scholarly journals Study of a Fractional-Order Chaotic System Represented by the Caputo Operator

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Ndolane Sene
Keyword(s):  
Fractional Derivative ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Caputo Fractional Derivative ◽  
Numerical Scheme ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Numerical Discretization ◽  
Good Agreement

This paper is presented on the theory and applications of the fractional-order chaotic system described by the Caputo fractional derivative. Considering the new fractional model, it is important to establish the presence or absence of chaotic behaviors. The Lyapunov exponents in the fractional context will be our fundamental tool to arrive at our conclusions. The variations of the model’s parameters will generate chaotic behavior, in general, which will be established using the Lyapunov exponents and bifurcation diagrams. For the system’s phase portrait, we will present and apply an interesting fractional numerical discretization. For confirmation of the results provided in this paper, the circuit schematic is drawn and simulated. As it will be observed, the results obtained after the simulation of the numerical scheme and with the Multisim are in good agreement.

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.

scholarly journals Theory and applications of new fractional-order chaotic system under Caputo operator

2021 ◽  
Vol 12 (1) ◽  
pp. 20-38
Author(s):  
Ndolane Sene
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Caputo Derivative ◽  
Phase Portraits ◽  
Chaotic Circuit ◽  
Chaotic Model ◽  
The Stability ◽  
The Impact ◽  
Schematic Circuit ◽  
Good Agreement

This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.

scholarly journals Метод расчета спектра показателей Ляпунова для систем с запаздыванием

2018 ◽  
Vol 44 (9) ◽  
pp. 19
Author(s):  
А.Д. Колоскова ◽  
О.И. Москаленко ◽  
А.А. Короновский
Keyword(s):  
Lyapunov Exponents ◽  
Delayed Feedback ◽  
Model Systems ◽  
Delay Systems ◽  
Control Parameters ◽  
Bifurcation Diagrams ◽  
Good Agreement

AbstractA method for calculating the spectrum of Lyapunov exponents for delay systems is proposed. To validate the method, a delayed-feedback oscillator and the Mackay–Glass equation are considered as model systems. For both systems, bifurcation diagrams and spectra of Lyapunov exponents are constructed as functions of one of the control parameters. The results are shown to be in good agreement with each other, which indicates the efficacy of the proposed method.

scholarly journals Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

2017 ◽  
Vol 22 (4) ◽  
pp. 503-513 ◽  
Author(s):  
Fei Wang ◽  
Yongqing Yang
Keyword(s):  
Nonlinear Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Barbalat’S Lemma ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
The Stability ◽  
The Relationship ◽  
Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

NUMERICAL AND EXPERIMENTAL STUDIES OF SELF-SYNCHRONIZATION AND SYNCHRONIZED CHAOS

1992 ◽  
Vol 02 (03) ◽  
pp. 645-657 ◽  
Author(s):  
M. DE SOUSA VIEIRA ◽  
P. KHOURY ◽  
A. J. LICHTENBERG ◽  
M. A. LIEBERMAN ◽  
W. WONCHOBA ◽  
...  
Keyword(s):  
Chaotic Behavior ◽  
Experimental Studies ◽  
Chaotic Signal ◽  
Chaotic Synchronization ◽  
Parameter Variation ◽  
Phase Locked Loops ◽  
Bifurcation Diagrams ◽  
Digital Phase Locked Loops ◽  
Chaotic Carrier ◽  
Good Agreement

We study self-synchronization of digital phase-locked loops (DPLL's) and the chaotic synchronization of DPLL's in a communication system which consists of three or more coupled DPLL's. Triangular wave signals, convenient for experiments, are employed. Numerical and experimental studies of two loops are in good agreement, giving bifurcation diagrams that show quasiperiodic, locked, and chaotic behavior. The approach to chaos does not show the full bifurcation sequence of sinusoidal signals. For studying synchronization to a chaotic signal, the chaotic carrier is generated in a subsystem of two or more self-synchronized DPLL's where one of the loops is stable and the other is unstable. The receiver consists of a stable loop. We verified numerically and experimentally that the receiver may synchronize with the transmitter if the stable loop in the transmitter and receiver are nearly identical and the synchronization degrades with noise and parameter variation. We studied the phase space where synchronization occurs, and quantify the deviation from synchronization using the concept of mutual information.

scholarly journals Analysis of a New Quadratic 3D Chaotic Attractor

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shahed Vahedi ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Bifurcation Diagrams ◽  
Basic Properties ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

A new three-dimensional chaotic system is introduced. Basic properties of this system show that its corresponding attractor is topologically different from some well-known systems. Next, detailed information on dynamic of this system is obtained numerically by means of Lyapunov exponents spectrum, bifurcation diagrams, and 0-1 chaos indicator test. We finally prove existence of this chaotic attractor theoretically using Shil’nikov theorem and undetermined coefficient method.

scholarly journals CHAOTIC BEHAVIOR OF BHALEKAR–GEJJI DYNAMICAL SYSTEM UNDER ATANGANA–BALEANU FRACTAL FRACTIONAL OPERATOR

Fractals ◽  
2021 ◽  
pp. 2240005
Author(s):  
SHABIR AHMAD ◽  
AMAN ULLAH ◽  
ALI AKGÜL ◽  
THABET ABDELJAWAD
Keyword(s):  
Fractal Dimension ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Existence Theory ◽  
Complex Behavior ◽  
Nonlinear Functional ◽  
Proposed Model ◽  
Fractal Derivative ◽  
2D And 3D

In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the new fractal–fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.

A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050070 ◽  
Author(s):  
CONG WU
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Comparison Principle ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Order Systems ◽  
Caputo Fractional Derivatives ◽  
Full Result

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

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