scholarly journals Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jingwei Deng ◽  
Weiyuan Ma ◽  
Kaiying Deng ◽  
Yingxing Li
Keyword(s):  
Dynamical Systems ◽  
Fractional Calculus ◽  
Comparison Principle ◽  
Direct Method ◽  
Physical Space ◽  
Central Issue ◽  
Lyapunov Direct Method ◽  
Fractional Systems ◽  
Fractional System ◽  
Theoretical Results

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. This paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.

scholarly journals Stability for Caputo Fractional Differential Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Sung Kyu Choi ◽  
Bowon Kang ◽  
Namjip Koo
Keyword(s):  
Comparison Principle ◽  
Direct Method ◽  
Stability Of Solutions ◽  
Lyapunov Direct Method ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential

We introduce the notion ofh-stability for fractional differential systems. Then we investigate the boundedness andh-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results.

Liouville theorem for fractional Hénon–Lane–Emden systems on a half space

2019 ◽  
Vol 150 (6) ◽  
pp. 3060-3073
Author(s):  
Phuong Le
Keyword(s):  
Strong Solution ◽  
Half Space ◽  
Liouville Theorem ◽  
Direct Method ◽  
Fractional Systems ◽  
Fractional System ◽  
Beta 2 ◽  
Image Position ◽  
A Direct Method

AbstractThis paper is concerned with the fractional system \begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}where n ⩾ 2, 0 < α, β < 2, a > −α, b > −β and p, q ⩾ 1. By exploiting a direct method of scaling spheres for fractional systems, we prove that if $p \leqslant \frac {n+\alpha +2a}{n-\beta }$, $q \leqslant \frac {n+\beta +2b}{n-\alpha }$, $p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$ and (u, v) is a nonnegative strong solution of the system, then u ≡ v ≡ 0.

scholarly journals Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 500 ◽  
Author(s):  
Guotao Wang ◽  
Jianfang Qin ◽  
Huanhe Dong ◽  
Tingting Guan
Keyword(s):  
Dynamic System ◽  
Fractional Order ◽  
Comparison Principle ◽  
Nonlinear Dynamic ◽  
Direct Method ◽  
Nonlinear Dynamic System ◽  
Nonautonomous System ◽  
Lyapunov Direct Method

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

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.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

scholarly journals Results on the approximate controllability of fractional hemivariational inequalities of order $1< r<2$

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. Mohan Raja ◽  
V. Vijayakumar ◽  
Le Nhat Huynh ◽  
R. Udhayakumar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Fixed Point ◽  
Control Systems ◽  
Approximate Controllability ◽  
Hemivariational Inequalities ◽  
Evolution Inclusions ◽  
Fractional Systems ◽  
Cosine Families ◽  
Fixed Point Approach ◽  
Semilinear Control Systems ◽  
Theoretical Results

AbstractIn this paper, we investigate the approximate controllability of fractional evolution inclusions with hemivariational inequalities of order $1< r<2$ 1 < r < 2 . The main results of this paper are verified by using the fractional theories, multivalued analysis, cosine families, and fixed-point approach. At first, we discuss the existence of the mild solution for the class of fractional systems. After that, we establish the approximate controllability of linear and semilinear control systems. Finally, an application is presented to illustrate our theoretical results.

Characterization of power systems near their stability boundary by Lyapunov direct method

2014 ◽  
Vol 47 (3) ◽  
pp. 9087-9092 ◽  
Author(s):  
Igor B. Yadykin ◽  
Dmitry E. Kataev ◽  
Alexey B. Iskakov ◽  
Vladislav K. Shipilov
Keyword(s):  
Power Systems ◽  
Direct Method ◽  
Stability Boundary ◽  
Lyapunov Direct Method

Stability and resonance conditions of second-order fractional systems

2016 ◽  
Vol 24 (4) ◽  
pp. 659-672 ◽  
Author(s):  
Elena Ivanova ◽  
Xavier Moreau ◽  
Rachid Malti
Keyword(s):  
Second Order ◽  
Damping Factor ◽  
Fractional Differentiation ◽  
Resonance Amplitude ◽  
Fractional Systems ◽  
Integral Number ◽  
At Resonance ◽  
Fractional System ◽  
The Stability ◽  
Normalized Gain

The interest of studying fractional systems of second order in electrical and mechanical engineering is first illustrated in this paper. Then, the stability and resonance conditions are established for such systems in terms of a pseudo-damping factor and a fractional differentiation order. It is shown that a second-order fractional system might have a resonance amplitude either greater or less than one. Moreover, three abaci are given allowing the pseudo-damping factor and the differentiation order to be determined for, respectively, a desired normalized gain at resonance, a desired phase at resonance, and a desired normalized resonant frequency. Furthermore, it is shown numerically that the system root locus presents a discontinuity when the fractional differentiation order is an integral number.

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