scholarly journals Stability Analysis of Perturbed Linear Non-integer Differential Systems

2021 ◽  
pp. 24-29
Author(s):  
Ubong D. Akpan
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential System ◽  
Caputo Derivative ◽  
Differential Systems ◽  
Fractional Differential System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
The Stability

In this work, the effect of perturbation on linear fractional differential system is studied. The analysis is done using Riemann-Liouville derivative and the conclusion extended to using Caputo derivative since the result is similar. Conditions for determining the stability and asymptotic stability of perturbed linear fractional differential system are given.

scholarly journals On the Stability Analysis of Linear Unperturbed Non-integer Differential Systems

2021 ◽  
pp. 85-89
Author(s):  
Ubong D. Akpan
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential System ◽  
Differential Systems ◽  
Caputo Derivatives ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
The Stability

In this paper, the stability of non-integer differential system is studied using Riemann-Liouville and Caputo derivatives. The stability notion for determining the stability/asymptotic stability or otherwise fractional differential system is given. Example is provided to demonstrate the effectiveness of the result.

scholarly journals Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Fengrong Zhang ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Caputo Derivative ◽  
Sufficient Conditions ◽  
Comparison Method ◽  
Asymptotical Stability ◽  
Sufficient Condition ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
The Stability

This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.

Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Changpin Li ◽  
Zhiqiang Li
Keyword(s):  
Differential System ◽  
Integral Transforms ◽  
Nonlinear Case ◽  
Algebraic Decay ◽  
Future Studies ◽  
Linearization Theorem ◽  
Fractional Differential System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
The Stability

Abstract In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Equivalent Same-Order System for Multi-Rational-Order Fractional Differential System With Caputo Derivative

2011 ◽  
Author(s):  
Changpin Li ◽  
Fengrong Zhang
Keyword(s):  
Differential System ◽  
Caputo Derivative ◽  
Order System ◽  
Fractional Differential System ◽  
Higher Dimensional ◽  
Fractional Differential ◽  
The Stability ◽  
Rational Order ◽  
The Relationship ◽  
Generalized Fractional Derivative

The equivalent same-order system for multi-rational-order fractional differential system with Caputo derivative is studied. With the relationship between Caputo derivative and generalized fractional derivative, we can change the multi-order fractional differential system whose fractional order is rational into a much higher-dimensional fractional differential system with the same order lying in (0,1). Based on the equivalent results, the stability analysis of any multi-rational-order fractional differential system is given. Finally, several examples are provided to illustrate the results in this paper.

scholarly journals Equivalent system for a multiple-rational-order fractional differential system

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120156 ◽  
Author(s):  
Changpin Li ◽  
Fengrong Zhang ◽  
Jürgen Kurths ◽  
Fanhai Zeng
Keyword(s):  
Fractional Derivative ◽  
Differential System ◽  
Caputo Derivative ◽  
Original System ◽  
Dimensional System ◽  
Equivalent System ◽  
Fractional Differential System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Rational Order

The equivalent system for a multiple-rational-order (MRO) fractional differential system is studied, where the fractional derivative is in the sense of Caputo or Riemann–Liouville. With the relationship between the Caputo derivative and the generalized fractional derivative, we can change the MRO fractional differential system with a Caputo derivative into a higher-dimensional system with the same Caputo derivative order lying in (0,1). The stability of the zero solution to the original system is studied through the analysis of its equivalent system. For the Riemann–Liouville case, we transform the MRO fractional differential system into a new one with the same order lying in (0,1), where the properties of the Riemann–Liouville derivative operator and the fractional integral operator are used. The corresponding stability is also studied. Finally, several numerical examples are provided to illustrate the derived results.

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

scholarly journals On the stability of impulsively perturbed differential systems

1977 ◽  
Vol 17 (3) ◽  
pp. 423-432 ◽  
Author(s):  
S.G. Pandit
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Stability Property ◽  
Unperturbed System ◽  
Uniform Asymptotic Stability ◽  
Differential Systems ◽  
Perturbed System ◽  
Gronwall Integral Inequality ◽  
The Stability ◽  
Points Of Discontinuity

This paper deals with the study of uniform asymptotic stability of the measure differential system Dx = F(t, x) + G(t, x)Du, where the symbol D stands for the derivative in the sense of distributions. The system is viewed as a perturbed system of the ordinary differential system x' = F(t, x), where the perturbation term G(t, x)Du is impulsive and the state of the system changes suddenly at the points of discontinuity of u. It is shown, under certain conditions, that the uniform asymptotic stability property of the unperturbed system is shared by the perturbed system. To do this, the well-known Gronwall integral inequality is generalized so as to be applicable to Lebesgue-Stieltjes integrals.

Global stability analysis of a fractional differential system in hepatitis B

2021 ◽  
Vol 143 ◽  
pp. 110619
Author(s):  
Lislaine Cristina Cardoso ◽  
Rubens Figueiredo Camargo ◽  
Fernando Luiz Pio dos Santos ◽  
José Paulo Carvalho Dos Santos
Keyword(s):  
Stability Analysis ◽  
Hepatitis B ◽  
Global Stability ◽  
Differential System ◽  
Global Stability Analysis ◽  
Fractional Differential System ◽  
Fractional Differential
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