scholarly journals Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Part II: The Noncommensurate Case

2021 ◽  
Vol 31 (12) ◽  
pp. 2150187
Author(s):  
Marius-F. Danca
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Lorenz System ◽  
Bifurcation Parameter ◽  
Fractional Order Systems ◽  
Matlab Code ◽  
Matlab Program ◽  
Caputo’S Derivative ◽  
Fractional Differential

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [ Danca & Kuznetsov, 2018 ], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

scholarly journals Matlab Code for Lyapunov Exponents of Fractional-Order Systems

2018 ◽  
Vol 28 (05) ◽  
pp. 1850067 ◽  
Author(s):  
Marius-F. Danca ◽  
Nikolay Kuznetsov
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Computing Time ◽  
Fractional Order Systems ◽  
Variational System ◽  
Matlab Code ◽  
Matlab Program ◽  
Caputo’S Derivative ◽  
Fractional Differential ◽  
Predictor Corrector

In this paper, the Benettin–Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. First, it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams–Bashforth–Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams–Bashforth–Moulton method, is utilized. Four representative examples are considered.

scholarly journals On the fractional differential equations with not instantaneous impulses

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 676-684 ◽  
Author(s):  
Xianmin Zhang ◽  
Praveen Agarwal ◽  
Zuohua Liu ◽  
Xianzhen Zhang ◽  
Wenbin Ding ◽  
...  
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Order Systems ◽  
Fractional Differential

AbstractBased on some previous works, an equivalent equations is obtained for the differential equations of fractional-orderq ∈(1, 2) with non-instantaneous impulses, which shows that there exists the general solution for this impulsive fractional-order systems. Next, an example is used to illustrate the conclusion.

scholarly journals A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aziz Khan ◽  
Hashim M. Alshehri ◽  
J. F. Gómez-Aguilar ◽  
Zareen A. Khan ◽  
G. Fernández-Anaya
Keyword(s):  
Fractional Order ◽  
Existence And Uniqueness ◽  
Variable Order ◽  
Fractional Order Systems ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractThis paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals NUMERICAL SOLUTION FOR TIME-FRACTIONAL MURRAY REACTION-DIFFUSION EQUATIONS VIA REDUCED DIFFERENTIAL TRANSFORM METHOD

2021 ◽  
Author(s):  
Muhammed Yiğider ◽  
Serkan Okur
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Differential Transform Method ◽  
Reaction Diffusion Equations ◽  
Transform Method ◽  
Differential Transform ◽  
Fractional Differential ◽  
Reduced Differential Transform Method ◽  
Time Fractional Differential Equations

In this study, solutions of time-fractional differential equations that emerge from science and engineering have been investigated by employing reduced differential transform method. Initially, the definition of the derivatives with fractional order and their important features are given. Afterwards, by employing the Caputo derivative, reduced differential transform method has been introduced. Finally, the numerical solutions of the fractional order Murray equation have been obtained by utilizing reduced differential transform method and results have been compared through graphs and tables. Keywords: Time-fractional differential equations, Reduced differential transform methods, Murray equations, Caputo fractional derivative.

Periodic impulsive fractional differential equations

2017 ◽  
Vol 8 (1) ◽  
pp. 482-496 ◽  
Author(s):  
Michal Fečkan ◽  
Jin Rong Wang
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Fractional Differential Equations ◽  
Periodic Boundary ◽  
Lorenz System ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential ◽  
Existence Of Periodic Solutions ◽  
Lower Limits ◽  
Stability Results

Abstract This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.

Sign in / Sign up

Export Citation Format

Share Document