scholarly journals Chaos Synchronization between Two Different Fractional Systems of Lorenz Family

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
A. E. Matouk
Keyword(s):  
Theoretical Analysis ◽  
Laplace Transform ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Order Parameter ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Chen System ◽  
Fractional Systems ◽  
Lü System

This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.

CHAOS SYNCHRONIZATION OF FRACTIONAL-ORDER DIFFERENTIAL SYSTEMS

2006 ◽  
Vol 20 (07) ◽  
pp. 791-803 ◽  
Author(s):  
CHANGPIN LI ◽  
WEIHUA DENG
Keyword(s):  
Laplace Transform ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Fractional Systems ◽  
Coupling Methods ◽  
The Laplace Transform ◽  
Fractional Orders

Chaos synchronization of the Duffing, Lorenz and Rössler systems with fractional orders are studied theoretically and numerically. Three methods are applied in this paper: combination of active-passive decomposition (APD) and one-way coupling methods, Pecora–Carroll method, bidirectional coupling method. The sufficient conditions of achieving synchronization between two identical fractional systems are derived by using the Laplace transform theory. Numerical simulations demonstrate the effectiveness of the proposed synchronization schemes for these fractional systems.

scholarly journals Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Jianeng Tang
Keyword(s):  
Time Delay ◽  
Laplace Transform ◽  
Numerical Simulations ◽  
Active Control ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Control Technique ◽  
The Laplace Transform

Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.

Adaptive Synchronization of Fractional Hyper-Chaotic System with Unknown Parameters

2013 ◽  
Vol 850-851 ◽  
pp. 868-871 ◽  
Author(s):  
Li Xin Yang ◽  
Wan Sheng He ◽  
Jin Ping Jia ◽  
Fan Di Zhang
Keyword(s):  
Fractional Order ◽  
Tracking Control ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Adaptive Synchronization ◽  
Uncertain Parameters ◽  
Unknown Parameters ◽  
Fractional Systems ◽  
Control Approach ◽  
The Stability

In this paper, chaos synchronization of the modified Sprott E system is investigated. Based on the stability theorem for fractional systems, tracking control approach is used for the fractional-order systems with uncertain parameters. Meanwhile, suitable adaptive synchronization controller and recognizing rules of the uncertain parameters are designed. Numerical simulation results show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order hyper-chaotic systems.

scholarly journals A New Scheme on Synchronization of Commensurate Fractional-Order Chaotic Systems Based on Lyapunov Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Hua Wang ◽  
Hang-Feng Liang ◽  
Peng Zan ◽  
Zhong-Hua Miao
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lyapunov Equation ◽  
Practical Method ◽  
Fractional Order Systems ◽  
Model Uncertainties ◽  
External Disturbances ◽  
Zero Items

This paper proposes a new fractional-order approach for synchronization of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. A simple but practical method to synchronize many familiar fractional-order chaotic systems has been put forward. A new theorem is proposed for a class of cascade fractional-order systems and it is applied in chaos synchronization. Combined with the fact that the states of the fractional chaotic systems are bounded, many coupled items can be taken as zero items. Then, the whole system can be simplified greatly and a simpler controller can be derived. Finally, the validity of the presented scheme is illustrated by numerical simulations of the fractional-order unified system.

Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control

2013 ◽  
Vol 805-806 ◽  
pp. 1975-1978
Author(s):  
Jia Neng Tang
Keyword(s):  
Time Delay ◽  
Laplace Transform ◽  
Numerical Simulations ◽  
Active Control ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Control Technique ◽  
The Laplace Transform

In this paper, based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method.

SYNCHRONIZATION AND GENERALIZED SYNCHRONIZATION OF FRACTIONAL ORDER CHAOTIC SYSTEMS

2009 ◽  
Vol 23 (13) ◽  
pp. 1695-1714 ◽  
Author(s):  
XING-YUAN WANG ◽  
JING ZHANG
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Systems ◽  
State Observer ◽  
Generalized Synchronization ◽  
Fractional Order Systems ◽  
The Stability

In this paper, based on the modified state observer method, synchronization and generalized synchronization of a class of fractional order chaotic systems are presented. The two synchronization approaches are theoretically and numerically studied and two simple criterions are proposed. By using the stability theory of linear fractional order systems, suitable conditions for achieving synchronization and generalized synchronization are given. Numerical simulations coincide with the theoretical analysis.

