scholarly journals Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 331 ◽  
Author(s):  
Yu Chen ◽  
JinRong Wang
Keyword(s):  
Fractional Order ◽  
Continuous Dependence ◽  
Sufficient Conditions ◽  
Initial Point ◽  
Impulsive Differential Equations ◽  
Junction Points ◽  
Random Impulse ◽  
Continuous Dependence Results ◽  
Theoretical Results ◽  
Continuous Dependence Of Solutions

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

scholarly journals New Methods of Finite-Time Synchronization for a Class of Fractional-Order Delayed Neural Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Weiwei Zhang ◽  
Jinde Cao ◽  
Ahmed Alsaedi ◽  
Fuad E. Alsaadi
Keyword(s):  
Neural Networks ◽  
Fractional Order ◽  
Finite Time ◽  
Time Synchronization ◽  
Sufficient Conditions ◽  
Time Interval ◽  
Finite Time Interval ◽  
Delayed Neural Networks ◽  
New Methods ◽  
Theoretical Results

Finite-time synchronization for a class of fractional-order delayed neural networks with fractional order α, 0<α≤1/2 and 1/2<α<1, is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequality skills, two sufficient conditions are considered to ensure synchronization of fractional-order delayed neural networks in a finite-time interval. Numerical example is given to verify the feasibility of the theoretical results.

scholarly journals Robust Synchronization Controller Design for a Class of Uncertain Fractional Order Chaotic Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Lin Wang ◽  
Chunzhi Yang
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Closed Loop ◽  
Controller Design ◽  
Sufficient Conditions ◽  
Closed Loop System ◽  
Synchronization Problem ◽  
Robust Synchronization ◽  
Lyapunov Stability Criterion ◽  
Theoretical Results

Synchronization problem for a class of uncertain fractional order chaotic systems is studied. Some fundamental lemmas are given to show the boundedness of a complicated infinite series which is produced by differentiating a quadratic Lyapunov function with fractional order. By using the fractional order extension of the Lyapunov stability criterion and the proposed lemma, stability of the closed-loop system is analyzed, and two sufficient conditions, which can enable the synchronization error to converge to zero asymptotically, are driven. Finally, an illustrative example is presented to confirm the proposed theoretical results.

Dynamical analysis of fractional-order differentiated Cournot triopoly game

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1367-1380
Author(s):  
Abdulrahman Al-khedhairi
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Periodic Forcing ◽  
Nash Equilibrium Point ◽  
Dynamical Behaviors ◽  
Theoretical Results

The objective of the article is to study the dynamics of the proposed fractional-order Cournot triopoly game. Sufficient conditions for the existence and uniqueness of the triopoly game solution are obtained. Stability analysis of equilibrium points of the fractional-order game is also discussed. The conditions for the presence of Nash equilibrium point along with its global stability analysis are studied. The interesting dynamical behaviors of the arbitrary-order Cournot triopoly game are discussed. Moreover, the effects of seasonal periodic forcing on the game’s behaviors are examined. The 0–1 test is used to distinguish between regular and irregular dynamics of system behaviors. Numerical analysis is used to verify the theoretical results that are obtained, and revealed that the nonautonomous fractional-order model induces more complicated dynamics in the Cournot triopoly game behavior and the seasonally forced game exhibits more complex dynamics than the unforced one.

scholarly journals Synchronization of bidirectional N-coupled fractional-order chaotic systems with ring connection based on antisymmetric structure

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Cuimei Jiang ◽  
Akbar Zada ◽  
M. Tamer Şenel ◽  
Tongxing Li
Keyword(s):  
Fractional Order ◽  
Lyapunov Functions ◽  
Chaotic Systems ◽  
Design Method ◽  
Sufficient Conditions ◽  
Order Error ◽  
Synchronization Problem ◽  
Direct Design ◽  
Error System ◽  
Theoretical Results

Abstract This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results.

scholarly journals Stability Results for a Coupled System of Impulsive Fractional Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 927 ◽  
Author(s):  
Akbar Zada ◽  
Shaheen Fatima ◽  
Zeeshan Ali ◽  
Jiafa Xu ◽  
Yujun Cui
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Coupled System ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Stability Results ◽  
Theoretical Results

In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s fixed point theorem. The main theoretical results are well illustrated with the help of an example.

scholarly journals Dynamics of an Impulsive Stochastic Nonautonomous Chemostat Model with Two Different Growth Rates in a Polluted Environment

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yajie Li ◽  
Xinzhu Meng
Keyword(s):  
Growth Rates ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Positive Periodic Solution ◽  
Stochastic Noise ◽  
Polluted Environment ◽  
Chemostat Model ◽  
Lyapunov Functions Method ◽  
Theoretical Results

This paper proposes a novel impulsive stochastic nonautonomous chemostat model with the saturated and bilinear growth rates in a polluted environment. Using the theory of impulsive differential equations and Lyapunov functions method, we first investigate the dynamics of the stochastic system and establish the sufficient conditions for the extinction and the permanence of the microorganisms. Then we demonstrate that the stochastic periodic system has at least one nontrivial positive periodic solution. The results show that both impulsive toxicant input and stochastic noise have great effects on the survival and extinction of the microorganisms. Furthermore, a series of numerical simulations are presented to illustrate the performance of the theoretical results.

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.

scholarly journals Robust Stability Analysis of Fractional-Order Hopfield Neural Networks with Parameter Uncertainties

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Shuo Zhang ◽  
Yongguang Yu ◽  
Wei Hu
Keyword(s):  
Neural Networks ◽  
Fractional Order ◽  
Robust Stability ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Neural System ◽  
Hopfield Neural Networks ◽  
Parameter Uncertainties ◽  
Robust Stability Analysis ◽  
Theoretical Results

The issue of robust stability for fractional-order Hopfield neural networks with parameter uncertainties is investigated in this paper. For such neural system, its existence, uniqueness, and global Mittag-Leffler stability of the equilibrium point are analyzed by employing suitable Lyapunov functionals. Based on the fractional-order Lyapunov direct method, the sufficient conditions are proposed for the robust stability of the studied networks. Moreover, robust synchronization and quasi-synchronization between the class of neural networks are discussed. Furthermore, some numerical examples are given to show the effectiveness of our obtained theoretical results.

Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term

2020 ◽  
Vol 70 (5) ◽  
pp. 1165-1182
Author(s):  
George E. Chatzarakis ◽  
George M. Selvam ◽  
Rajendran Janagaraj ◽  
George N. Miliaras
Keyword(s):  
Fractional Order ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Damping Term ◽  
Transformation Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Oscillation Criteria ◽  
The Difference ◽  
Theoretical Results

AbstractThe aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form$$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$In the above equation α (0 < α ≤ 1) is the fractional order, $\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δα is the difference operator of the Riemann-Liouville (R-L) derivative of order α. We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results.

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