Bivariate Module-Phase Synchronization of a Fractional-Order Lorenz System in Different Dimensions

2013 ◽  
Vol 8 (3) ◽  
Author(s):  
Xing-Yuan Wang ◽  
Hao Zhang
Keyword(s):  
Numerical Simulation ◽  
Phase Space ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Lorenz System ◽  
Phase Synchronization ◽  
High Dimensional ◽  
Different Dimensions ◽  
Simulation Results ◽  
First Time

Based on the classic Lorenz system, this paper studies the problem of bivariate module-phase synchronizations in a fractional-order Lorenz system, bivariate module-phase synchronizations in a fractional-order spatiotemporal coupled Lorenz system, and malposed module-phase synchronization in a fractional-order spatiotemporal coupled Lorenz system. It is the first time, to our knowledge, that module-phase synchronization in fractional-order high-dimensional systems is applied. According to the fractional calculus techniques and spatiotemporal theory, we design controllers and achieve synchronizations both in module space and phase space at the same time. In the simulation, we discuss the bivariate module-phase synchronization and malposed module-phase synchronization. The numerical simulation results demonstrate the validity of controllers.

CHAOTIC SYNCHRONIZATION OF FRACTIONAL-ORDER SPATIOTEMPORAL COUPLED LORENZ SYSTEM

2012 ◽  
Vol 23 (10) ◽  
pp. 1250067 ◽  
Author(s):  
XING-YUAN WANG ◽  
HAO ZHANG
Keyword(s):  
Fractional Order ◽  
Chaos Theory ◽  
Lorenz System ◽  
Systems Design ◽  
Lyapunov Stability Theory ◽  
Chaotic Synchronization ◽  
Spatiotemporal Chaos ◽  
Nonlinear Controller ◽  
Simulation Results ◽  
First Time

By utilizing the fractional calculus techniques and spatiotemporal chaos theory, this paper brings Lorenz system to fractional-order spatiotemporal coupled differential equation for the first time, and proposes the fractional-order spatiotemporal coupled Lorenz system. Based on that, we study the problem of chaotic synchronization of fractional-order spatiotemporal coupled Lorenz systems, design the linear controller and nonlinear controller by utilizing the Lyapunov stability theory and prove the correctness in theory. The numerical simulation results demonstrate the validity of controllers in high-dimension fractional-order spatiotemporal coupled Lorenz system.

GENERALIZED SYNCHRONIZATION OF FRACTIONAL ORDER HYPERCHAOTIC LORENZ SYSTEM

2009 ◽  
Vol 23 (17) ◽  
pp. 2167-2178 ◽  
Author(s):  
TIANSHU WANG ◽  
XINGYUAN WANG
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Lorenz System ◽  
Order System ◽  
Generalized Synchronization ◽  
Response System ◽  
Hyperchaotic Lorenz System ◽  
Simulation Results ◽  
The Stability ◽  
Predictor Corrector

In this paper, a type of new fractional order hyperchaotic Lorenz system is proposed. Based on the fractional calculus predictor-corrector algorithm, the fractional order hyperchaotic Lorenz system is investigated numerically, and the simulation results show that the lowest orders for hyperchaos in hyperchaotic Lorenz system is 3.884. According to the stability theory of fractional order system, an improved state-observer is designed, and the response system of generalized synchronization is obtained analytically, whose feasibility is proved theoretically. The synchronization method is adopted to realize the generalized synchronization of 3.884-order hyperchaotic Lorenz system, and the numerical simulation results verify the effectiveness.

ACTIVE TRACKING CONTROL OF THE HYPERCHAOTIC LORENZ SYSTEM

2008 ◽  
Vol 22 (19) ◽  
pp. 1859-1865 ◽  
Author(s):  
XINGYUAN WANG ◽  
DAHAI NIU ◽  
MINGJUN WANG
Keyword(s):  
Numerical Simulation ◽  
Tracking Control ◽  
Chaotic Systems ◽  
Lorenz System ◽  
The Self ◽  
Hyperchaotic System ◽  
Reference Signals ◽  
Tracking Controller ◽  
Hyperchaotic Lorenz System ◽  
Simulation Results

A nonlinear active tracking controller for the four-dimensional hyperchaotic Lorenz system is designed in the paper. The controller enables this hyperchaotic system to track all kinds of reference signals, such as the sinusoidal signal. The self-synchronization of the hyperchaotic Lorenz system and the different-structure synchronization with other chaotic systems can also be realized. Numerical simulation results show the effectiveness of the controller.

