Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation

The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.     

A novel fractional-order hyperchaotic system and its synchronization

In this paper,a novel hyperchaotic system with one nonlinear term and its fractional order system are proposed.Furthermore,synchronization between two fractional-order systems with different fractional-order values is achieved.The proposed synchronization scheme is simple and theoretically rigorous.Numerical simulations are in agreement with the theoretical analysis.     

Dynamic analysis of the fractional-order Liu system and its synchronization

Based on the stability theory of fractional-order systems, the dynamic behaviors of fractional-order Liu system are studied theoretically. Coupling synchronization of two identical fractional-order chaotic systems is also studied and a simple criterion is presented. Numerical simulations show the effectiveness of our methods.     

Chaotic dynamics of the fractional-order Lü system and its synchronization

In this Letter we numerically investigate the chaotic behaviors of the fractional-order Lü system. A striking finding is that the lowest order for this system to have chaos is 0.3, which is the lowest-order chaotic system among all the found chaotic systems reported in the literature to date. Period-doubling routes to chaos in the fractional-order Lü system are also found. Master–slave synchronization of chaotic fractional-order Lü systems with linear coupling is also studied.     

Chaotic dynamics of the fractional-order Ikeda delay system and its synchronization

In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be able to appear in this system is found to be 0.1. Master--slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.     

分数阶Lorenz系统的分析及电路实现

频域传递函数近似方法不仅是常用的 分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性. 关键词： 分数阶系统 Lorenz系统 分岔分析 电路实现     

一个新的分数阶混沌系统的翼倍增及滑模同步

首先,提出了一个新的分数阶混沌系统,通过对系统第二个等式的线性项x作绝对值运算,并分析了其唯一的参数k,该参数在一定区间内取值时可将混沌吸引子由两个翼的结构变换为四翼的拓扑结构,从而实现翼倍增. 其次,分别采用Matlab和Multisim对新的分数阶系统及其翼倍增系统进行了数值模拟和电路仿真,电路仿真结果和数值模拟结果相一致. 最后,基于滑模变结构控制理论和分数阶稳定性定理,为新的分数阶系统及其翼倍增系统设计了新的分数阶积分滑模控制器实现系统的同步,仿真结果和理论分析相一致,证实了所设计滑模控制器的有效性. 关键词： 分数阶 翼倍增 滑模控制 同步     

Q-S synchronization of the fractional-order unified system

Concept of Q-S synchronization for fractional-order systems is introduced and Q-S synchronization of the fractional-order unified system is investigated in this paper. On the basis of the stability theory of the fractional-order system, two suitable control schemes are designed to achieve Q-S synchronization of the fractional-order unified systems under the given observable variables of drive system and the response system. Theoretical analysis and numerical simulations are shown to demonstrate the validity and feasibility of the proposed method.     

基于状态观测器的分数阶时滞混沌系统同步研究

研究分数阶时滞混沌系统同步问题,基于状态观测器方法和分数阶系统稳定性理论,设计分数阶时滞混沌系统同步控制器,使得分数阶时滞混沌系统达到同步,同时给出了数学证明过程.该同步控制器采用驱动系统和响应系统的输出变量进行设计,无需驱动系统和响应系统的状态变量,简化了控制器的设计,提高了控制器的实用性.利用Lyapunov稳定性理论和分数阶线性矩阵不等式,研究并给出了同步控制器参数的选择条件.以分数阶时滞Chen混沌系统为例,设计基于状态观测器的同步控制器,实现了分数阶时滞Chen混沌系统同步,并将其应用于保密通信系统中.仿真结果证明了该同步方法的有效性.     

Limit cycles in a forced Lorenz system

We consider a Lorenz system where the control parameter is sinusoidally modulated. Limit cycles appear where previously there were strange attractors. Perturbation theory is used to estimate the critical amplitude of the modulation for which limit cycles appear.     

