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.

Hyperchaos behaviors and chaos synchronization of two unidirectional coupled simplified Lorenz systems

To design a hyperchaotic generator and apply chaos into secure communication, a linear unidirectional coupling control is applied to two identical simplified Lorenz systems. The dynamical evolution process of the coupled system is investigated with variations of the system parameter and coupling coefficients. Particularly, the influence of coupling strength on dynamics of the coupled system is analyzed in detail. The range of the coupling strength in which the coupled system can generate hyperchaos or realize synchronization is determined, including phase portraits, Lyapunov exponents, and Poincare section. And the critical value of the system parameter between hyperchaos and synchronization is also found with fixed coupled strength. In addition, abundant dynamical behaviors such as four-wing hyperchaotic, two-wing chaotic, single-wing coexisting attractors and periodic orbits are observed and chaos synchronization error curves are also drawn by varying system parameter c. Numerical simulations are implemented to verify the results of these investigations.

Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with Caputo Fractional Derivative

Although some numerical methods of the fractional-order chaotic systems have been announced,high-precision numerical methods have always been the direction that researchers strive to pursue.Based on this problem,this paper introduces a high-precision numerical approach.Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method.We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies.We investigate the influence of α_(1),α_(2),α_(3) on the numerical solution of fractional-order Lorenz chaotic systems.The simulation results of integer order are in good agreement with those of othermethods.The simulation results of numerical experiments demonstrate the effectiveness of the present method.

Horseshoe and entropy in a fractional-order unified system

This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincare section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

简化Lorenz混沌系统的投影同步控制

针对最新研究的简化Lorenz混沌系统模型,分别采用比例投影同步和广义函数投影同步控制策略,研究了两个不同初值的简化Lorenz混沌系统的同步问题。基于罗斯-霍尔维兹准则和Lyapunov稳定性原理,推导了实现两混沌系统投影同步的同步定理,确定了实现投影同步的控制参数的取值范围;分析了控制参数对同步系统性能的影响规律。基于Matlab/Simulink的动态仿真研究表明,该投影同步控制方法是有效的。

简化Lorenz系统多翅膀混沌吸引子的设计与电路实现

基于简化Lorenz系统,通过设计非线性函数,得到单方向多翅膀混沌吸引子.在此基础上,通过引入切换控制函数,得到了网格多翅膀混沌吸引子.通过相图、耗散性、平衡点、分岔图和Poincaré截面方法,分析了系统的动力学特性.结果表明,该多翅膀系统具有丰富的动力学行为.设计并实现了网格多翅膀系统的模拟电路,通过示波器观测到网格多翅膀混沌吸引子,验证了该系统的物理可实现性,电路实验结果与数值仿真结果相一致.

简化Lorenz混沌仿真和控制实验平台开发

采用Python语言建立简化Lorenz混沌仿真和控制实验平台,能够进行简化Lorenz混沌的仿真和镇定控制。采用Tkinter建立软件的GUI界面,并采用Matplotlib进行图形的绘制,可以修改混沌系统的参数和初始状态以及控制器的参数。采用主动控制器进行简化Lorenz混沌的镇定控制,状态变量渐进收敛到零。该实验平台可以进行简化Lorenz混沌的仿真和镇定控制,能够提高学生创新实验技能和工程实践能力。

基于忆阻器反馈的Lorenz超混沌系统及其电路实现

采用二次型磁控忆阻器作为系统的正反馈项,设计了一个超混沌电路,建立了该系统的无量纲数学模型,探讨了忆阻器混沌系统与原混沌系统的不同之处.分析了系统的平衡点集和稳定性,发现系统继承了原系统的对称性,确定了系统参数所对应的稳定和不稳定区域分布,得到了系统的稳定和不稳定平衡点集.采用分岔图、Lyapunov指数谱、Poincaré截面等分析方法,研究了系统的动力学行为随系统参数和忆阻器初始状态而变化的情况,观察到了混沌系统随忆阻器初值不同引起的吸引子共存和状态转移现象,结合相图与谱熵算法分析了状态转移现象.设计并实现了该系统的模拟电子电路,实验结果表明,电路实验结果与数值仿真结果相吻合,为忆阻器混沌电路的实际应用奠定了基础.

