Exponential Stability Of Equilibrium Point Research Articles

Multistability analysis of quaternion-valued neural networks with cosine activation functions

It is significant to design a system with high storage capacity for associative memory and pattern recognition. To address this issue, this paper first proposes a quaternion-valued neural network (QVNN) model with multiple equilibrium points in which the cosine function is used as the activation function of QVNN. Then, based on the Brouwer fixed point theorem and the geometric properties of the activation function, sufficient conditions for QVNN to have unique equilibrium points, finite equilibrium points, and countable infinite equilibrium points are obtained, respectively. Furthermore, sufficient conditions for the exponential stability of equilibrium points are derived, and the attraction basins of the stable equilibrium points are given. Finally, two numerical examples are given to confirm the validity of the proposed theoretical results.

The multistability of delayed competitive neural networks with piecewise non‐monotonic activation functions

This paper addresses the problem of multistability of competitive neural networks with nonlinear, non‐monotonic piecewise activation functions and time‐varying delays. Several sufficient conditions are proposed to guarantee the existence of equilibrium points and the locally exponential stability of equilibrium points, where is a positive integer and determined by the property of activation functions and the parameters of neural networks. The quantitative relationship between the equilibrium points of the system and the zero roots of the bounding functions is given. In addition, the attraction basins of the exponentially stable equilibrium points are obtained. Finally, a numerical simulation is given to illustrate the effectiveness of the obtained results.

Study on asymptotic behavior of stochastic Lotka\u2013Volterra system in a polluted environment

A three-species non-autonomous stochastic Lotka–Volterra food web system in a polluted environment is proposed, and the existence of positive periodic solutions of this system is established by constructing a proper Lyapunov function. Then the extinction property and its threshold between persistence and extinction are discussed by using Itô’s formula and the strong law of large numbers of martingale, and the sufficient condition of a.s. exponential stability of equilibrium point is obtained. Finally, the conclusions are tested by several numerical simulations.

Stability analysis of delayed fuzzy Cohen‐Grossberg neural networks with discontinuous activations

SummaryIn this article, we investigate the dynamical behavior of a class of delayed fuzzy Cohen‐Grossberg neural networks (FCGNNs) with discontinuous activation functions subject to time delays and fuzzy terms. By using the inequality analysis technique and the M‐matrix theory, sufficient and proper conditions are given in order to establish the existence, convergence, and global exponential stability of equilibrium point of the system. In particular, we discuss the impact of discontinuous neuron activations on the existence and exponential stability of equilibrium point for FCGNNs. Two numerical examples are provided to substantiate the theoretical results.

Exponential stability of impulsive complex-valued neural networks with time delay

This paper studies the stability problem of impulsive complex-valued neural networks with time delay. Firstly, impulses are viewed as disturbance. Criteria on uniqueness and global exponential stability of equilibrium point are established using the Lyapunov function method. A modified version of Halanay-type inequality is used to make the results in Song et al. (2016) much better. Secondly, impulses are used as control input. Sufficient conditions are derived to ensure the exponential stability of the equilibrium point by using the Razumikhin technique. This result explains the dynamics of example 2 in Song et al. (2016), which does not give any theoretical result on impulsive control problem. Two numerical examples are presented to demonstrate the effectiveness of the theorems.

Global exponential stability and lag synchronization for delayed memristive fuzzy Cohen–Grossberg BAM neural networks with impulses

This paper investigates the stability and lag synchronization for memristor-based fuzzy Cohen–Grossberg bidirectional associative memory (BAM) neural networks with mixed delays (asynchronous time delays and continuously distributed delays) and impulses. By applying the inequality analysis technique, homeomorphism theory and some suitable Lyapunov–Krasovskii functionals, some new sufficient conditions for the uniqueness and global exponential stability of equilibrium point are established. Furthermore, we obtain several sufficient criteria concerning globally exponential lag synchronization for the proposed system based on the framework of Filippov solution, differential inclusion theory and control theory. In addition, some examples with numerical simulations are given to illustrate the feasibility and validity of obtained results.

