On the Three-Dimensional Fractional-Order Hénon Map with Lorenz-Like Attractors

2020 ◽  
Vol 30 (11) ◽  
pp. 2050217
Author(s):  
Amina-Aicha Khennaoui ◽  
Adel Ouannas ◽  
Zaid Odibat ◽  
Viet-Thanh Pham ◽  
Giuseppe Grassi
Keyword(s):  
Fractional Order ◽  
Chaotic Map ◽  
Three Dimensional ◽  
Synchronization Control ◽  
Fractional Order Systems ◽  
Bifurcation Diagrams ◽  
Hénon Map ◽  
Henon Map ◽  
Control Scheme ◽  
Fractional Orders

A three-dimensional (3D) Hénon map of fractional order is proposed in this paper. The dynamics of the suggested map are numerically illustrated for different fractional orders using phase plots and bifurcation diagrams. Lorenz-like attractors for the considered map are realized. Then, using the linear fractional-order systems stability criterion, a controller is proposed to globally stabilize the fractional-order Hénon map. Furthermore, synchronization control scheme has been designed to exhibit a synchronization behavior between a given 2D fractional-order chaotic map and the 3D fractional-order Hénon map. Numerical simulations are also performed to verify the main results of the study.

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.

The novel control method of three dimensional discrete hyperchaotic Hénon map

2014 ◽  
Vol 247 ◽  
pp. 487-493 ◽  
Author(s):  
Shao-Fu Wang ◽  
Xiao-Cong Li ◽  
Fei Xia ◽  
Zhan-Shan Xie
Keyword(s):  
Control Method ◽  
Three Dimensional ◽  
The Novel ◽  
Hénon Map ◽  
Henon Map

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 Bifurcation analysis of the three-dimensional Hénon map

2017 ◽  
Vol 10 (3) ◽  
pp. 625-645 ◽  
Author(s):  
Ming Zhao ◽  
◽  
Cuiping Li ◽  
Jinliang Wang ◽  
Zhaosheng Feng ◽  
...  
Keyword(s):  
Bifurcation Analysis ◽  
Three Dimensional ◽  
Hénon Map ◽  
Henon Map

scholarly journals Robust Synchronization Control of Uncertain Fractional-Order Chaotic Systems via Disturbance Observer

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Kaijuan Xue ◽  
Yongbing Huangfu
Keyword(s):  
Fractional Order ◽  
Disturbance Observer ◽  
Chaotic Systems ◽  
Control Method ◽  
Observer Design ◽  
Synchronization Error ◽  
Synchronization Control ◽  
Compact Set ◽  
Control Scheme ◽  
Unknown Disturbances

This paper studies the synchronization of two different fractional-order chaotic systems through the fractional-order control method, which can ensure that the synchronization error converges to a sufficiently small compact set. Afterwards, the disturbance observer of the synchronization control scheme based on adaptive parameters is designed to predict unknown disturbances. The Lyapunov function method is used to verify the appropriateness of the disturbance observer design and the convergence of the synchronization error, and then the feasibility of the control scheme is obtained. Finally, our simulation studies verify and clarify the proposed method.

scholarly journals Hybrid Stability Checking Method for Synchronization of Chaotic Fractional-Order Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Seng-Kin Lao ◽  
Lap-Mou Tam ◽  
Hsien-Keng Chen ◽  
Long-Jye Sheu
Keyword(s):  
Fractional Order ◽  
Design Stage ◽  
Fractional Order Systems ◽  
State Variables ◽  
Driving System ◽  
Control Scheme ◽  
The Stability ◽  
Synchronization Scheme ◽  
Hyperchaotic Systems ◽  
Stability Checking

A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits.

THREE-DIMENSIONAL HÉNON-LIKE MAPS AND WILD LORENZ-LIKE ATTRACTORS

2005 ◽  
Vol 15 (11) ◽  
pp. 3493-3508 ◽  
Author(s):  
S. V. GONCHENKO ◽  
I. I. OVSYANNIKOV ◽  
C. SIMÓ ◽  
D. TURAEV
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Maximal Lyapunov Exponent ◽  
Hénon Map ◽  
Henon Map ◽  
Quadratic Map ◽  
Different Types ◽  
Natural Way

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ1, is positive. Concerning the next Lyapunov exponent, Λ2, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ2| < ρ, where ρ is some tolerance ranging between 10-5 and 10-6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

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