Exponential Stability Of Periodic Solution Research Articles

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.

Periodic solution problems of neutral-type stochastic neural networks with time-varying delays

This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems.

Exponential stability of periodic solution for stochastic neural networks involving multiple time-varying delays

<abstract><p>This paper discusses the exponential stability of periodic solutions for stochastic neural networks with multiple time-varying delays. For these networks, sufficient conditions in the linear matrix inequality forms are rare in the literature. We constructed an appropriate Lyapunov-Krasovskii functional to eliminate the items with multiple delays and establish some sufficient conditions in linear matrix inequality forms, to ensure exponential stability of the periodic solutions. Several examples are provided to demonstrate that our results are effective and less conservative than previous ones.</p></abstract>

Existence and exponential stability of periodic solutions of Nicholson-type systems with nonlinear density-dependent mortality and linear harvesting

In this work we study a Nicholson-type periodic system with variable delay, density-dependent mortality and linear harvesting rate. Using the topological degree and Lyapunov stability theories, we obtain sufficient conditions that allow us to demonstrate the existence of periodic solutions for the Nicholson-type system and, under suitable conditions, the uniqueness and local exponential stability of the periodic solution is established. We illustrate our results with an example and numerical simulations.

Global exponential periodicity of nonlinear neural networks with multiple time-varying delays

<abstract><p>Global exponential periodicity of nonlinear neural networks with multiple time-varying delays is investigated. Such neural networks cannot be written in the vector-matrix form because of the existence of the multiple delays. It is noted that although the neural network with multiple time-varying delays has been investigated by Lyapunov-Krasovskii functional method in the literature, the sufficient conditions in the linear matrix inequality form have not been obtained. Two sets of sufficient conditions in the linear matrix inequality form are established by Lyapunov-Krasovskii functional and linear matrix inequality to ensure that two arbitrary solutions of the neural network with multiple delays attract each other exponentially. This is a key prerequisite to prove the existence, uniqueness, and global exponential stability of periodic solutions. Some examples are provided to demonstrate the effectiveness of the established results. We compare the established theoretical results with the previous results and show that the previous results are not applicable to the systems in these examples.</p></abstract>

Impulsive control for stationary oscillation of nonlinear delay systems and applications

<abstract><p>In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.</p></abstract>

Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument

In this paper, we introduce a Cohen-Grossberg neural networks model with piecewise alternately advanced and retarded argument. Some sufficient conditions are established for the existence and global exponential stability of periodic solutions. The approaches are based on employing Brouwer's fixed-point theorem and an integral inequality of Gronwall type with deviating argument. The criteria given are easily verifiable, possess many adjustable parameters, and depend on piecewise constant argument deviations, which provide flexibility for the design and analysis of Cohen-Grossberg neural networks model. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.

Existence and global exponential stability of periodic solutions for feedback control complex dynamical networks with time-varying delays

This paper focuses on the existence and global exponential stability of periodic solutions for feedback control complex dynamical networks with time-varying delays (FCCDND). Combining the continuation theorem of coincidence degree theory, a combinatorial identity about Kirchhoff’s matrix tree theorem in graph theory, and Lyapunov method, a novel method is established to investigate the existence and global exponential stability of periodic solutions for FCCDND. Finally, the effectiveness and practicability of our results are illustrated by a numerical example and its simulation.

Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications

Abstract This paper is concerned with the existence and global exponential stability of the periodic solution of delayed Cohen–Grossberg neural networks (CGNNs) with discontinuous activation functions. The activations considered herein are non-decreasing but not required to be Lipschitz or continuous. Based on differential inclusion theory, Lyapunov functional theory and Leary–Schauder alternative theorem, some sufficient criteria are derived to ensure the existence and global exponential stability of the periodic solution. In order to show the superiority of the obtained results, an application and some detailed comparisons between some existing related results and our results are presented. Finally, some numerical examples are also illustrated.

Stability analysis for quaternion-valued inertial memristor-based neural networks with time delays

Quaternion-valued inertial memristor-based neural networks with time-varying delays are found to be effective in demonstrating complex dynamic behaviors. Considering the stability of the periodic solution for a novel neural network, we construct a class quaternion-valued inertial memristor-based neural network model by combining with the cellular neural networks and the inertial item with time-varying delay. Based on the continuation theorem of Mawhin’s coincidence degree theory, differential inclusion theory, and inequality techniques, some sufficient criteria for the existence of periodic solutions are obtained. Meanwhile, through constructing a novel Lyapunov functional method, several sufficient conditions have been derived to ensure the global exponential stability of periodic solutions. At last, an example is provided to illustrate the obtained theoretical results of the quaternion-valued inertial memristor-based neural networks model.

