scholarly journals Complex Dynamics of the Fractional-Order Rössler System and Its Tracking Synchronization Control

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Huihai Wang ◽  
Shaobo He ◽  
Kehui Sun
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Complex Dynamics ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Chaotic Signals ◽  
Rossler System ◽  
Master System ◽  
Rössler System ◽  
Tracking Synchronization

Numerical analysis of fractional-order chaotic systems is a hot topic of recent years. The fractional-order Rössler system is solved by a fast discrete iteration which is obtained from the Adomian decomposition method (ADM) and it is implemented on the DSP board. Complex dynamics of the fractional-order chaotic system are analyzed by means of Lyapunov exponent spectra, bifurcation diagrams, and phase diagrams. It shows that the system has rich dynamics with system parameters and the derivative order q. Moreover, tracking synchronization controllers are theoretically designed and numerically investigated. The system can track different signals including chaotic signals from the fractional-order master system and constant signals. It lays a foundation for the application of the fractional-order Rössler system.

HYPERCHAOS IN THE FRACTIONAL-ORDER RÖSSLER SYSTEM WITH LOWEST-ORDER

2009 ◽  
Vol 19 (01) ◽  
pp. 339-347 ◽  
Author(s):  
DONATO CAFAGNA ◽  
GIUSEPPE GRASSI
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Point Of View ◽  
Hyperchaotic System ◽  
Adomian Decomposition ◽  
Rossler System ◽  
Fractional Differential ◽  
Rössler System

This Letter analyzes the hyperchaotic dynamics of the fractional-order Rössler system from a time-domain point of view. The approach exploits the Adomian decomposition method (ADM), which generates series solution of the fractional differential equations. A remarkable finding of the Letter is that hyperchaos occurs in the fractional Rössler system with order as low as 3.12. This represents the lowest order reported in literature for any hyperchaotic system studied so far.

DIVERSE STRUCTURE SYNCHRONIZATION OF FRACTIONAL ORDER HYPER-CHAOTIC SYSTEMS

2013 ◽  
Vol 27 (11) ◽  
pp. 1350034 ◽  
Author(s):  
XING-YUAN WANG ◽  
GUO-BIN ZHAO ◽  
YU-HONG YANG
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Control Method ◽  
Rossler System ◽  
Rössler System ◽  
Predictor Corrector ◽  
Fractional Orders ◽  
Active Control Method

This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor–corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient.

scholarly journals Dynamics Analysis and Synchronous Control of Fractional-Order Entanglement Symmetrical Chaotic Systems

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 1996
Author(s):  
Tengfei Lei ◽  
Beixing Mao ◽  
Xuejiao Zhou ◽  
Haiyan Fu
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Adomian Decomposition Method ◽  
Solution Algorithm ◽  
Fractional Order Systems ◽  
Synchronous Control ◽  
Adomian Decomposition ◽  
Lyapunov Exponent Spectrum

In this paper, the Adomian decomposition method (ADM) semi-analytical solution algorithm is applied to solve a fractional-order entanglement symmetrical chaotic system. The dynamics of the system are analyzed by the Lyapunov exponent spectrum, bifurcation diagrams, poincaré diagrams, and chaos diagrams. The results show that the systems have rich dynamics. Meanwhile, sliding mode synchronizations of fractional-order chaotic systems are investigated theoretically and numerically. The results show the effectiveness of the proposed method and potential application value of fractional-order systems.

NUMERICAL EXPERIMENTS ON THE HYPERCHAOTIC CHEN SYSTEM BY THE ADOMIAN DECOMPOSITION METHOD

2008 ◽  
Vol 05 (03) ◽  
pp. 403-412 ◽  
Author(s):  
M. MOSSA AL-SAWALHA ◽  
M. S. M. NOORANI ◽  
I. HASHIM
Keyword(s):  
Numerical Experiments ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Dimensional System ◽  
Chen System ◽  
System A ◽  
Adomian Decomposition ◽  
Rossler System ◽  
Quadratic Nonlinearities ◽  
Rössler System

The aim of this paper is to investigate the accuracy of the Adomian decomposition method (ADM) for solving the hyperchaotic Chen system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the decomposition solutions and the fourth order Runge–Kutta (RK4) solutions are made. We look particularly at the accuracy of the ADM as the hyperchaotic Chen system has higher Lyapunov exponents than the hyperchaotic Rössler system. A comparison with the hyperchaotic Rössler system is given.