分数阶Chua's系统错位同步无感模块化电路实现及应用

基于分数阶混沌系统稳定性理论, 设计高效的非线性控制器, 实现初始值不同的两个分数阶Chua's系统错位投影同步. 根据分数阶复频域近似方法, 提出分数阶系统的等效电路, 实现分数阶Chua's系统错位投影同步的无感模块化电路. 最后,利用改进的混沌掩盖通信原理, 将以上同步方案应用于混沌保密通信中, 在发送端使用分数阶混沌序列对有用信号加密传送, 从接收端可以无失真地恢复出有用信号. 数值仿真与电路仿真证实了提出方案的可行性. 关键词： 分数阶Chua's系统 错位投影同步 无感模块化电路 保密通信     

分数阶Liu系统与分数阶统一系统中的混沌现象及二者的异结构同步

对近几年提出的Liu混沌系统和统一混沌系统，研究了其分数阶系统的混沌动力学行为，发现低于三阶的两系统均存在混沌吸引子，且存在混沌的最低阶数仅为0.3，并计算了存在混沌时系统的最大Lyapunov指数，证明了混沌的存在性；利用Active控制技术实现了分数阶Liu系统与分数阶Lorenz系统及分数阶Lü系统的异结构同步.理论分析及数值实验都证明了该同步方案的有效性. 关键词： 分数阶Liu系统 分数阶统一系统 混沌 异结构同步     

Adaptive control and synchronization of a fractional-order chaotic system

In this paper, the chaotic dynamics of a three-dimensional fractional-order chaotic system is investigated. The lowest order for exhibiting chaos in the fractional-order system is obtained. Adaptive schemes are proposed for control and synchronization of the fractional-order chaotic system based on the stability theory of fractional-order dynamic systems. The presented schemes, which contain only a single-state variable, are simple and flexible. Numerical simulations are used to demonstrate the feasibility of the presented methods.     

分数阶统一混沌系统的自适应同步

提出了分数阶统一混沌系统的自适应同步方法, 给出了自适应同步控制器和参数自适应率. 设计的单一控制器简单且只含有一个驱动变量. 所用同步方法适用于一类分数阶混沌系统的同步, 且具有较强的抗噪声能力, 具有较高的实用价值. 数值模拟结果表明了该方法的有效性. 关键词： 分数阶 统一混沌系统 自适应同步     

Lorenz系统族采样同步研究

从对连续时间混沌系统进行数字测量的角度，提出了连续时间混沌系统的采样同步问题. 利用控制理论的结果，设计了Lorenz系统族的采样同步系统. 仅需测量一个状态分量，就可在采样时刻实现同步. 数值模拟例证了同步的性能. 关键词： 混沌同步 Lorenz系统族 采样同步     

Dynamical analysis of a new fractional-order Rabinovich system and its fractional matrix projective synchronization

This paper constructs a new physical system, i.e., the fractional-order Rabinovich system, and investigates its stability, chaotic behaviors, chaotic control and matrix projective synchronization. Firstly, two Lemmas of the new system's stability at three equilibrium points are given and proved. Next, the largest Lyapunov exponent, the corresponding bifurcation diagram and the chaotic behaviors are studied. Then, the linear and nonlinear feedback controllers are designed to realize the system's local asymptotical stability and the global asymptotical stability, respectively. It's particularly significant that, the fractional matrix projective synchronization between two Rabinovich systems is achieved and two kinds of proofs are provided for Theorem 4.1. Especially, under certain degenerative conditions, the fractional matrix projective synchronization can be reduced to the complete synchronization, anti-synchronization, projective synchronization and modified projective synchronization of the fractional-order Rabinovich systems. Finally, all the theoretical analysis is verified by numerical simulation.     

Projective synchronization of a new hyperchaotic Lorenz system

This Letter mainly concerns projective synchronization (PS) of a new hyperchaotic Lorenz system. PS with both identical and different scaling factors between two hyperchaotic Lorenz systems are realized. A general sufficient condition for PS in a certain class of chaotic (hyperchaotic) system with uncertainties is obtained by using adaptive control. Numerical simulations are performed to verify and illustrate the analytical results.     

一个分数阶新超混沌系统的同步

研究了分数阶混沌系统的同步问题,以分数阶新超混沌系统为例,设计了三种同步方法,运用拉普拉斯变换理论和分数阶线性稳定理论给以证明,数值仿真或电路仿真证实了所提同步方法的有效性和可行性. 关键词： 分数阶 混沌同步 新超混沌系统     

Designing synchronization schemes for fractional-order chaotic system via a single state fractional-order controller

In this paper the synchronization of fractional-order chaotic systems is studied and a new single state fractional-order chaotic controller for chaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can apply to an arbitrary three-dimensional fractional chaotic system whether the system is incommensurate or commensurate. This approach is universal, simple and theoretically rigorous. Numerical simulations of several fractional-order chaotic systems demonstrate the universality and the effectiveness of the proposed method.     

A new image encryption algorithm based on fractional-order hyperchaotic Lorenz system

We propose a new image encryption algorithm on a basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.