scholarly journals Special Characteristics and Synchronizations of Multi Hybrid-Order Chaotic Systems

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 664
Author(s):  
Jiaxun Liu ◽  
Zuoxun Wang ◽  
Fangfang Zhang ◽  
Yankai Yin ◽  
Fengying Ma
Keyword(s):  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Combination Synchronization ◽  
Simulation Results ◽  
Total Dimension ◽  
Hybrid Forms

Based on advantages of integer and fractional chaotic systems, hybrid chaotic systems and their definitions and some fundamental concepts are proposed, such as hybrid degree (HD), the lowest order (LO) and the total dimension order (TDO). The preliminary properties of hybrid Lorenz systems and hybrid forms of some classic chaotic systems are studied. Then, the relations between HD, LO and TDO with different parameters is investigated in chaotic systems. To be specific, HD is associated with fractional order. It is a directional method to search LO and TDO in chaotic systems. Finally, based on the incommensurate fractional stability theory, we accomplish combination synchronization for three different hybrid order chaotic systems. The simulation results verify the effectiveness of the synchronization controller.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Synchronization of Fractional-Order Chaotic Systems with Uncertain Parameters

2010 ◽  
Vol 171-172 ◽  
pp. 723-727
Author(s):  
Hong Zhang ◽  
Qiu Mei Pu
Keyword(s):  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
System Stability ◽  
Order System ◽  
Uncertain Parameters ◽  
Mode Theory ◽  
Simulation Results ◽  
The Stability

For the synchronization of fractional-order chaotic systems with uncertain parameters, a controller based on sliding mode theory is presented. Based on the stability theory of fractional-order system, stability of the proposed method is analyzed. The theory is successfully applied to synchronize fractional Newton-Leipnik chaotic systems with uncertain parameters. The simulation results show the effectiveness of the proposed controller.

scholarly journals Complex Modified Projective Synchronization of Fractional-Order Complex-Variable Chaotic System with Unknown Complex Parameters

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 407
Author(s):  
Zhang ◽  
Feng ◽  
Yang
Keyword(s):  
Fractional Order ◽  
Complex Variable ◽  
Chaotic Systems ◽  
Projective Synchronization ◽  
Order Complex ◽  
Modified Projective Synchronization ◽  
Simulation Results ◽  
Synchronization Scheme ◽  
Complex Valued ◽  
Variable Inequality

This paper investigates the problem of complex modified projective synchronization (CMPS) of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. By a complex-variable inequality and a stability theory for fractional-order nonlinear systems, a new scheme is presented for constructing CMPS of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued systems but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme.

GENERALIZED SYNCHRONIZATION OF FRACTIONAL ORDER CHAOTIC SYSTEMS

2011 ◽  
Vol 25 (09) ◽  
pp. 1283-1292 ◽  
Author(s):  
MING-JUN WANG ◽  
XING-YUAN WANG
Keyword(s):  
Theoretical Analysis ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Chaotic Synchronization ◽  
Generalized Synchronization ◽  
Fractional Order Systems ◽  
Different Dimensions ◽  
The Stability ◽  
Synchronization Scheme

In the paper, generalized chaotic synchronization of a class of fractional order systems is studied. Based on the stability theory of linear fractional order systems, a generalized synchronization scheme is presented, and theoretical analysis is provided to verify its feasibility. The proposed method can realize generalized synchronization not only of fractional order systems with same dimension, but also of systems with different dimensions. Besides, the function relation of generalized synchronization can be linear or nonlinear. Numerical simulations show the effectiveness of the scheme.

scholarly journals Combination synchronization of fractional order n-chaotic systems using active backstepping design

2019 ◽  
Vol 8 (1) ◽  
pp. 597-608 ◽  
Author(s):  
Vijay K. Yadav ◽  
S. Das
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Design Method ◽  
Backstepping Design ◽  
Fractional Order Systems ◽  
Combination Synchronization

Abstract In this article, a scheme using active backstepping design method is proposed to achieve combination synchronization of n number of fractional order chaotic systems. In the proposed method the controllers are designed with the help of a new lemma and Lyapunov function in a systematic way. Synchronization among three/four fractional order systems have been shown as examples of synchronization of n-chaotic systems. Numerical simulation and graphical results clearly exhibit that the method of this new procedure is easy to implement and reliable for synchronization of fractional order chaotic systems.

CHAOTIC DYNAMICS AND SYNCHRONIZATION OF FRACTIONAL-ORDER CHUA'S CIRCUITS WITH A PIECEWISE-LINEAR NONLINEARITY

2005 ◽  
Vol 19 (20) ◽  
pp. 3249-3259 ◽  
Author(s):  
JUN GUO LU
Keyword(s):  
Lyapunov Exponent ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Synchronization Method ◽  
Simulation Results

In this paper, we numerically investigate the chaotic behaviors of the fractional-order Chua's circuit with a piecewise-linear nonlinearity. We find that chaos exists in the fractional-order Chua's circuit with order less than 3. The lowest order we find to have chaos is 2.7 in the homogeneous fractional-order Chua's circuit and 2.8 in the unhomogeneous fractional-order Chua's circuit. Our results are validated by the existence of a positive Lyapunov exponent. A chaos synchronization method is also presented for synchronizing the homogeneous fractional-order chaotic Chua's systems. The approach, based on stability theory of fractional-order linear systems, is simple and theoretically rigorous. It does not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method.

Dual Combination Synchronization of Six Chaotic Systems

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Junwei Sun ◽  
Suxia Jiang ◽  
Guangzhao Cui ◽  
Yanfeng Wang
Keyword(s):  
Adaptive Control ◽  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Combination Synchronization

Based on combination synchronization of three chaotic systems and combination–combination synchronization of four chaotic systems, a novel scheme of dual combination synchronization is investigated for six chaotic systems in the paper. Using combined adaptive control and Lyapunov stability theory of chaotic systems, some sufficient conditions are attained to realize dual combination synchronization of six chaotic systems. The corresponding theoretical proofs and numerical simulations are presented to demonstrate the effectiveness and correctness of the dual combination synchronization. Due to the complexity of dual combination synchronization, it will be more secure and interesting to transmit and receive signals in application of communication.

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