Synchronization in Integer and Fractional Order Chaotic Systems

2011 ◽  
pp. 127-151
Author(s):  
Ahmed E. Matouk
Keyword(s):  
Lyapunov Function ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Fractional Order Systems ◽  
Chen System ◽  
Van Der Pol ◽  
Coupling Technique ◽  
The Laplace Transform

In this chapter, the author introduces the basic methods of chaos synchronization in integer order systems, such as Pecora and Carroll method and One-Way coupling technique, applying these synchronization methods to the modified autonomous Duffing-Van der Pol system (MADVP). The conditional Lyapunov exponents (CLEs) are also calculated for the drive and response MADVP systems which match with the analytical results given by Pecora and Carroll method. Based on Lyapunov stability theory, chaos synchronization is achieved for two coupled MADVP systems by finding a suitable Lyapunov function. Moreover, synchronization in fractional order chaotic systems is also introduced. The conditions of Pecora and Carroll method and One-Way coupling method in fractional order systems are also investigated. In addition, chaos synchronization is achieved for two coupled fractional order MADVP systems using One-Way coupling technique. Furthermore, synchronization between two different fractional order chaotic systems is studied; the fractional order Lü system is controlled to be the fractional order Chen system. The analytical conditions for the synchronization of this pair of different fractional order chaotic systems are derived by utilizing the Laplace transform theory. Numerical simulations are carried out to show the effectiveness of all the proposed synchronization techniques.

THE SYNCHRONIZATION FOR AUTONOMOUS CHAOTIC SYSTEMS WITH DISTURBANCE OF PARAMETER

2012 ◽  
Vol 26 (09) ◽  
pp. 1250058
Author(s):  
XINGYUAN WANG ◽  
BING XU
Keyword(s):  
Numerical Simulations ◽  
Active Control ◽  
Chaotic Systems ◽  
Lorenz System ◽  
The Self ◽  
Hyperchaotic System ◽  
Chen System ◽  
Adaptive Parameter ◽  
Lü System ◽  
System A

This paper analyzes the synchronizations for autonomous chaotic systems with disturbance of parameter, achieves the self-synchronization of Lü system, a modified coupled dynamos system and a four-dimensional hyperchaotic system by the methods of active control and adaptive parameter. In addition, the synchronization of Lorenz system and Chen system is presented. Numerical simulations are provided for illustration and verification of the proposed method.

ANTI-SYNCHRONIZATION OF THREE-DIMENSIONAL AUTONOMOUS CHAOTIC SYSTEMS VIA ACTIVE CONTROL

2007 ◽  
Vol 21 (17) ◽  
pp. 3017-3027
Author(s):  
XINGYUAN WANG ◽  
YONG WANG
Keyword(s):  
Numerical Simulations ◽  
Active Control ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Chen System ◽  
Lü System ◽  
Lu System

This paper analyzes anti-synchronization of three-dimensional autonomous chaotic systems and achieves the anti-synchronization of a class of three-dimensional autonomous chaotic systems, i.e., Lorenz system, Chen system, and Lü system with one another via active control. Numerical simulations are demonstrated to verify the effectiveness of the proposed method.

Fractional order generalized thermoelastic response in a half space due to a periodically varying heat source

2018 ◽  
Vol 14 (1) ◽  
pp. 2-15 ◽  
Author(s):  
Jitesh Tripathi ◽  
Shrikant Warbhe ◽  
K.C. Deshmukh ◽  
Jyoti Verma
Keyword(s):  
Mathematical Model ◽  
Heat Source ◽  
Laplace Transform ◽  
Half Space ◽  
Fractional Order ◽  
Order Parameter ◽  
Order Theory ◽  
Numerical Inversion ◽  
Content Type ◽  
Copper Material

Purpose The present work is concerned with the solution of a fractional-order thermoelastic problem of a two-dimensional infinite half space under axisymmetric distributions in which lower surface is traction free and subjected to a periodically varying heat source. The thermoelastic displacement, stresses and temperature are determined within the context of fractional-order thermoelastic theory. To observe the variations of displacement, temperature and stress inside the half space, the authors compute the numerical values of the field variables for copper material by utilizing Gaver-Stehfast algorithm for numerical inversion of Laplace transform. The effects of fractional-order parameter on the variations of field variables inside the medium are analyzed graphically. The paper aims to discuss these issues. Design/methodology/approach Integral transform technique and Gaver-Stehfast algorithm are applied to prepare the mathematical model by considering the periodically varying heat source in cylindrical co-ordinates. Findings This paper studies a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under fractional-order theory of thermoelasticity proposed by Sherief (Ezzat and El-Karamany, 2011b). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Ezzat and El-Karamany, 2011d; Ezzat, 2012; Ezzat, El Karamany, Ezzat, 2012) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (El-Karamany et al., 2011) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter. Research limitations/implications Constructed purely on theoretical mathematical model by considering different parameters and the functions. Practical implications The system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations. Originality/value In this problem, the authors have used the time fractional-order theory of thermoelasticity to solve the problem for a half space with a periodically varying heat source to control the speed of wave propagation in terms of heat and elastic waves for different conductivity like weak conductivity, moderate conductivity and super conductivity which is a new and novel contribution.

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