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 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.

Characteristic Analysis and DSP Realization of Fractional-Order Simplified Lorenz System Based on Adomian Decomposition Method

2015 ◽  
Vol 25 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Huihai Wang ◽  
Kehui Sun ◽  
Shaobo He
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Lorenz System ◽  
Adomian Decomposition Method ◽  
Qr Factorization ◽  
Iteration Step ◽  
Phase Portraits ◽  
Characteristic Analysis ◽  
Adomian Decomposition ◽  
Simplified Lorenz System

By adopting Adomian decomposition method, the fractional-order simplified Lorenz system is solved and implemented on a digital signal processor (DSP). The Lyapunov exponent (LE) spectra of the system is calculated based on QR-factorization, and it accords well with the corresponding bifurcation diagrams. We analyze the influence of the parameter and the fractional derivative order on the system characteristics by color maximum LE (LEmax) and chaos diagrams. It is found that the smaller the order is, the larger the LEmax is. The iteration step size also affects the lowest order at which the chaos exists. Further, we implement the fractional-order simplified Lorenz system on a DSP platform. The phase portraits generated on DSP are consistent with the results that were obtained by computer simulations. It lays a good foundation for applications of the fractional-order chaotic systems.

scholarly journals Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 1 ◽  
Author(s):  
Han-Ping Hu ◽  
Jia-Kun Wang ◽  
Fei-Long Xie
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Hopfield Neural Network ◽  
Projective Synchronization ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Intermittent Chaos

In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.

Hybrid Projective Synchronization for the Fractional-Order Chen-Lee System and its Circuit Realization

2013 ◽  
Vol 300-301 ◽  
pp. 1573-1578
Author(s):  
Seng Kin Lao ◽  
Hsien Keng Chen ◽  
Lap Mou Tam ◽  
Long Jye Sheu
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Body Motion ◽  
Projective Synchronization ◽  
Fractional Order Systems ◽  
Control Of Chaos ◽  
Circuit Realization ◽  
Hybrid Projective Synchronization ◽  
The Time Domain ◽  
Theoretical Analyses

The growing interest shows the importance of the control of chaos in fractional-order systems in recent years. This paper investigates in the hybrid projective synchronization of two chaotic systems with fractional-order, which were derived from Euler equations of rigid body motion. Theoretical analyses of the proposed methods are validated by numerical simulation in the time domain. Moreover, the synchronization system is realized using electronic circuits with fractance in the frequency domain.

Dynamics analysis of fractional-order Hopfield neural networks

2020 ◽  
pp. 2050083
Author(s):  
Iqbal M. Batiha ◽  
Ramzi B. Albadarneh ◽  
Shaher Momani ◽  
Iqbal H. Jebril
Keyword(s):  
Neural Network ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Phase Portraits ◽  
Dynamics Analysis ◽  
Fractional Order Systems ◽  
Runge Kutta Method ◽  
Intermittent Chaos ◽  
Predictor Corrector

This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor–Corrector Adams–Bashforth–Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge–Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin–Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag–Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents’ diagrams.

scholarly journals Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Junhai Ma ◽  
Hongliang Tu
Keyword(s):  
Nash Equilibrium ◽  
Bounded Rationality ◽  
Electricity Market ◽  
Complex Dynamics ◽  
Fractal Dimensions ◽  
Practical Significance ◽  
Stable Region ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Game Model

According to a triopoly game model in the electricity market with bounded rational players, a new Cournot duopoly game model with delayed bounded rationality is established. The model is closer to the reality of the electricity market and worth spreading in oligopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Its complex dynamics is demonstrated by means of the largest Lyapunov exponent, bifurcation diagrams, phase portraits, and fractal dimensions. Since the output adjustment speed parameters are varied, the stability of Nash equilibrium gives rise to complex dynamics such as cycles of higher order and chaos. Furthermore, by using the straight-line stabilization method, the chaos can be eliminated. This paper has an important theoretical and practical significance to the electricity market under the background of developing new energy.

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 Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Argha Mondal ◽  
Sanjeev Kumar Sharma ◽  
Ranjit Kumar Upadhyay ◽  
Arnab Mondal
Keyword(s):  
Fractional Order ◽  
Complex Dynamics ◽  
Firing Frequency ◽  
Order System ◽  
Neuron Models ◽  
Order Model ◽  
Constant Input ◽  
Wide Range ◽  
Excitable Systems ◽  
Firing Properties

Abstract Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < α ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

Quantitative Analysis of Singularities for Fractional Order Systems

2015 ◽  
Author(s):  
Yan Li ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Intrinsic Property ◽  
Time Domain Analysis ◽  
Order System ◽  
Fractional Order Systems ◽  
Stability And Control ◽  
Practical Strategy ◽  
The Time Domain ◽  
And Control

The singularity is an intrinsic property for various fractional order systems. This paper focuses on the time domain analysis of typical “non-proper” fractional order transfer functions, which plays the crucial role in the implementation, stability and control of fractional order systems. To this end, the fractional order system is converted into a weak singularity integro-differential equation, where the non-proper property can be clearly presented. A practical strategy is shown to find out the poles in the first Riemann plane, which is especially applicable to small commensurate order problems. The distributed order and order sensitivity problems are discussed as well. A number of examples are illustrated by using some reliable fractional order numerical methods.

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