scholarly journals Elegant Chaos in Fractional-Order System without Equilibria

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Donato Cafagna ◽  
Giuseppe Grassi
Keyword(s):  
Numerical Methods ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Dynamics ◽  
Fractional Order System ◽  
Order System ◽  
System Parameters ◽  
System Equations ◽  
Predictor Corrector

A new fractional-order chaotic system with no equilibria is presented. The proposed system can be considered elegant in the sense given by Sprott (2010), since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. The chaotic dynamics are analyzed using the predictor-corrector algorithm when the fractional-order of the derivative is 0.98. Finally, the presence of chaos is validated by applying different numerical methods.

scholarly journals Parallel Algorithm for Numerical Methods Applied to Fractional-order System

2020 ◽  
Vol 21 (4) ◽  
pp. 701-707
Author(s):  
Florin Rosu
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Parallel Algorithm ◽  
Fractional Order ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Predictor Corrector

A parallel algorithm is presented that approximates a solution for fractional-order systems. The algorithm isimplemented in CUDA, using the specific GPU capabilities. The numerical methods used are Adams-Bashforth-Moulton (ABM) predictor-corrector scheme and Diethelm’s numerical method. A comparison is done between these numerical methods that adapts the same algorithm for the approximation of the solution.

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 Chaos Synchronization between Fractional-Order Unified Chaotic System and Rossler Chaotic System

2012 ◽  
Vol 562-564 ◽  
pp. 2088-2091
Author(s):  
Xian Yong Wu ◽  
Yi Long Cheng ◽  
Kai Liu ◽  
Xin Liang Yu ◽  
Xian Qian Wu
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Chaotic Attractors ◽  
System Parameters ◽  
Unified Chaotic System ◽  
Simulation Results ◽  
Drive Systems ◽  
Asymptotic Synchronization

The chaotic dynamics of the unified chaotic system and the Rossler system with different fractional-order are studied in this paper. The research shows that the chaotic attractors can be found in the two systems while the orders of the systems are less than three. Asymptotic synchronization of response and drive systems is realized by active control through designing proper controller when system parameters are known. Theoretical analysis and simulation results demonstrate the effective of this method.

Chaotic Dynamics of Fractional-Order Liu System

2011 ◽  
Vol 55-57 ◽  
pp. 1327-1331 ◽  
Author(s):  
Xin Gao
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic Dynamics ◽  
Fractional Order System ◽  
Order System ◽  
Liu System

In this paper, we numerically investigate the chaotic behaviors of a new fractional-order system. We find that chaotic behaviors exist in the fractional-order system with order less than 3. The lowest order we find to have chaos is 2.4 in such system. In addition, we numerically simulate the continuances of the chaotic behaviors in the fractional-order system with orders from 2.7 to 3. Our investigations are validated through numerical simulations.

The Synchronization of a Fractional-Order Chaotic System

2013 ◽  
Vol 655-657 ◽  
pp. 1488-1491
Author(s):  
Fan Di Zhang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Fractional Order System ◽  
Order System ◽  
Integer Order

In this paper, the synchronization of fractional-orderchaotic system is studied. Based on the fractional stability theory, suitable controller is designed to realize the synchronization between fractional-order system and a integer-order system. Numerical simulations show that the effectiveness and feasibility of the controllers .

scholarly journals Hidden Coexisting Attractors in a Fractional-Order System without Equilibrium: Analysis, Circuit Implementation, and Finite-Time Synchronization

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Guangchao Zheng ◽  
Ling Liu ◽  
Chongxin Liu
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Finite Time ◽  
Secure Communication ◽  
Time Synchronization ◽  
Basins Of Attraction ◽  
Fractional Order System ◽  
Equilibrium Analysis ◽  
Order System ◽  
Coexisting Attractors

In this paper, a novel three-dimensional fractional-order chaotic system without equilibrium, which can present symmetric hidden coexisting chaotic attractors, is proposed. Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction. In particular, the system can generate diverse coexisting attractors varying with different orders, which presents ample and complex dynamic characteristics. And there is great potential for secure communication. Then electronic circuit of the fractional-order system is designed to help verify its effectiveness. What is more, taking the disturbances into account, a finite-time synchronization of the fractional-order chaotic system without equilibrium is achieved and the improved controller is proven strictly by applying finite-time stable theorem. Eventually, simulation results verify the validity and rapidness of the proposed method. Therefore, the fractional-order chaotic system with hidden attractors can present better performance for practical applications, such as secure communication and image encryption, which deserve further investigation.

One-to-four-wing hyperchaotic fractional-order system and its circuit realization

Circuit World ◽  
2020 ◽  
Vol 46 (2) ◽  
pp. 107-115
Author(s):  
Xiang Li ◽  
Zhijun Li ◽  
Zihao Wen
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Complexity Analysis ◽  
Initial Conditions ◽  
Fractional Order System ◽  
Order System ◽  
Hyperchaotic System ◽  
Circuit Realization ◽  
Content Type

Purpose This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems. Design/methodology/approach A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components. Findings The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system. Originality/value The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

scholarly journals Coexisting Three-Scroll and Four-Scroll Chaotic Attractors in a Fractional-Order System by a Three-Scroll Integer-Order Memristive Chaotic System and Chaos Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Xikui Hu ◽  
Ping Zhou
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Control ◽  
Initial Conditions ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Integer Order ◽  
Memristive System ◽  
Fractional Orders

Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-orders q. Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.

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