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.

scholarly journals Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nadjette Debbouche ◽  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi ◽  
Mohammed K. A. Kaabar ◽  
...  
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Hopfield Neural Network ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Neural Network System ◽  
The Stability ◽  
Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

Anti-Synchronization Control of the Complex Lu System

2014 ◽  
Vol 721 ◽  
pp. 269-272
Author(s):  
Fan Di Zhang
Keyword(s):  
Complex System ◽  
Fractional Order ◽  
Control Strategy ◽  
Stability Theorem ◽  
Projective Synchronization ◽  
Synchronization Control ◽  
Fractional Order Systems ◽  
Lü System ◽  
The Stability ◽  
Lu System

This paper propose fractional-order Lu complex system. Moreover, projective synchronization control of the fractional-order hyper-chaotic complex Lu system is studied based on feedback technique and the stability theorem of fractional-order systems, the scheme of anti-synchronization for the fractional-order hyper-chaotic complex Lu system is presented. Numerical simulations on examples are presented to show the effectiveness of the proposed control strategy.

BIFURCATIONS AND CHAOS IN FRACTIONAL-ORDER SIMPLIFIED LORENZ SYSTEM

2010 ◽  
Vol 20 (04) ◽  
pp. 1209-1219 ◽  
Author(s):  
KEHUI SUN ◽  
XIA WANG ◽  
J. C. SPROTT
Keyword(s):  
Fractional Order ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Fractional Order Systems ◽  
Wide Range ◽  
The Time Domain ◽  
Fractional Orders ◽  
Simplified Lorenz System

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.

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 Dynamic Behavior Analysis and Robust Synchronization of a Novel Fractional-Order Chaotic System with Multiwing Attractors

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chenhui Wang
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Stability Criteria ◽  
Minimum Order ◽  
3 Dimensional ◽  
Robust Synchronization ◽  
New Type ◽  
The Stability

To enrich the types of multiwing chaotic attractors in fractional-order chaotic systems (FOCSs), a new type of 3-dimensional FOCSs is designed in this study. The most important contribution of this FOCS consists in the coexistence of multiple multiwing chaotic attractors, including 2-wing, 3-wing, and 4-wing attractors. It is also indicated that the minimum order that the system can exhibit chaotic behavior is 0.84. Then, based on certain fractional stability criteria, a robust synchronization controller is derived for this kind of FOCSs with multiwing chaotic attractors and parametric uncertainties, and the stability of the synchronization error is proven strictly. Meanwhile, the theoretical analysis is tested by simulation results.

scholarly journals The Dynamics and Synchronization of a Fractional-Order System with Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoya Yang ◽  
Xiaojun Liu ◽  
Honggang Dang ◽  
Wansheng He
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Complex Variables ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
System Parameters ◽  
The Stability

A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

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.

Adaptive Synchronization of the Fractional-Order Sprott N System

2013 ◽  
Vol 850-851 ◽  
pp. 872-875
Author(s):  
Hong Gang Dang
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Phase Plane ◽  
Adaptive Synchronization ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
The Stability

In this paper, adaptive synchronization of the fractional-order Sprott N system is investigated. Firstly, the chaotic attractors on different phase plane of the system are got by means of numerical simulation. Then based on the stability theory of fractional-order systems, the adaptive synchronization of the system is realized. Numerical simulations are used to demonstrate the effectiveness for the controllers.

The Synchronization of Two Different Fractional-Order Systems

2014 ◽  
Vol 926-930 ◽  
pp. 3318-3321
Author(s):  
Xiao Jun Liu
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Two Systems ◽  
The Stability

In this paper, the synchronization for two fractional-order systems is investigated. Firstly, the chaotic attractors of two systems are got by means of numerical simulation. Then based on the stability theory of fractional-order systems, the synchronization of these two systems is realized. Numerical simulations are used to demonstrate the effectiveness for the controllers.

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.

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