Global Exponential Stability of Impulsive Fuzzy High-Order BAM Neural Networks With Continuously Distributed Delays.

This paper investigates the stability of equilibrium point and periodic solution for impulsive fuzzy high-order bidirectional associative memory neural networks with continuously distributed delays. By applying the inequality analysis technique, -matrix, and Banach contraction mapping principle and constructing some suitable Lyapunov functionals, some sufficient conditions for the uniqueness and global exponential stability of equilibrium point and global exponential stability of periodic solutions are established. In addition, three examples with numerical simulations are presented to demonstrate the feasibility and effectiveness of the theoretical results.

The Analysis of Stochastic Lotka–Volterra Model in Polluted Environment

Population systems often subject to both white noise and environment pollution, so a stochastic non-autonomous omnivory Lotka–Volterra model in a polluted environment is investigated in this paper. We establish the sufficient conditions for the existence of positive periodic solution and prove it by constructing Lyapunov function. According to Ito’s formula and the strong law of large numbers for martingales, we also discuss the extinction of population. The sufficient conditions for almost sure exponential stability of equilibrium point \(E^*(0,0,0,S^*,T^*)\) are obtained. Finally, we illustrate our results by some examples with the help of computer simulation.

On the periodic dynamics of memristor-based neural networks with leakage and time-varying delays

In this paper, the periodic dynamics have been studied for a general kind of memristor-based neural networks with leakage and time-varying delays. Some new sufficient conditions have been derived ensuring that the existence, uniqueness and globally exponential stability of the periodic solution for the neural network by using differential inclusions theory, the topological degree theory in set-valued analysis and Lyapunov function technique and so on. As a special case, we have shown that the existence, uniqueness and global exponential stability of equilibrium point for the autonomous neural networks with leakage delays.

Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays

This paper investigates the stability problem for a class of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. By employing the idea of vector Lyapunov function, M-matrix theory and inequality technique, several sufficient conditions are obtained to ensure the global exponential stability of equilibrium point. When the impulsive effects are not considered, several sufficient conditions are also given to guarantee the existence, uniqueness and global exponential stability of equilibrium point. Two examples are given to illustrate the effectiveness and lower level of conservatism of the proposed criteria in comparison with some existing results.

Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects

In this paper, the global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects is discussed. By employing Lyapunov functional method and using matrix inequality technique, several sufficient conditions in complex-valued linear matrix inequality form are obtained to ensure the existence, uniqueness and global exponential stability of equilibrium point for the considered neural networks. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The proposed stability results are less conservative than some recently known ones in the literatures, which is demonstrated via two examples with simulations.

LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique

The paper is concerned with a class of neutral BAM neural networks with delays in leakage terms. The existence of equilibrium point dependent on time delays of system (1.1) is obtained by establishing and applying two new inequalities. Sufficient condition is established to ensure the global exponential stability of the above neutral networks by establishing and employing above two new inequalities and new LMI methods. The results of this paper are new and complementary to the previously known results.

Exponential stability of a class of competitive neural networks with multi-proportional delays

In this paper, the exponential stability of a class of competitive neural networks with multi-proportional delays is studied. First, through suitable transformations, a class of competitive neural networks with multi-proportional delays can be equivalently turned into a class of competitive neural networks with multi-constant delays and variable coefficients. By using fixed point theorem, the existence and uniqueness of equilibrium point of the system is proved. Furthermore by constructing appropriate delay differential inequality, two delay-independent and delay-independent sufficient conditions for the exponential stability of equilibrium point are obtained. Finally, several examples and their simulations are given to illustrate the effectiveness of the obtained results.

Global Exponential Stability of Cohen-Grossberg Neural Networks with Piecewise Constant Argument of Generalized Type and Impulses.

