OBSERVER-BASED SYNCHRONIZATION FOR A CLASS OF FRACTIONAL CHAOTIC SYSTEMS VIA A SCALAR SIGNAL: RESULTS INVOLVING THE EXACT SOLUTION OF THE ERROR DYNAMICS

2011 ◽  
Vol 21 (03) ◽  
pp. 955-962 ◽  
Author(s):  
DONATO CAFAGNA ◽  
GIUSEPPE GRASSI
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Exact Analytical Solution ◽  
Fractional Order Systems ◽  
Fractional Error ◽  
Transmitted Signal ◽  
Error Dynamics

This paper deals with chaos synchronization for a class of fractional-order systems characterized by one nonlinearity. In particular, an observer-based approach is illustrated, which presents two remarkable features: (i) it provides an exact analytical solution of the fractional error dynamics, written in terms of Mittag-Leffler function; (ii) it enables synchronization to be achieved using a scalar transmitted signal. Finally, a synchronization example based on fractional Chua's system is illustrated, with the aim to show the capabilities of the developed approach.

Exact Analytical Solution of Fractional Relaxation-Oscillation Equation by Adomian Decomposition and He’s Variational Technique

2009 ◽  
Author(s):  
K. C. Basak ◽  
P. C. Ray ◽  
R. K. Bera
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Order ◽  
Decomposition Method ◽  
Exact Analytical Solution ◽  
Relaxation Oscillation ◽  
Variational Technique ◽  
Adomian Decomposition ◽  
Fractional Relaxation ◽  
Oscillation Equation

Exact solution of linear fractional relaxation-oscillation equation is obtained by the decomposition method of Adomian and also by He’s variational method for fractional order α, for 1 < α ≤ 2. Surface plots of the above solution are drawn for different values of fractional order α and time t. Amplitude of the oscillation increases with α but it decreases as time increases.

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.

CHAOS SYNCHRONIZATION OF FRACTIONAL ORDER UNIFIED CHAOTIC SYSTEM VIA NONLINEAR CONTROL

2011 ◽  
Vol 25 (03) ◽  
pp. 407-415 ◽  
Author(s):  
XIANG RONG CHEN ◽  
CHONG XIN LIU
Keyword(s):  
Nonlinear Control ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Control Method ◽  
Fractional Order Systems ◽  
Unified Chaotic System ◽  
Simulation Results ◽  
Nonlinear Control Method ◽  
The Stability

Based on the stability theory of fractional order systems, an effective but theoretically rigorous nonlinear control method is proposed to synchronize the fractional order chaotic systems. Using this method, chaos synchronization between two identical fractional order unified systems is studied. Simulation results are shown to illustrate the effectiveness of this method.

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.

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.

scholarly journals Finite Time Synchronization for Fractional Order Sprott C Systems with Hidden Attractors

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Cui Yan ◽  
He Hongjun ◽  
Lu Chenhui ◽  
Sun Guan
Keyword(s):  
Fractional Order ◽  
Finite Time ◽  
Chaotic Systems ◽  
Time Synchronization ◽  
Engineering Practice ◽  
Fractional Order Systems ◽  
Spectral Entropy ◽  
Hidden Attractors ◽  
Finite Time Stability ◽  
Peculiar Phenomenon

Fractional order systems have a wider range of applications. Hidden attractors are a peculiar phenomenon in nonlinear systems. In this paper, we construct a fractional-order chaotic system with hidden attractors based on the Sprott C system. According to the Adomain decomposition method, we numerically simulate from several algorithms and study the dynamic characteristics of the system through bifurcation diagram, phase diagram, spectral entropy, and C0 complexity. The results of spectral entropy and C0 complexity simulations show that the system is highly complex. In order to apply such research results to engineering practice, for such fractional-order chaotic systems with hidden attractors, we design a controller to synchronize according to the finite-time stability theory. The simulation results show that the synchronization time is short and the robustness is stable. This paper lays the foundation for the study of fractional order systems with hidden attractors.

Hybrid Projective Synchronization for the Fractional-Order Chen-Lee System and its Circuit Realization

2013 ◽  
Vol 300-301 ◽  
pp. 1573-1578
Author(s):  
Seng Kin Lao ◽  
Hsien Keng Chen ◽  
Lap Mou Tam ◽  
Long Jye Sheu
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Body Motion ◽  
Projective Synchronization ◽  
Fractional Order Systems ◽  
Control Of Chaos ◽  
Circuit Realization ◽  
Hybrid Projective Synchronization ◽  
The Time Domain ◽  
Theoretical Analyses

The growing interest shows the importance of the control of chaos in fractional-order systems in recent years. This paper investigates in the hybrid projective synchronization of two chaotic systems with fractional-order, which were derived from Euler equations of rigid body motion. Theoretical analyses of the proposed methods are validated by numerical simulation in the time domain. Moreover, the synchronization system is realized using electronic circuits with fractance in the frequency domain.

FUZZY CHAOS SYNCHRONIZATION VIA SAMPLED DRIVING SIGNALS

2004 ◽  
Vol 14 (08) ◽  
pp. 2721-2733 ◽  
Author(s):  
JUAN GONZALO BARAJAS-RAMÍREZ ◽  
GUANRONG CHEN ◽  
LEANG S. SHIEH
Keyword(s):  
Continuous Time ◽  
System Approach ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Disturbance Attenuation ◽  
Fuzzy Observer ◽  
Master System ◽  
Error Dynamics ◽  
Takagi Sugeno ◽  
Driving Signal

In this paper, a methodology to design a system that robustly synchronizes a master chaotic system from a sampled driving signal is developed. The method is based on the fuzzy Takagi–Sugeno representation of chaotic systems, from which a continuous-time fuzzy observer is designed as the solution of an LMI minimization problem such that the error dynamics have H∞disturbance attenuation performance. Then, from the dual-system approach, the fuzzy observer is digitally redesigned such that the performance is maintained for the sampled master system. The effectiveness of the proposed synchronization methodology is finally illustrated via numerical simulations of the chaotic Chen's system.

Sign in / Sign up

Export Citation Format

Share Document