scholarly journals On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ndolane Sene ◽  
Ameth Ndiaye
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Order Derivative ◽  
Order System ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
The Stability

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

scholarly journals Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ndolane Sene
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Fractional Chaotic System ◽  
The Stability ◽  
Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization

2016 ◽  
Vol 26 (13) ◽  
pp. 1650222 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsonbaty ◽  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Control Technique ◽  
Order System ◽  
Qualitative Behavior ◽  
Hyperchaotic System

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

scholarly journals Dynamics Analysis and Control of a Five-Term Fractional-Order System

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Li-xin Yang ◽  
Xiao-jun Liu
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Phase Portraits ◽  
Order System ◽  
Bifurcation Diagrams ◽  
Compound Structure ◽  
Dynamical Properties ◽  
The Stability ◽  
And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

scholarly journals Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ping Zhou ◽  
Rui Ding
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
Lorenz Chaotic System ◽  
Control And Synchronization

The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

scholarly journals Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Attiq ul Rehman ◽  
Ram Singh ◽  
Thabet Abdeljawad ◽  
Eric Okyere ◽  
Liliana Guran
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Modeling Analysis ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Temporary Immunity

AbstractThe present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number $\mathcal{R}_{0}$ R 0 is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

scholarly journals The Dynamics and Synchronization of a Fractional-Order System with Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoya Yang ◽  
Xiaojun Liu ◽  
Honggang Dang ◽  
Wansheng He
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Complex Variables ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
System Parameters ◽  
The Stability

A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

Dynamics analysis of fractional-order Hopfield neural networks

2020 ◽  
pp. 2050083
Author(s):  
Iqbal M. Batiha ◽  
Ramzi B. Albadarneh ◽  
Shaher Momani ◽  
Iqbal H. Jebril
Keyword(s):  
Neural Network ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Phase Portraits ◽  
Dynamics Analysis ◽  
Fractional Order Systems ◽  
Runge Kutta Method ◽  
Intermittent Chaos ◽  
Predictor Corrector

This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor–Corrector Adams–Bashforth–Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge–Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin–Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag–Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents’ diagrams.

LAG SYNCHRONIZATION OF THE FRACTIONAL-ORDER SYSTEM VIA NONLINEAR OBSERVER

2011 ◽  
Vol 25 (29) ◽  
pp. 3951-3964 ◽  
Author(s):  
HAO ZHU ◽  
ZHONGSHI HE ◽  
SHANGBO ZHOU
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Stability Theorem ◽  
Nonlinear Observer ◽  
Order System ◽  
Fractional Order Systems ◽  
Lag Synchronization ◽  
Systematic Method ◽  
Fractional System ◽  
The Stability

In this paper, based on the idea of nonlinear observer, lag synchronization of chaotic fractional system with commensurate and incommensurate order is studied by the stability theorem of linear fractional-order systems. The theoretical analysis of fractional-order systems in this paper is a systematic method. This technique is applied to achieve the lag synchronization of fractional-order Rössler's system, verified by numerical simulation.

scholarly journals A New Fractional-Order Chaotic Complex System and Its Antisynchronization

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Cuimei Jiang ◽  
Shutang Liu ◽  
Chao Luo
Keyword(s):  
Complex System ◽  
Fractional Order ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Chaotic Complex System ◽  
The Stability ◽  
Complex Lorenz System

We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

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