A study of a modified nonlinear dynamical system with fractal-fractional derivative

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sunil Kumar ◽  
R.P. Chauhan ◽  
Shaher Momani ◽  
Samir Hadid
Keyword(s):  
Dynamical System ◽  
Fractional Derivative ◽  
Active Control ◽  
Numerical Scheme ◽  
Chaos Control ◽  
Nonlinear Dynamical System ◽  
Control Technique ◽  
Complex Behavior ◽  
Content Type ◽  
Control And Synchronization

Purpose This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control. Design/methodology/approach The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems. Findings Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved. Originality/value This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

scholarly journals Adaptive Feedback Control for Chaos Control and Synchronization for New Chaotic Dynamical System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
M. M. El-Dessoky ◽  
M. T. Yassen
Keyword(s):  
Dynamical System ◽  
Adaptive Control ◽  
Feedback Control ◽  
Chaos Control ◽  
Control Technique ◽  
Unknown Parameters ◽  
Chaotic Dynamical System ◽  
Adaptive Feedback ◽  
Adaptive Feedback Control ◽  
Control And Synchronization

This paper investigates the problem of chaos control and synchronization for new chaotic dynamical system and proposes a simple adaptive feedback control method for chaos control and synchronization under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Based on Lyapunov's stability theory, adaptive control law is derived such that the trajectory of the new system with unknown parameters is globally stabilized to the origin. In addition, an adaptive control approach is proposed to make the states of two identical systems with unknown parameters asymptotically synchronized. Numerical simulations are shown to verify the analytical results.

Boundedness of a Class of Complex Lorenz Systems

2021 ◽  
Vol 31 (07) ◽  
pp. 2150101
Author(s):  
Xu Zhang
Keyword(s):  
Dynamical System ◽  
Current Literature ◽  
Chaos Control ◽  
Type Systems ◽  
Chaotic Attractors ◽  
Strange Nonchaotic Attractors ◽  
Control And Synchronization ◽  
Ultimate Bound

The estimate of the ultimate bound for a dynamical system is an important problem, which is useful for chaos control and synchronization. In this paper, the estimated ultimate bound of a class of complex Lorenz systems is provided, which extends the parameter regions identified in the current literature on this problem. Based on these results, a kind of complex Lorenz-type systems is constructed, which might have many or infinitely many strange nonchaotic attractors, chaotic attractors, or an infinitely-many-scroll attractor.

scholarly journals Dynamics, Chaos Control, and Synchronization in a Fractional-Order Samardzija-Greller Population System with Order Lying in (0, 2)

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
A. Al-khedhairi ◽  
S. S. Askar ◽  
A. E. Matouk ◽  
A. Elsadany ◽  
M. Ghazel
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Control Method ◽  
Control Technique ◽  
Linear Feedback ◽  
Nonlinear Feedback ◽  
Population System ◽  
Wide Range ◽  
Control And Synchronization ◽  
Fractional Parameter

This paper demonstrates dynamics, chaos control, and synchronization in Samardzija-Greller population model with fractional order between zero and two. The fractional-order case is shown to exhibit rich variety of nonlinear dynamics. Lyapunov exponents are calculated to confirm the existence of wide range of chaotic dynamics in this system. Chaos control in this model is achieved via a novel linear control technique with the fractional order lying in (1, 2). Moreover, a linear feedback control method is used to control the fractional-order model to its steady states when 0<α<2. In addition, the obtained results illustrate the role of fractional parameter on controlling chaos in this model. Furthermore, nonlinear feedback synchronization scheme is also employed to illustrate that the fractional parameter has a stabilizing role on the synchronization process in this system. The analytical results are confirmed by numerical simulations.

scholarly journals Chaos Control and Synchronization in Fractional-Order Lorenz-Like System

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Sachin Bhalekar
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Stable Region ◽  
Linear Feedback ◽  
Effective Dimension ◽  
Nonlinear Feedback ◽  
Linear Feedback Controller ◽  
Control And Synchronization

The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006). The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback.

scholarly journals Coexisting attractors, chaos control and synchronization in a self-exciting homopolar dynamo system

2020 ◽  
Vol 13 (2) ◽  
pp. 167-179
Author(s):  
Xingrong Chen ◽  
Li Xiao ◽  
Sifeu Takougang Kingni ◽  
Irene Moroz ◽  
Zhouchao Wei ◽  
...  
Keyword(s):  
Chaotic System ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Chaos Control ◽  
Lyapunov Stability Theory ◽  
Control Signal ◽  
Coexisting Attractors ◽  
Content Type ◽  
Single Controller ◽  
Control And Synchronization

PurposeThe purpose of this paper is to investigate coexisting attractors, chaos control and synchronization in a self-exciting homopolar dynamo system in this paper.Design/methodology/approachTwo single controllers are designed and added to the proposed 3D autonomous chaotic system, and its stability at zero equilibrium point is guaranteed by applying an appropriate control signal based on the Lyapunov stability theory.FindingsNumerical simulations reveal that the proposed 3D dynamo system exhibits periodic oscillations, double-scroll chaotic attractors and coexisting attractors. Finally, a single controller is designed for the global asymptotic synchronization of a unidirectionally coupled identical 3D autonomous chaotic system.Originality/valueThe derived results of this paper are new and complement some earlier works. The innovation concludes two points in this paper; coexisting attractors are foundthe and an appropriate control signal based on the Lyapunov stability theory is established. The ideas of this paper can be applied to investigate some other homopolar dynamo systems.

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