=16 cells. Since time delay is unavoidable during the process of information transmission, the goal of this article is to research the multistability of DM-CNNs with time delay, and, meanwhile, to increase the storage capacity of DM-delay (D)CNNs. Depending on the different constitutive relations of memristors, two cases of the multistability for DM-DCNNs are discussed. After determining the constitutive relations, the number of EPs of DM-DCNNs is increased to 3ⁿ with n cells by means of the appropriate state-space decomposition and the Brouwer's fixed point theorem. Furthermore, the enlarged attraction domains of EPs can be obtained, and 2ⁿ of these EPs are locally exponentially stable in two cases. Compared with standard CNNs, the dynamic behavior of DM-DCNNs shows an outstanding merit. That is, the value of voltage and current approach to zero when the system becomes stable, and the memristor provides a nonvolatile memory to store the computation results. Finally, two numerical simulations are presented to illustrate the effectiveness of the theoretical results, and the applications of associative memories are shown at the end of this article.

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.

Multistability of Switched Neural Networks With Gaussian Activation Functions Under State-Dependent Switching

This article presents theoretical results on the multistability of switched neural networks with Gaussian activation functions under state-dependent switching. It is shown herein that the number and location of the equilibrium points of the switched neural networks can be characterized by making use of the geometrical properties of Gaussian functions and local linearization based on the Brouwer fixed-point theorem. Four sets of sufficient conditions are derived to ascertain the existence of 7^p₁5^p₂3^p₃ equilibrium points, and 4^p₁3^p₂2^p₃ of them are locally stable, wherein p₁ , p₂ , and p₃ are nonnegative integers satisfying 0 ≤ p₁+p₂+p₃ ≤ n and n is the number of neurons. It implies that there exist up to 7ⁿ equilibria, and up to 4ⁿ of them are locally stable when p₁=n . It also implies that properly selecting p₁ , p₂ , and p₃ can engender a desirable number of stable equilibria. Two numerical examples are elaborated to substantiate the theoretical results.

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.

Analysis of Equilibria for a Class of Recurrent Neural Networks With Two Subnetworks

This article is concerned with the problem of the number and dynamical properties of equilibria for a class of connected recurrent networks with two switching subnetworks. In this network model, parameters serve as switches that allow two subnetworks to be turned ON or OFF among different dynamic states. The two subnetworks are described by a nonlinear coupled equation with a complicated relation among network parameters. Thus, the number and dynamical properties of equilibria have been very hard to investigate. By using Sturm's theorem, together with the geometrical properties of the network equation, we give a complete analysis of equilibria, including the existence, number, and dynamical properties. Necessary and sufficient conditions for the existence and exact number of equilibria are established. Moreover, the dynamical property of each equilibrium point is discussed without prior assumption of their locations. Finally, simulation examples are given to illustrate the theoretical results in this article.

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.

Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function

This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3ⁿ equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2ⁿ of 3ⁿ total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.

A Unified Framework Design for Finite-Time and Fixed-Time Synchronization of Discontinuous Neural Networks

In this article, the problems of finite-time/fixed-time synchronization have been investigated for discontinuous neural networks in the unified framework. To achieve the finite-time/fixed-time synchronization, a novel unified integral sliding-mode manifold is introduced, and corresponding unified control strategies are provided; some criteria are established for selecting suitable parameters for solving the related issue, namely, the dynamics of neural network can reach the designed sliding-mode manifold in finite/fixed time, and stay on it thereafter. Moreover, the estimations of setting time are given out. The established unified framework can bring in various protocols by choosing the different parameters of controllers and sliding-mode manifold, which extend previous related results. Finally, some numerical examples are introduced to show the effectiveness and superiority of resulting conclusions.

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.

Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances

This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5ⁿ regions in which 3ⁿ ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3ⁿ from 2ⁿ in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.

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.

Global Exponential Synchronization of Coupled Delayed Memristive Neural Networks With Reaction-Diffusion Terms via Distributed Pinning Controls

This article presents new theoretical results on global exponential synchronization of nonlinear coupled delayed memristive neural networks with reaction-diffusion terms and Dirichlet boundary conditions. First, a state-dependent memristive neural network model is introduced in terms of coupled partial differential equations. Next, two control schemes are introduced: distributed state feedback pinning control and distributed impulsive pinning control. A salient feature of these two pinning control schemes is that only partial information on the neighbors of pinned nodes is needed. By utilizing the Lyapunov stability theorem and Divergence theorem, sufficient criteria are derived to ascertain the global exponential synchronization of coupled neural networks via the two pining control schemes. Finally, two illustrative examples are elaborated to substantiate the theoretical results and demonstrate the advantages and disadvantages of the two control schemes.

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.

Novel results on finite-time stabilization of state-based switched chaotic inertial neural networks with distributed delays

The p-norm finite-time stabilization (FTS) issue of a class of state-based switched inertial chaotic neural networks (SBSCINNs) with distributed time-varying delays is investigated. By using a suitable variable transformation, such second-order SBSCINNs are turned into the first-order differential equations. Then some novel criteria are obtained to stabilize SBSCINNs in a finite time based on the theory of finite-time control and non-smooth analysis together with designing two proper delay-dependent feedback controllers. Besides, the settling time of FTS is also estimated and discussed. Finally, the validity and practicability of the deduced theoretical results are verified by examples and applications.