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 A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 564 ◽  
Author(s):  
Jesus Munoz-Pacheco ◽  
Ernesto Zambrano-Serrano ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Equilibrium System ◽  
Hidden Attractors ◽  
Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

scholarly journals Dynamics Analysis and Control of a Five-Term Fractional-Order System

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Li-xin Yang ◽  
Xiao-jun Liu
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Phase Portraits ◽  
Order System ◽  
Bifurcation Diagrams ◽  
Compound Structure ◽  
Dynamical Properties ◽  
The Stability ◽  
And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

CHAOS IN DIFFUSIONLESS LORENZ SYSTEM WITH A FRACTIONAL ORDER AND ITS CONTROL

2012 ◽  
Vol 22 (04) ◽  
pp. 1250088 ◽  
Author(s):  
YONG XU ◽  
RENCAI GU ◽  
HUIQING ZHANG ◽  
DONGXI LI
Keyword(s):  
Feedback Control ◽  
Control Problem ◽  
Fractional Order ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Control Technique ◽  
Order System ◽  
System Parameters ◽  
Simulation Results ◽  
The Stability

This paper aims to investigate the phenomenon of Diffusionless Lorenz system with fractional-order. We discuss the stability of equilibrium points of the fractional-order system theoretically, and analyze the chaotic behaviors and typical bifurcations numerically. We find rich dynamics in fractional-order Diffusionless Lorenz system with appropriate fractional order and system parameters. Besides, the control problem of fractional-order Diffusionless Lorenz system is examined using feedback control technique, and simulation results show the effectiveness of the method.

scholarly journals Bifurcations of a New Fractional-Order System with a One-Scroll Chaotic Attractor

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Chaotic Attractor ◽  
System Parameter ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Order Case ◽  
The Stability ◽  
The One ◽  
Derivative Order

In this paper, a new fractional-order system which has a chaotic attractor of the one-scroll structure is presented. Firstly, the stability of the equilibrium points of the system is investigated. And based on the stability analysis, the generation conditions of the one-scroll structure for the attractor are determined. In a commensurate-order case, bifurcations with the variation of a system parameter are investigated as derivative orders decrease from 0.99. In an incommensurate-order case, bifurcations with the variation of a derivative order are analyzed as other orders decrease from 1.

Synchronization of a Fractional-Order System with Unknown Parameters

2013 ◽  
Vol 850-851 ◽  
pp. 796-799
Author(s):  
Xiao Ya Yang
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Attractor ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Unknown Parameters ◽  
The Stability

In this paper, synchronization of a fractional-order system with unknown parameters is studied. The chaotic attractor of the system is got by means of numerical simulation. Then based on the stability theory of fractional-order systems, suitable synchronization controllers and parameter identification rules for the unknown parameters are designed. Numerical simulations are used to demonstrate the effectiveness of 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.

Stability and resonance analysis of a general non-commensurate elementary fractional-order system

2020 ◽  
Vol 23 (1) ◽  
pp. 183-210 ◽  
Author(s):  
Shuo Zhang ◽  
Lu Liu ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Order Parameters ◽  
Fractional Order System ◽  
Order System ◽  
Stability Conditions ◽  
Order Restriction ◽  
Resonance Analysis ◽  
General Kind ◽  
The Stability

AbstractThe elementary fractional-order models are the extension of first and second order models which have been widely used in various engineering fields. Some important properties of commensurate or a few particular kinds of non-commensurate elementary fractional-order transfer functions have already been discussed in the existing studies. However, most of them are only available for one particular kind elementary fractional-order system. In this paper, the stability and resonance analysis of a general kind non-commensurate elementary fractional-order system is presented. The commensurate-order restriction is fully released. Firstly, based on Nyquist’s Theorem, the stability conditions are explored in details under different conditions, namely different combinations of pseudo-damping (ζ) factor values and order parameters. Then, resonance conditions are established in terms of frequency behaviors. At last, an example is given to show the stable and resonant regions of the studied systems.

LAG SYNCHRONIZATION OF THE FRACTIONAL-ORDER SYSTEM VIA NONLINEAR OBSERVER

2011 ◽  
Vol 25 (29) ◽  
pp. 3951-3964 ◽  
Author(s):  
HAO ZHU ◽  
ZHONGSHI HE ◽  
SHANGBO ZHOU
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Stability Theorem ◽  
Nonlinear Observer ◽  
Order System ◽  
Fractional Order Systems ◽  
Lag Synchronization ◽  
Systematic Method ◽  
Fractional System ◽  
The Stability

In this paper, based on the idea of nonlinear observer, lag synchronization of chaotic fractional system with commensurate and incommensurate order is studied by the stability theorem of linear fractional-order systems. The theoretical analysis of fractional-order systems in this paper is a systematic method. This technique is applied to achieve the lag synchronization of fractional-order Rössler's system, verified by numerical simulation.

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 Some Fundamental Properties on the Sampling Free Nabla Laplace Transform

2019 ◽  
Author(s):  
Yiheng Wei ◽  
Yuquan Chen ◽  
Yong Wang ◽  
YangQuan Chen
Keyword(s):  
System Theory ◽  
Laplace Transform ◽  
Fractional Order ◽  
Fractional Order System ◽  
The Other ◽  
Order System ◽  
Fractional Order Systems ◽  
Fundamental Properties ◽  
Other Hand

Abstract Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the Ntransform more effective and efficient. On the other hand, they enrich the discrete fractional order system theory.

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