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.

Finite-Time Stabilization of Competitive Neural Networks With Time-Varying Delays

This article investigates finite-time stabilization of competitive neural networks with discrete time-varying delays (DCNNs). By virtue of comparison strategies and inequality techniques, finite-time stabilization of the underlying DCNNs is analyzed by designing a discontinuous state feedback controller, which simplifies the controller design and proof processes of some existing results. Meanwhile, global exponential stabilization of the DCNNs is provided under a continuous state feedback controller. In addition, global exponential stability of the DCNNs is shown as an M-matrix, which contains some published outcomes as special cases. Finally, three examples are given to illuminate the validity of the theories.

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.

Monostability and Multistability for Almost-Periodic Solutions of Fractional-Order Neural Networks With Unsaturating Piecewise Linear Activation Functions

Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.

Multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions

This paper is concerned with multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions. Under some reasonable conditions, the addressed networks have (2m+1)ⁿ equilibrium points. (m+1)ⁿ of which are locally asymptotically stable, and the others are unstable. The attraction basins of the locally asymptotically stable equilibrium points are given in the form of hyperspherical regions. These results here, which include existence, uniqueness, locally asymptotical stability, instability and attraction basins of the multiple equilibrium points, generalize and improve the earlier publications. Finally, an illustrative example with numerical simulation is given to show the feasibility and the effectiveness of the theoretical results. The theoretical results and illustrative example indicate that the activation functions improve the storage capacity of neural networks significantly.

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

This paper is concerned with the multistability of switched neural networks with piecewise linear activation functions under state-dependent switching. Under some reasonable assumptions on the switching threshold and activation functions, by using the state-space decomposition method, contraction mapping theorem, and strictly diagonally dominant matrix theory, we can characterize the number of equilibria as well as analyze the stability/instability of the equilibria. More interesting, we can find that the switching threshold plays an important role for stable equilibria in the unsaturation regions of activation functions, and the number of stable equilibria of an n -neuron switched neural network with state-dependent parameters increases to 3ⁿ from 2ⁿ in the conventional one. Furthermore, for two-neuron switched neural networks, the precise attraction basin of each stable equilibrium point can be figured out, and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines. Two simulation examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.

Multiple ψ -Type Stability of Cohen-Grossberg Neural Networks With Both Time-Varying Discrete Delays and Distributed Delays

In this paper, multiple ψ -type stability of Cohen-Grossberg neural networks (CGNNs) with both time-varying discrete delays and distributed delays is investigated. By utilizing ψ -type functions combined with a new ψ -type integral inequality for treating distributed delay terms, some sufficient conditions are obtained to ensure that multiple equilibrium points are ψ -type stable for CGNNs with discrete and distributed delays, where the distributed delays include bounded and unbounded delays. These conditions of CGNNs with different output functions are less restrictive. More specifically, the algebraic criteria of the generalized model are applicable to several well-known neural network models by taking special parameters, and multiple different output functions are introduced to replace some of the same output functions, which improves the diversity of output results for the design of neural networks. In addition, the estimation of relative convergence rate of ψ -type stability is determined by the parameters of CGNNs and the selection of ψ -type functions. As a result, the existing results on multistability and monostability can be improved and extended. Finally, some numerical simulations are presented to illustrate the effectiveness of the obtained results.

Multistability Analysis of Quaternion-Valued Neural Networks With Time Delays

This paper addresses the multistability issue for quaternion-valued neural networks (QVNNs) with time delays. By using the inequality technique, sufficient conditions are proposed for the boundedness and the global attractivity of delayed QVNNs. Based on the geometrical properties of the activation functions, several criteria are obtained to ensure the existence of equilibrium points, of which are locally stable. Two numerical examples are provided to illustrate the effectiveness of the obtained results.

Global Mittag-Leffler stability analysis of fractional-order impulsive neural networks with one-side Lipschitz condition

This paper is concerned with the stability analysis issue of fractional-order impulsive neural networks. Under the one-side Lipschitz condition or the linear growth condition of activation function, the existence of solution is analyzed respectively. In addition, the existence, uniqueness and global Mittag-Leffler stability of equilibrium point of the fractional-order impulsive neural networks with one-side Lipschitz condition are investigated by the means of contraction mapping principle and Lyapunov direct method. Finally, an example with numerical simulation is given to illustrate the validity and feasibility of the proposed results.

Multistability of Delayed Recurrent Neural Networks with Mexican Hat Activation Functions

This letter studies the multistability analysis of delayed recurrent neural networks with Mexican hat activation function. Some sufficient conditions are obtained to ensure that an [Formula: see text]-dimensional recurrent neural network can have [Formula: see text] equilibrium points with [Formula: see text], and [Formula: see text] of them are locally exponentially stable. Furthermore, the attraction basins of these stable equilibrium points are estimated. We show that the attraction basins of these stable equilibrium points can be larger than their originally partitioned subsets. The results of this letter improve and extend the existing stability results in the literature. Finally, a numerical example containing different cases is given to illustrate the theoretical results.