In this letter, we consider a model of Cohen-Grossberg neural networks with piecewise constant argument of generalized type and impulses. Sufficient conditions ensuring the existence and uniqueness of solutions are obtained. Based on constructing a new differential inequality with piecewise constant argument and impulse and using the Lyapunov function method, we derive sufficient conditions ensuring the global exponential stability of equilibrium point, with approximate exponential convergence rate. An example is given to illustrate the validity and advantage of the theoretical results.

Impulsive effects on stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays

In this paper, the impulsive effects on the stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays are investigated. Several sufficient conditions are obtained ensuring the existence, uniqueness and global exponential stability of equilibrium point for the considered neural networks by the idea of vector Lyapunov function method, M-matrix theory and analytic technique. Moreover, the estimation for exponential convergence rate index is proposed. An example with simulations is provided to verify the effectiveness of the obtained results.

Global exponential stability and periodic solutions of high-order bidirectional associative memory (BAM) neural networks with time delays and impulses

High-order bidirectional associative memory (BAM) neural networks with mixed features such as time delays and impulses play an increasingly important role in the design and implementation of neural network systems. It exhibits advantage of stronger approximation property, faster convergence rate, greater storage capacity and higher fault tolerance than low-order neural networks. In this paper, issues of both stability and periodicity for a class of high-order BAM neural networks with time delays and impulses are investigated. With M-matrix theory, linear matrix inequality technique, fixed point theorem and Lyapunov approach, we derive new sufficient conditions for achieving global exponential stability of equilibrium point and the existence and global exponential stability of periodic solutions for the addressed high-order BAM neural networks. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Global exponential stability via inequality technique for inertial BAM neural networks with time delays

In this paper, the existence and global exponential stability of equilibrium point for inertial BAM neural networks with time delays are discussed. Firstly, by using homeomorphism theory and inequality technique, the LMI-based sufficient condition on the existence and uniqueness of equilibrium point for above inertial BAM neural networks is obtained. Secondly, a LMI-based condition which can ensure the global exponential stability of equilibrium point for the system is obtained by using LMI method and inequality technique. In our results, the boundedness assumption on the activation functions in Ke and Miao (2013) [19,20] is removed. Hence, our result on global exponential stability of equilibrium point for above system is less conservative.

New results on the stability of fuzzy cellular neural networks with time-varying leakage delays

In this paper, fuzzy cellular neural networks with time-varying delays in the leakage terms are studied. Without assuming the boundedness on the activation functions, some sufficient conditions for the existence, uniqueness and global exponential stability of equilibrium point are established by using the fixed point theorem and constructing suitable Lyapunov functional, which are new, and they improve previously known results. Moreover, an example is given to demonstrate the feasibility of our method.

On the periodic dynamics of memristor-based neural networks with time-varying delays

In this paper, we study the existence, uniqueness and stability of periodic solution for a wide class of memristor-based neural networks with time-varying delays. By employing the topological degree theory in set-valued analysis, differential inclusions theory and a new Lyapunov function method, we prove that the neural network has a unique periodic solution, which is globally exponentially stable. Moreover, we prove the existence, uniqueness and global exponential stability of equilibrium point for time-varying delayed memristor-based neural networks with constant coefficients. The obtained results improve and extend previous works on memristor-based or usual neural network dynamical systems with continuous or discontinuous right-hand side. Finally, two numerical examples are provided to show the applicability and effectiveness of our main results.

Existence and Exponential Stability of Equilibrium Point for Fuzzy BAM Neural Networks with Infinitely Distributed Delays and Impulses on Time Scales

By using the fixed point theorem and constructing a Lyapunov functional, we establish some sufficient conditions on the existence, uniqueness, and exponential stability of equilibrium point for a class of fuzzy BAM neural networks with infinitely distributed delays and impulses on time scales. We also present a numerical example to show the feasibility of obtained results. Our example also shows that the described time and continuous neural time networks have the same dynamic behaviours for the stability.