New findings on global exponential stability of inertial neural networks with both time-varying and distributed delays

ABSTRACT In this manuscript, inertial neural networks with both time- varying and distributed delays are studied. Applying inequality techniques and Lyapunov function approach, a new sufficient condition that guarantees the existence and exponential stability of periodic solutions for the addressed networks is presented. The obtained results supplement some earlier publications that deal with the periodic solutions of inertial neural networks with time- varying delays. Computer simulations are displayed to check the derived analytical results.

Periodic solutions for stochastic Cohen–Grossberg neural networks with time-varying delays

Abstract This paper is concerned with the periodic solutions for a class of stochastic Cohen–Grossberg neural networks with time-varying delays. Since there is a non-linearity in the leakage terms of stochastic Cohen–Grossberg neural networks, some techniques are needed to overcome the difficulty in dealing with the nonlinearity. By applying fixed points principle and Gronwall–Bellman inequality, some sufficient conditions on the existence and exponential stability of periodic solution for the stochastic neural networks are established. Moreover, a numerical example is presented to validate the theoretical results. Our results are also applicable to the existence and exponential stability of periodic solution for the corresponding deterministic systems.

Existence, Uniqueness and Exponential Stability of Periodic Solution for Discrete-Time Delayed BAM Neural Networks Based on Coincidence Degree Theory and Graph Theoretic Method

In this work, a general class of discrete time bidirectional associative memory (BAM) neural networks (NNs) is investigated. In this model, discrete and continuously distributed time delays are taken into account. By utilizing this novel method, which incorporates the approach of Kirchhoff’s matrix tree theorem in graph theory, Continuation theorem in coincidence degree theory and Lyapunov function, we derive a few sufficient conditions to ensure the existence, uniqueness and exponential stability of the periodic solution of the considered model. At the end of this work, we give a numerical simulation that shows the effectiveness of this work.

Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative

On the basis of some crucial properties for one-parameter and two-parameter Mittag-Leffler functions, the existence, uniqueness and global exponential stability of periodic solution are discussed for a class of semilinear impulsive fractional functional differential equations with piecewise Caputo derivative. Some better result is achieved and it improves and extends some existing research finding.

Novel Sufficient Conditions on Periodic Solutions for Discrete-Time Neutral-Type Neural Networks

In this paper, we consider the existence and global exponential stability of periodic solutions for a class of delayed discrete-time neutral-type neural networks. Novel sufficient conditions to guarantee the existence and global exponential stability of periodic solutions are established for above discrete-time neutral-type neural networks by combining Mawhin’s continuation theorem of coincidence degree theory with graph theory as well as Lyapunov sequence method. Our results on the existence and global exponential stability of periodic solutions are more concise and easily verified than those obtained in Du et al. (J Frankl Inst 353:448–461, 2016).

New Results on Periodicity of Non-autonomous Inertial Neural Networks Involving Non-reduced Order Method

This article mainly explores a class of non-autonomous inertial neural networks with time-varying delays and coefficients. By combining Lyapunov function method with differential inequality approach, some novel assertions are gained to guarantee the existence and exponential stability of periodic solutions for the addressed model. An example and its numerical simulations are given to support the proposed approach. The obtained results play an important role in designing the inertial neural networks and complement the earlier publications.

The existence and exponential stability of periodic solution for impulsive delay differential equations

AbstractIn this work, we first establish a new generalized Halanay inequality with impulses, which have two advantages: one is that impulses can be applied to stabilize the unstable continuous system, the other is that the requirements on the coefficients of inequality are more relaxed than those in previous works. Then, by using the inequality and Banach fixed point theorem, we will obtain some sufficient conditions ensuring the existence and exponential stability of periodic solution for impulsive delay differential equations. The sufficient conditions are easily checked in practice and have a wider adaptive range. Some examples are given to illustrate our results.

Periodic solutions for quaternion-valued fuzzy cellular neural networks with time-varying delays

In this paper, quaternion-valued fuzzy cellular neural networks (QVFCNNs) with time-varying delays are considered. First, we decompose QVFCNNs into their equivalent real-valued systems according to Hamilton’s multiplication rules. Then, we establish the existence and global exponential stability of periodic solutions of QVFCNNs by using the Schauder fixed point theorem and by constructing an appropriate Lyapunov function. Our results are completely new and supplementary to the known results. Finally, we give a numerical example to illustrate the effectiveness of our results.

Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method

This paper, mainly explores a class of non-autonomous inertial neural networks with proportional delays and time-varying coefficients. By combining Lyapunov function method with differential inequality approach, non-reduced order method is used to establish some novel assertions on the existence and generalized exponential stability of periodic solutions for the addressed model. In addition, an example and its numerical simulations are given to support the proposed approach.

Control and stabilization of the periodic fifth order Korteweg-de Vries equation

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in Hs(𝕋), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.