scholarly journals Complex Dynamics and Hard Limiter Control of a Fractional-Order Buck-Boost System

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Bo Yan ◽  
Shaojie Wang ◽  
Shaobo He
Keyword(s):  
Fractional Order ◽  
Complex Dynamics ◽  
Adomian Decomposition Method ◽  
Nonlinear Circuits ◽  
Fractional Order System ◽  
Order System ◽  
Control Analysis ◽  
Adomian Decomposition ◽  
Current Limiter ◽  
And Control

Chaos and control analysis for the fractional-order nonlinear circuits is a recent hot topic. In this study, a fractional-order model is deduced from a Buck-Boost converter, and its discrete solution is obtained based on the Adomian decomposition method (ADM). Chaotic dynamic characteristics of the fractional-order system are investigated by the bifurcation diagram, 0-1 test, spectral entropy (SE) algorithm, and NIST test. Meanwhile, the control of the fractional-order Buck-Boost model is discussed through two different ways, namely, the intensity feedback and the hard limiter control. Specifically, the hard limiter control can be realized using a current limiter in the circuit, where the current limiter device is applied to control the branch current. The results show that the proposed fractional-order system has complex dynamic behaviors and potential application values in the engineering field.

AN APPLICATION OF ADOMIAN DECOMPOSITION FOR ANALYSIS OF FRACTIONAL-ORDER CHAOTIC SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350050 ◽  
Author(s):  
R. CAPONETTO ◽  
S. FAZZINO
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Evolution Equations ◽  
Materials Science ◽  
Nonlinear Evolution ◽  
Adomian Decomposition Method ◽  
Analytic Solutions ◽  
Nonlinear Evolution Equations ◽  
Fractional Order Differential Equations ◽  
Adomian Decomposition

Fractional-order differential equations are interesting for their applications in the construction of mathematical models in finance, materials science or diffusion. Various methods for obtaining analytic solutions of nonlinear evolution equations have been proposed. In this paper, an application of the Adomian decomposition method is employed for simulation and analysis of fractional-order chaotic systems. The results reveal that the proposed method is very effective and simple and leads to accurate, approximately convergent solutions to nonlinear equations.

Local Bifurcations and Chaos in the Fractional Rössler System

2018 ◽  
Vol 28 (08) ◽  
pp. 1850098 ◽  
Author(s):  
Jan Čermák ◽  
Luděk Nechvátal
Keyword(s):  
Fractional Order ◽  
Control Method ◽  
Chaotic Behavior ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Rossler System ◽  
Master System ◽  
Rössler System ◽  
Active Control Method

The paper discusses the fractional Rössler system and the dependence of its dynamics on some entry parameters. An explicit algorithm for a priori determination of fractional Hopf bifurcations is derived and scenarios documenting a route of the system from stability to chaos are performed with respect to a varying system’s fractional order as well as to a varying system’s coefficient. Contrary to the existing results, the searched values of the fractional Hopf bifurcations follow directly from a revealed analytical dependence between these two systems’ entries. Their various critical values are established and confirmed by numerical experiments demonstrating not only the loss of stability of an equilibrium point, but also other phenomena of transition to chaotic behavior. In addition, we suggest an active control method for synchronization of two chaotic fractional-order Rössler systems. Our theoretical analysis enables to synchronize them for any value of a free parameter under which the master system displays a chaotic behavior.

Implementation of fractional optimal control problems in real-world applications

2020 ◽  
Vol 23 (6) ◽  
pp. 1783-1796
Author(s):  
Neelam Singha
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Adomian Decomposition Method ◽  
Necessary Optimality Conditions ◽  
Order Model ◽  
Fractional Optimal Control ◽  
Adomian Decomposition ◽  
Real World Applications ◽  
Fractional Optimal Control Problems ◽  
And Control

Abstract In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order optimal control problem. The key motive is to perform numerical simulations that shall facilitate us in understanding the behavior of state and control variables. Further, the graphical interpretation of solutions effectively validates the applicability of the present analysis to investigate the growth of cancer cells in the presence of medical treatment.

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