An Estimate of the Bound of the Lyapunov Exponents for Caputo-Hadamard Fractional Differential System

2021 ◽  
Author(s):  
Changpin Li ◽  
Chuntao Yin
Keyword(s):  
Lyapunov Exponents ◽  
Differential System ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Initial Value ◽  
Numerical Examples ◽  
Fractional Differential System ◽  
Grönwall Inequality ◽  
Fractional Differential ◽  
Theoretical Results

Abstract This paper is devoted to estimating the bound of the Lyapunov exponents for the Caputo-Hadamard fractional differential system. First, using the Gronwall inequality, we analyze the continuous dependence of the solution to the Caputo-Hadamard fractional initial value problem. Then, we define the Lyapunov exponents for the Caputo-Hadamard fractional differential system and estimate their bounds. Besides, numerical examples are displayed which support 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 New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

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 Stability results for the linear degenerate fractional differential system

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Zhixin Zhang ◽  
Jia-Bao Liu ◽  
Jinde Cao ◽  
Wei Jiang ◽  
Ahmed Alsaedi ◽  
...  
Keyword(s):  
Differential System ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Stability Results ◽  
Linear Degenerate

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

2015 ◽  
Vol 5 (4) ◽  
pp. 301-311 ◽  
Author(s):  
Lijun Yi
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Error Bound ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Time Stepping ◽  
Theoretical Results

AbstractThe h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

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