Asymptotic Stability Of Dynamic Equations With Two Fractional Terms: Continuous Versus Discrete Case

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Jan Čermák ◽  
Tomáš Kisela
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivatives ◽  
Fractional Difference ◽  
Caputo Fractional Derivatives ◽  
Stability Regions ◽  
The Stability ◽  
Fractional Terms ◽  
Discrete Stability ◽  
Stability Sets ◽  
Asymptotic Stability Conditions

AbstractThe paper discusses asymptotic stability conditions for the linear fractional difference equation∇with real coefficients a, b and real orders α > β > 0 such that α/β is a rational number. For given α, β, we describe various types of discrete stability regions in the (a, b)-plane and compare them with the stability regions recently derived for the underlying continuous patternDinvolving two Caputo fractional derivatives. Our analysis shows that discrete stability sets are larger and their structure much more rich than in the case of the continuous counterparts.

scholarly journals Boundary Criteria for the Stability of Delay Differential-Algebraic Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Leping Sun ◽  
Yuhao Cong
Keyword(s):  
Asymptotic Stability ◽  
Differential Algebraic Equations ◽  
Analytical Function ◽  
Algebraic Equations ◽  
Stability Criteria ◽  
Stability Regions ◽  
The Stability ◽  
Delay Differential

This paper is concerned with the asymptotic stability of delay differential-algebraic equations. Two stability criteria described by evaluating a corresponding harmonic analytical function on the boundary of a certain region are presented. Stability regions are also presented so as to show the method geometrically. Our results are not reported.

scholarly journals Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
MohammadHossein Derakhshan ◽  
Azim Aminataei
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Time Varying ◽  
Lyapunov Direct Method ◽  
Fractional Differential Systems ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems.

On the subject of influence of dissipative and constant of moments on the stability of uniform rotations of non-free two elastically connected gyroscopes of Lagrange

2021 ◽  
Vol 34 ◽  
pp. 50-61
Author(s):  
Yurii Kononov ◽  
Yaroslav Sviatenko
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Stability Conditions ◽  
Inertial Coordinate System ◽  
Resisting Medium ◽  
Angular Velocities ◽  
The Subject ◽  
The Stability ◽  
Spherical Hinge ◽  
Asymptotic Stability Conditions

The conditions for asymptotic stability of uniform rotations in a resisting medium of two heavy Lagrange gyroscopes connected by an elastic spherical hinge are obtained in the form of a system of three inequalities. The bottom gyroscope has a fixed point. The rotation of the gyroscopes is maintained by constant moments in the inertial coordinate system. The influence of the elasticity of the hinge on the stability conditions is estimated. It is shown that for a sufficiently high rigidity of the hinge, the asymptotic stability conditions are determined by only one inequality, which coincides with the inequality obtained for the case of a cylindrical hinge. When the angular velocities of the gyroscopes' own rotations coincide, this inequality coincides with the well--known condition for one gyroscope. Cases of degeneration of an elastic spherical hinge into a spherical inelastic, cylindrical and universal elastic hinge (Hooke's hinge) are considered. For the Hooke hinge, it is shown that there is no asymptotic stability at a sufficiently high angular velocity of gyroscopes rotation.

scholarly journals Some Numerical Examples on the Stability of Fractional Linear Dynamical Systems

2018 ◽  
Vol 17 (3) ◽  
pp. 51-66
Author(s):  
S Priyadharsini
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Linear Models ◽  
Sufficient Conditions ◽  
Characteristic Polynomials ◽  
Caputo Fractional Derivatives ◽  
Linear Dynamical Systems ◽  
Numerical Examples ◽  
Eigen Values ◽  
The Stability

The concept of stability of a class of fractional-order linear system is considered in this paper. Existing sufficient conditions are assumed to guarantee the stability of linear models with the Caputo fractional derivatives. The results have been developed by using the concept of Laplace transform, and approximations of Mittag-Leffler.  Furthermore, results concerning asymptotical stability of linear fractional-order models are also achieved. The proposed method is based upon Eigen values and the characteristic polynomials. Numerical illustrations are specified to exhibit effectiveness of the proposed method.

scholarly journals On the Stability of Some Discrete Fractional Nonautonomous Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Fahd Jarad ◽  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Kübra Biçen
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Direct Method ◽  
Uniform Stability ◽  
Lyapunov Direct Method ◽  
Uniform Asymptotic Stability ◽  
Fractional Difference ◽  
Nonautonomous Systems ◽  
The Stability

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

scholarly journals On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Dynamic Systems ◽  
Fractional Derivatives ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Time Varying ◽  
Caputo Fractional Derivatives ◽  
Nonnegative Solutions ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

scholarly journals Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 58
Author(s):  
Osman Tunç ◽  
Özkan Atan ◽  
Cemil Tunç ◽  
Jen-Chih Yao
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Qualitative Properties ◽  
Caputo Fractional Derivatives ◽  
Fractional Differential ◽  
Weaker Conditions ◽  
Qualitative Properties Of Solutions ◽  
Qualitative Analyses

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.

scholarly journals Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

2014 ◽  
Vol 4 (3) ◽  
pp. 242-266 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Derivatives ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Caputo Fractional Derivatives ◽  
Numerical Examples ◽  
One Dimensional ◽  
Compact Difference ◽  
Spatial Discretisation ◽  
The Stability

AbstractSome efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

scholarly journals Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1431
Author(s):  
Bilal Basti ◽  
Nacereddine Hammami ◽  
Imadeddine Berrabah ◽  
Farid Nouioua ◽  
Rabah Djemiat ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Biological Models ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Single Form ◽  
Stability Results

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

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