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.

scholarly journals Dynamical Behaviors of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Li Wan ◽  
Qinghua Zhou ◽  
Jizi Li
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Asymptotic Stability ◽  
Lyapunov Method ◽  
Reaction Diffusion ◽  
Ultimate Boundedness ◽  
Dynamical Behaviors ◽  
Weak Attractor ◽  
Stochastic Reaction ◽  
Theoretical Results

This paper investigates dynamical behaviors of stochastic Cohen-Grossberg neural network with delays and reaction diffusion. By employing Lyapunov method, Poincaré inequality and matrix technique, some sufficient criteria on ultimate boundedness, weak attractor, and asymptotic stability are obtained. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

scholarly journals Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yueding Yuan ◽  
Zhiming Guo
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Careful Analysis ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Delayed Reaction ◽  
Fluctuation Method

We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

Stability analysis of linear distributed order fractional systems with distributed delays

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Doychin Boyadzhiev ◽  
Hristo Kiskinov ◽  
Magdalena Veselinova ◽  
Andrey Zahariev
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Density Function ◽  
Global Asymptotic Stability ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Distributed Delays ◽  
Fractional Systems ◽  
Distributed Order

AbstractIn the present work we study linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivatives, with respect to a nonnegative density function. For the initial problem of this kind of systems, existence, uniqueness and a priory estimate of the solution are proved. As an application of the obtained results, we establish sufficient conditions for global asymptotic stability of the zero solution of the investigated types of systems.

Dynamic Analysis of a Delayed Fractional-Order SIR Model with Saturated Incidence and Treatment Functions

2018 ◽  
Vol 28 (14) ◽  
pp. 1850180 ◽  
Author(s):  
Xinhe Wang ◽  
Zhen Wang ◽  
Xia Huang ◽  
Yuxia Li
Keyword(s):  
Asymptotic Stability ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Local Asymptotic Stability ◽  
Proposed Model ◽  
Saturated Incidence ◽  
Theoretical Results

In this paper, a delayed fractional-order SIR (susceptible, infected, and removed) epidemic model with saturated incidence and treatment functions is presented. Firstly, the non-negativity and boundedness of solutions of the proposed model are proved. Next, some sufficient conditions are established to ensure the local asymptotic stability of the disease-free equilibrium point [Formula: see text] and the endemic equilibrium point [Formula: see text] for any delay. Meanwhile, global asymptotic stability of the endemic equilibrium point [Formula: see text] is investigated by constructing a suitable Lyapunov function. Some sufficient conditions are established for the global asymptotic stability of this endemic equilibrium point. Finally, some numerical simulations are illustrated to verify the correctness of the theoretical results.

Global asymptotic stability for a SEI reaction–diffusion model of infectious diseases with immigration

2018 ◽  
Vol 11 (03) ◽  
pp. 1850044
Author(s):  
Salem Abdelmalek ◽  
Samir Bendoukha
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Reaction Diffusion ◽  
Stability Of Solutions ◽  
Reaction Diffusion Model ◽  
Globally Stable

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

scholarly journals Dynamic Behaviors of a Single Species Stage Structure Model with Michaelis–Menten-TypeJuvenile Population Harvesting

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1281
Author(s):  
Xiangqin Yu ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Single Species ◽  
Structure Model ◽  
Boundary Equilibrium ◽  
Juvenile Population ◽  
Nonlinear Harvesting ◽  
Theoretical Results

A single species stage structure model with Michaelis–Menten-type juvenile population harvesting is proposed and investigated. The existence and local stability of the model equilibria are studied. It shows that for the model, two cases of bistability may exist. Some conditions for the global asymptotic stability of the boundary equilibrium are derived by constructing some suitable Lyapunov functions. After that, based on the Bendixson–Dulac discriminant, we obtain the sufficient conditions for the global asymptotic stability of the internal equilibrium. Our study shows that nonlinear harvesting can make the dynamics of the system more complex than linear harvesting; for example, the system may admit the bistable stability property. Numeric simulations support our theoretical results.

scholarly journals Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao
Keyword(s):  
Asymptotic Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Stability Criteria ◽  
Differential Systems ◽  
Discrete Delays ◽  
Transcendental Equations ◽  
The Stability ◽  
Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

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