scholarly journals Attracting Sets Of Nonlinear Difference Equations With Time-Varying Delays

2018 ◽  
Vol 14 (2) ◽  
pp. 7975-7982
Author(s):  
Danhua He
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Nonlinear Difference Equations ◽  
Halanay Inequality ◽  
Attracting Set

In this paper, a class of nonlinear difference equations with time-varying delays is considered. Based on a generalized discrete Halanay inequality, some sufficient conditions for the attracting set and the global asymptotic stability of the nonlinear difference equations with time-varying delays are obtained.

Dynamical behavior of a P-dimensional system of nonlinear difference equations

2021 ◽  
Vol 71 (4) ◽  
pp. 903-924
Author(s):  
Yacine Halim ◽  
Asma Allam ◽  
Zineb Bengueraichi
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Nonlinear Difference Equations

Abstract In this paper, we study the periodicity, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of p nonlinear difference equations x n + 1 ( 1 ) = A + x n − 1 ( 1 ) x n ( p ) , x n + 1 ( 2 ) = A + x n − 1 ( 2 ) x n ( p ) , … , x n + 1 ( p − 1 ) = A + x n − 1 ( p − 1 ) x n ( p ) , x n + 1 ( p ) = A + x n − 1 ( p ) x n ( p − 1 ) $$\begin{equation*}x^{(1)}_{n+1}=A+\dfrac{x^{(1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(2)}_{n+1}=A+\dfrac{x^{(2)}_{n-1}}{x^{(p)}_{n}},\quad\ldots,\quad x^{(p-1)}_{n+1}=A+\dfrac{x^{(p-1)}_{n-1}}{x^{(p)}_{n}},\quad x^{(p)}_{n+1}=A+\dfrac{x^{(p)}_{n-1}}{x^{(p-1)}_{n}} \end{equation*} $$ where n ∈ ℕ0, p ≥ 3 is an integer, A ∈ (0, +∞) and the initial conditions x − 1 ( j ) $x_{-1}^{(j)}$ , x 0 ( j ) $x_{0}^{(j)}$ , j = 1, 2, …, p are positive numbers.

scholarly journals Asymptotic Stability for a Class of Nonlinear Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Zhi-wei Wang ◽  
Fei Gong ◽  
Rui-fang Wang
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Fractional Difference ◽  
Fractional Difference Equation

We study the global asymptotic stability of the equilibrium point for the fractional difference equationxn+1=(axn-lxn-k)/(α+bxn-s+cxn-t),n=0,1,…, where the initial conditionsx-r,x-r+1,…,x1,x0are arbitrary positive real numbers of the interval(0,α/2a),l,k,s,tare nonnegative integers,r=max⁡⁡{l,k,s,t}andα,a,b,care positive constants. Moreover, some numerical simulations are given to illustrate our results.

NOVEL STABILITY CONDITIONS FOR CELLULAR NEURAL NETWORKS WITH TIME DELAY

2001 ◽  
Vol 11 (07) ◽  
pp. 1853-1864 ◽  
Author(s):  
XIAOFENG LIAO ◽  
KWOK-WO WONG ◽  
JUEBANG YU
Keyword(s):  
Neural Networks ◽  
Time Delay ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Research Work ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Lyapunov Functionals ◽  
Time Varying ◽  
Time Varying Delay

In this paper, the global asymptotic stability of cellular neural networks with time delay is discussed using some novel Lyapunov functionals. Novel sufficient conditions for this type of stability are derived. They are less restrictive and more practical than those currently used. As a result, the design of cellular neural networks with time delay is refined. Our work can also be generalized to cellular neural networks with time-varying delay, a topic on which little research work has been done. By means of several different Lyapunov functionals, some sufficient conditions related to the global asymptotic stability for cellular neural networks with perturbations of time-varying delays are derived.

scholarly journals On the Uniform Exponential Stability of Time-Varying Systems Subject to Discrete Time-Varying Delays and Nonlinear Delayed Perturbations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Maher Hammami ◽  
Mohamed Ali Hammami ◽  
Manuel De la Sen
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Global Asymptotic Stability ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Uniform Exponential Stability ◽  
Time Varying Systems ◽  
Delayed Time

This paper addresses the problem of stability analysis of systems with delayed time-varying perturbations. Some sufficient conditions for a class of linear time-varying systems with nonlinear delayed perturbations are derived by using an improved Lyapunov-Krasovskii functional. The uniform global asymptotic stability of the solutions is obtained in terms of convergence toward a neighborhood of the origin.

scholarly journals Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Changjin Xu ◽  
Yusen Wu
Keyword(s):  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Inequality Theory

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

scholarly journals Global Asymptotic Stability of a Family of Nonlinear Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Maoxin Liao
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Recent Literature ◽  
Nonlinear Difference Equation ◽  
Nonlinear Difference Equations

In this note, we consider global asymptotic stability of the following nonlinear difference equationxn=(∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1))/(∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)),  n=0,1,…, whereki∈ℕ  (i=1,2,…,v),  v≥2,β1∈[-1,1],β2,β3,…,βv∈(-∞,+∞),x-m,x-m+1,…,x-1∈(0,∞), andm=max1≤i≤v{ki}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007).

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

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