Coexistence and local μ-stability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded time-varying delays

In this paper, the coexistence and dynamical behaviors of multiple equilibrium points are discussed for a class of memristive neural networks (MNNs) with unbounded time-varying delays and nonmonotonic piecewise linear activation functions. By means of the fixed point theorem, nonsmooth analysis theory and rigorous mathematical analysis, it is proven that under some conditions, such n-neuron MNNs can have 5ⁿ equilibrium points located in ℜⁿ, and 3ⁿ of them are locally μ-stable. As a direct application, some criteria are also obtained on the multiple exponential stability, multiple power stability, multiple log-stability and multiple log-log-stability. All these results reveal that the addressed neural networks with activation functions introduced in this paper can generate greater storage capacity than the ones with Mexican-hat-type activation function. Numerical simulations are presented to substantiate the theoretical results.

Complete stability of delayed recurrent neural networks with Gaussian activation functions

This paper addresses the complete stability of delayed recurrent neural networks with Gaussian activation functions. By means of the geometrical properties of Gaussian function and algebraic properties of nonsingular M-matrix, some sufficient conditions are obtained to ensure that for an n-neuron neural network, there are exactly 3^k equilibrium points with 0≤k≤n, among which 2^k and 3^k-2^k equilibrium points are locally exponentially stable and unstable, respectively. Moreover, it concludes that all the states converge to one of the equilibrium points; i.e., the neural networks are completely stable. The derived conditions herein can be easily tested. Finally, a numerical example is given to illustrate the theoretical results.

Dynamical Behaviors of Multiple Equilibria in Competitive Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such n -neuron competitive neural networks can have 5(n) equilibria, among which 3(n) equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the 3(n) locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

Multistability for Delayed Neural Networks via Sequential Contracting

In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.

Multistability and Instability of Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n -neuron neural networks can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.

Multistability in a class of stochastic delayed Hopfield neural networks

In this paper, multistability analysis for a class of stochastic delayed Hopfield neural networks is investigated. By considering the geometrical configuration of activation functions, the state space is divided into 2(n) + 1 regions in which 2(n) regions are unbounded rectangles. By applying Schauder's fixed-point theorem and some novel stochastic analysis techniques, it is shown that under some conditions, the 2(n) rectangular regions are positively invariant with probability one, and each of them possesses a unique equilibrium. Then by applying Lyapunov function and functional approach, two multistability criteria are established for ensuring these equilibria to be locally exponentially stable in mean square. The first multistability criterion is suitable to the case where the information on delay derivative is unknown, while the second criterion requires that the delay derivative be strictly less than one. For the constant delay case, the second multistability criterion is less conservative than the first one. Finally, an illustrative example is presented to show the effectiveness of the derived results.

Convergence and attractivity of memristor-based cellular neural networks with time delays

This paper presents theoretical results on the convergence and attractivity of memristor-based cellular neural networks (MCNNs) with time delays. Based on a realistic memristor model, an MCNN is modeled using a differential inclusion. The essential boundedness of its global solutions is proven. The state of MCNNs is further proven to be convergent to a critical-point set located in saturated region of the activation function, when the initial state locates in a saturated region. It is shown that the state convergence time period is finite and can be quantitatively estimated using given parameters. Furthermore, the positive invariance and attractivity of state in non-saturated regions are also proven. The simulation results of several numerical examples are provided to substantiate the results.

Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays

This paper deals with the problem of existence and uniform stability analysis of fractional-order complex-valued neural networks with constant time delays. Complex-valued recurrent neural networks is an extension of real-valued recurrent neural networks that includes complex-valued states, connection weights, or activation functions. This paper explains sufficient condition for the existence and uniform stability analysis of such networks. Three numerical simulations are delineated to substantiate the effectiveness of the theoretical results.

Dynamic analysis of periodic solution for high-order discrete-time Cohen-Grossberg neural networks with time delays

In this paper, we analyze the dynamic behavior of periodic solution for the high-order discrete-time Cohen-Grossberg neural networks (CGNNs) with time delays. First, the existence is studied based on the continuation theorem of coincidence degree theory and Young's inequality. And then, the criterion for the global exponential stability is given using Lyapunov method. Finally, simulation result shows the effectiveness of our proposed criterion.

Multistability of neural networks with Mexican-hat-type activation functions

In this paper, we are concerned with a class of neural networks with Mexican-hat-type activation functions. Due to the different structure from neural networks with saturated activation functions, a set of new sufficient conditions are presented to study the multistability, including the total number of equilibrium points, their locations, and stability. Furthermore, the attraction basins of stable equilibrium points are investigated for two-neuron neural networks. The investigation shows that the stable manifolds of unstable equilibrium points constitute the boundaries of attraction basins of stable equilibrium points. Several illustrative examples are given to verify the effectiveness of our results.