scholarly journals Stability and Stabilizing of Fractional Complex Lorenz Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Stability Theory ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Fractional Order Systems ◽  
Caputo Derivatives ◽  
Stability And Stabilization ◽  
The Stability

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

scholarly journals Stability of Fractional Order Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Margarita Rivero ◽  
Sergei V. Rogosin ◽  
José A. Tenreiro Machado ◽  
Juan J. Trujillo
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
State Of The Art ◽  
Point Of View ◽  
Fractional Order Systems ◽  
Short Period ◽  
Systems And Control ◽  
Different Types ◽  
The Stability ◽  
And Control

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

Investigation of Synchronization for a Fractional-Order Delayed System

2014 ◽  
Vol 687-691 ◽  
pp. 447-450 ◽  
Author(s):  
Hong Gang Dang ◽  
Wan Sheng He ◽  
Xiao Ya Yang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Fractional Order Systems ◽  
Delayed System ◽  
The Stability

In this paper, synchronization of a fractional-order delayed system is studied. Based on the stability theory of fractional-order systems, by designing appropriate controllers, the synchronization for the proposed system is achieved. Numerical simulations show the effectiveness of the proposed scheme.

scholarly journals Inverse Projective Synchronization between Two Different Hyperchaotic Systems with Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Yi Chai ◽  
Liping Chen ◽  
Ranchao Wu
Keyword(s):  
Fractional Order ◽  
Lorenz System ◽  
Projective Synchronization ◽  
Generalized Synchronization ◽  
Fractional Order Systems ◽  
Chen System ◽  
Fractional Order Differential Equations ◽  
Hyperchaotic Lorenz System ◽  
The Stability ◽  
Hyperchaotic Systems

This paper mainly investigates a novel inverse projective synchronization between two different fractional-order hyperchaotic systems, that is, the fractional-order hyperchaotic Lorenz system and the fractional-order hyperchaotic Chen system. By using the stability theory of fractional-order differential equations and Lyapunov equations for fractional-order systems, two kinds of suitable controllers for achieving inverse projective synchronization are designed, in which the generalized synchronization, antisynchronization, and projective synchronization of fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system are also successfully achieved, respectively. Finally, simulations are presented to demonstrate the validity and feasibility of the proposed method.

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.

Synchronization of a New Fractional Order Hyperchaotic System with Activation Feedback Control

2013 ◽  
Vol 397-400 ◽  
pp. 1278-1281
Author(s):  
Wei Wei Zhang ◽  
Ding Yuan Chen
Keyword(s):  
Feedback Control ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Current Paper ◽  
Hyperchaotic System ◽  
Fractional Order Systems ◽  
The Stability

In the current paper, a new fractional order hyperchaotic system is discussed. Using the activation feedback control, the synchronization of a new fractional order hyperchaotic system is implemented based on the stability theory of fractional order systems. Numerical simulations are demonstrated the effectiveness.

GENERALIZED SYNCHRONIZATION OF FRACTIONAL ORDER CHAOTIC SYSTEMS

2011 ◽  
Vol 25 (09) ◽  
pp. 1283-1292 ◽  
Author(s):  
MING-JUN WANG ◽  
XING-YUAN WANG
Keyword(s):  
Theoretical Analysis ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Chaotic Synchronization ◽  
Generalized Synchronization ◽  
Fractional Order Systems ◽  
Different Dimensions ◽  
The Stability ◽  
Synchronization Scheme

In the paper, generalized chaotic synchronization of a class of fractional order systems is studied. Based on the stability theory of linear fractional order systems, a generalized synchronization scheme is presented, and theoretical analysis is provided to verify its feasibility. The proposed method can realize generalized synchronization not only of fractional order systems with same dimension, but also of systems with different dimensions. Besides, the function relation of generalized synchronization can be linear or nonlinear. Numerical simulations show the effectiveness of the scheme.

D-Stability Based LMI Criteria of Stability and Stabilization for Fractional Order Systems

2015 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Feasible Solution ◽  
Order System ◽  
Linear Matrix ◽  
Fractional Order Systems ◽  
Matrix Inequalities ◽  
Integer Order ◽  
Stability And Stabilization ◽  
The Stability ◽  
Conditions Of Stability

This paper considers the stability and stabilization of fractional order systems (FOS) with the fractional order α: 0 < α < 1 case. The equivalence between stability of fractional order systems and D–stability of a matrix A in specific region is proven. The criteria of stability and stabilization of fractional order system are presented. The conditions are expressed in terms of linear matrix inequalities (LMIs) which can be easily calculated with standard feasible solution problem in MATLAB LMI toolbox. When α = 1, the results reduce to the conditions of stability and stabilization of integer order systems. Numerical examples are given to verify the effectiveness of the criteria. With the approach proposed in this paper, we can analyze and design fractional order systems in the same way as what we do to the integer order system state-space models.

Dynamics and Synchronization of the Fractional-Order Sprott E System

2013 ◽  
Vol 850-851 ◽  
pp. 876-879
Author(s):  
Hong Gang Dang
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Attractor ◽  
Fractional Order Systems ◽  
The Stability

In this paper, dynamics and synchronization of the fractional-order Sprott E system are investigated. Firstly, the chaotic attractor of the system is got by means of numerical simulation. Then based on the stability theory of fractional-order systems, the synchronization of the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the controllers.

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.

scholarly journals Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems

2019 ◽  
Vol 9 (20) ◽  
pp. 4348 ◽  
Author(s):  
Bo Li ◽  
Yun Wang ◽  
Xiaobing Zhou
Keyword(s):  
Theoretical Analysis ◽  
Fractional Order ◽  
Stability Theory ◽  
Time Delays ◽  
Multiple Time ◽  
Fractional Order Systems ◽  
Delayed Systems ◽  
Combination Synchronization ◽  
Multiple Time Delays ◽  
The Stability

Multi-switching combination synchronization of three fractional-order delayed systems is investigated. This is a generalization of previous multi-switching combination synchronization of fractional-order systems by introducing time-delays. Based on the stability theory of linear fractional-order systems with multiple time-delays, we propose appropriate controllers to obtain multi-switching combination synchronization of three non-identical fractional-order delayed systems. In addition, the results of our numerical simulations show that they are in accordance with the theoretical analysis.

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