Fractional Order Modeling and Analysis of Buck-Boost Converter

2015 ◽  
Vol 789-790 ◽  
pp. 842-848
Author(s):  
Li Feng Yi ◽  
Kai Ru Zhang ◽  
Jun Liu
Keyword(s):  
Mathematical Model ◽  
Theoretical Analysis ◽  
Fractional Calculus ◽  
Simulation Model ◽  
Fractional Order ◽  
Theoretical Foundation ◽  
Boost Converter ◽  
Modeling And Analysis ◽  
Simulation Results ◽  
Conduction Mode

Considered the theoretical foundation of fractional order, the fractional mathematical model of the Buck-Boost converter in continuous conduction mode operation is built and analyzed in theory. Based on the improved Oustaloup fractional calculus for filter algorithm, the simulation model is framed by using the Matlab/Simulink software. And the simulation results are compared with that of integer order. It proves the correctness of the fractional order mathematical model and the theoretical analysis.

A New Five Dimensional Hyperchaotic System and its Fractional Order Form

2013 ◽  
Vol 464 ◽  
pp. 375-380 ◽  
Author(s):  
Ling Liu ◽  
Chong Xin Liu ◽  
Yi Fan Liao
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Simulation Analysis ◽  
Three Dimensional ◽  
Hyperchaotic System ◽  
Order Form ◽  
Simulation Results ◽  
Predictor Corrector ◽  
New Hyperchaotic System

In this paper, a new five-dimensional hyperchaotic system by introducing two additional states feedback into a three-dimensional smooth chaotic system. With three nonlinearities, this system has more than one positive Lyapunov exponents. Based on the fractional derivative theory, the fractional-order form of this new hyperchaotic system has been investigated. Through predictor-corrector algorithm, the system is proved by numerical simulation analysis. Simulation results are provided to illustrate the performance of the fractional-order hyperchaotic attractors well.

Estimation of the Order and Parameters of a Fractional Order Model From a Noisy Step Response Data1

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Mahsan Tavakoli-Kakhki ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Closed Loop ◽  
Open Loop ◽  
Measurement Noise ◽  
Step Response ◽  
Order Model ◽  
Response Data ◽  
Fractional Order Model ◽  
Simulation Results

This paper deals with integral based methods to estimate the order and parameters of simple fractional order models from the extracted noisy step response data of a process. This data can be obtained from both open-loop and closed-loop tests. Numerical simulation results are presented to verify the robustness of these proposed methods in the presence of the measurement noise.

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.

CHAOTIC SYNCHRONIZATION OF FRACTIONAL-ORDER MODIFIED COUPLED DYNAMOS SYSTEM

2009 ◽  
Vol 23 (31) ◽  
pp. 5769-5777 ◽  
Author(s):  
XINGYUAN WANG ◽  
YIJIE HE
Keyword(s):  
Fractional Order ◽  
Lorenz System ◽  
Chaotic Synchronization ◽  
The Self ◽  
Simulation Results ◽  
System Designs

This paper studies the problem of chaotic synchronization of the fractional-order modified coupled dynamos system, designs the activate controllers, and then proves that the self-synchronization of the fractional-order modified coupled dynamos system and the fractional-order modified coupled dynamos system's different structure synchronization with the fractional-order Lorenz system can both arrive theoretically. The simulation results demonstrate the validity of the activate controller.

scholarly journals General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Xiaojun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Integral Transforms ◽  
Kernel Functions ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Laplace Integral ◽  
Fractional Calculus Operators ◽  
Anomalous Relaxation ◽  
First Time

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

A Fractional Order Circuit Model of the PT

2013 ◽  
Vol 860-863 ◽  
pp. 2304-2308
Author(s):  
Feng Gao ◽  
Gui Shu Liang ◽  
Xin Liu
Keyword(s):  
Theoretical Analysis ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Magnetization Curve ◽  
Circuit Model ◽  
Core Model ◽  
Iron Core ◽  
Saturation Phenomenon ◽  
Nonlinear Part ◽  
Simulation Results

This paper presents a new digital potential transformer (PT) model. Due to the effect of hysteresis and saturation phenomenon, the magnetization curve of iron core has a strong nonlinear. From the perspective of hysteresis effect of iron core, this article presents a PT core model using the theory of fractional calculus, then propose a new PT circuit model. The model simulates the nonlinear part of iron core using the fractional order expressions and can reflect the feature of saturation. Finally it shows the correctness and validity of the model through the theoretical analysis and simulation results.

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