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 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.

Novel Cryptography Technique via Chaos Synchronization of Fractional-Order Derivative Systems

2018 ◽  
pp. 404-437
Author(s):  
Alain Giresse Tene ◽  
Timoleon Crépin Kofane
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Order Derivative ◽  
Random Numbers ◽  
Generation Process ◽  
Initial Condition ◽  
Fractional Order Derivative ◽  
Q Matrix ◽  
Fractional System

Synchronization of fractional-order-derivative systems for cryptography purpose is still exploratory and despite an increase in cryptography research, several challenges remain in designing a powerful cryptosystem. This chapter addresses the problem of synchronization of fractional-order-derivative chaotic systems using random numbers generator for a novel technique to key distribution in cryptography. However, there is evidence that researchers have approached the problem using integer order derivative chaotic systems. Consequently, the aim of the chapter lies in coding and decoding a text via chaos synchronization of fractional-order derivative, the performance analysis and the key establishment scheme following an application on a text encryption using the chaotic Mathieu-Van Der Pol fractional system. In order to improve the level of the key security, the Fibonacci Q-matrix is used in the key generation process and the initial condition; the order of the derivative of the responder system secretly shared between the responder and the receiver are also involved. It followed from this study that compared to the existing cryptography techniques, this proposed method is found to be very efficient due to the fact that, it improves the key security.

Synchronization of Liu Chaotic System with Fractional-Order

2013 ◽  
Vol 336-338 ◽  
pp. 467-470
Author(s):  
Su Hai Huang
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Unknown Parameter ◽  
Chaos Synchronization ◽  
Sliding Mode ◽  
Projective Synchronization ◽  
Sliding Mode Controller ◽  
Adaptive Sliding Mode ◽  
Adaptive Sliding Mode Controller

This paper deals with chaos synchronization of the Liu chaotic system with fractional-order. Based on the fractional-order stability theory, an adaptive sliding mode controller has been constructed to realize projective synchronization of fractional-order Liu chaotic system with unknown parameter. An illustrative simulation result is given to demonstrate the effectiveness of the proposed sliding mode controller.

scholarly journals One Adaptive Synchronization Approach for Fractional-Order Chaotic System with Fractional-Order1

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ping Zhou ◽  
Rongji Bai
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Chaotic System ◽  
Stability Result ◽  
Adaptive Synchronization ◽  
Fractional Order Systems ◽  
Lorenz Chaotic System

Based on a new stability result of equilibrium point in nonlinear fractional-order systems for fractional-order lying in1<q<2, one adaptive synchronization approach is established. The adaptive synchronization for the fractional-order Lorenz chaotic system with fractional-order1<q<2is considered. Numerical simulations show the validity and feasibility of the proposed scheme.

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 Synchronization of the Fractional-Order Brushless DC Motors Chaotic System

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Shiyun Shen ◽  
Ping Zhou
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Direct Method ◽  
Fractional Order Systems ◽  
Lyapunov Direct Method ◽  
Brushless Dc Motors ◽  
Brushless Dc ◽  
Dc Motors ◽  
Synchronization Scheme

Based on the extension of Lyapunov direct method for nonlinear fractional-order systems, chaos synchronization for the fractional-order Brushless DC motors (BLDCM) is discussed. A chaos synchronization scheme is suggested. By means of Lyapunov candidate function, the theoretical proof of chaos synchronization is addressed. The numerical results show that the chaos synchronization scheme is valid.

scholarly journals Circuit Implementation Synchronization between Two Modified Fractional-Order Lorenz Chaotic Systems via a Linear Resistor and Fractional-Order Capacitor in Parallel Coupling

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Juan Liu ◽  
Xuefeng Cheng ◽  
Ping Zhou
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Lorenz Chaotic System ◽  
Simulation Results ◽  
Synchronization Scheme

In this study, a modified fractional-order Lorenz chaotic system is proposed, and the chaotic attractors are obtained. Meanwhile, we construct one electronic circuit to realize the modified fractional-order Lorenz chaotic system. Most importantly, using a linear resistor and a fractional-order capacitor in parallel coupling, we suggested one chaos synchronization scheme for this modified fractional-order Lorenz chaotic system. The electronic circuit of chaos synchronization for modified fractional-order Lorenz chaotic has been given. The simulation results verify that synchronization scheme is viable.

Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system

2020 ◽  
Vol 99 (4) ◽  
pp. 3143-3154 ◽  
Author(s):  
Mohammed F. Tolba ◽  
Hani Saleh ◽  
Baker Mohammad ◽  
Mahmoud Al-Qutayri ◽  
Ahmed S. Elwakil ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Order Derivative ◽  
Variable Order ◽  
Fractional Order Derivative
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