分数阶简化Lorenz系统的FPGA实现

目前针对分数阶混沌系统的研究大多数都是基于DSP(Digital Signal Processor)平台,由于分数阶混沌系统的复杂度较大,用DSP实现存在序列生成速度较慢的问题,只能应用于对速度要求不高的系统.针对该问题,研究了基于分数阶微积分Grunwald-Letnikov(GL)定义的分数阶简化Lorenz系统的FPGA(field programmable gate array)实现.通过最大Lyapunov指数和0~1测试验证了基于GL定义的分数阶简化Lorenz系统是混沌的.详细分析了基于GL定义的分数阶简化Lorenz系统的FPGA实现结构,并使用定点数格式实现了该系统.通过示波器观察FPGA输出结果与MATLAB仿真结果一致,从而进一步揭示了分数阶混沌系统的可实现性.

基于简化Lorenz分数阶混沌系统的数字图像加密新算法

提出一种基于简化Lorenz分数阶混沌系统的数字图像加密新算法.首先通过相位空间图、SE复杂度分析和C_0复杂度分析等表明简化Lorenz分数阶混沌系统具有复杂的空间结构和良好的伪随机性能等优良特性.然后基于简化Lorenz分数阶混沌系统设计一种灰度图像加密新算法,该算法采用置乱-扩散机制,在置乱阶段利用混沌系统产生坐标索引向量对明文图像进行依行依列地置乱,而在扩散阶段采用不同初值的Lorenz简化分数阶混沌系统产生密钥向量和初始向量对置乱后的像素灰度值进行扩散.有关本文提出算法的安全性能分析包括密钥空间分析、密钥敏感性分析和统计分析等均显示本文提出算法具有密钥空间大、密钥敏感性强,能够很好地抵抗选择密文攻击、选择明文攻击和已知明文攻击等各种已知攻击,从而该算法具有较高的安全性和良好的应用前景.

Implementation of Simplified Fractional-Order PID Controller Based on Modified Oustaloup's Recursive Filter

The aim of this paper is to simplify the design of fractional-order PID controllers.Because the analytical expressions and operations of fractional-order systems are complex,numerical approximation tool is needed for the simulation analysis and engineering practice of fractional-order control systems.The key to numerical approximation tool is the exact approximation of the fractional calculus operator.A commonly used method is to approximate the fractional calculus operator with an improved Oustaloup^recursive filter.Based on the modified Oustaloup5s recursive filter,a mathematical simplification method is proposed in this paper,and a simplified fractional-order PID controller(SFOC)is designed.The controller parameters are tuned by using genetic algorithm(GA).Effectiveness of the proposed control scheme is verified by simulation.The performance of SFOC has been compared with that of the integer-order PID controller and conventional fractional-order PID controller(CFOC).It is observed that SFOC requires smaller effort as compared with its integer and conventional fractional counterpart to achieve the same system performance.

分数阶Lorenz系统和SIR传染病模型的数值模拟

作者拟对Lorenz系统以及SIR模型阶数从整数阶到分数阶变化进行分析,探寻阶数变化对系统所带来的影响。先是针对基于Grünwald-Letnikov(GL)定义的分数阶简化Lorenz系统进行数值模拟,发现系统混沌现象的产生与其阶数有着必然的联系,而后对SIR模型进行了较为详细的分析,并利用求解分数阶微分方程组Euler法对该模型在Caputo定义下的方程组进行离散化处理,通过数值模拟,揭示了其在不同阶数下传染病的严重程度。

基于C_0算法的混沌系统复杂度特性分析

采用C0复杂度算法,分析了Logistic映射、简化Lorenz系统和超混沌Lorenz系统的复杂度特性,并与系统的Lyapunov指数谱和分岔图进行对比,结果表明,C0复杂度能正确反映系统的复杂度特性;三系统复杂度从大到小依次为Logistic系统、超混沌Lorenz系统和简化Lorenz系统.将C0复杂度算法与谱熵算法(SE)和强度统计算法(LMC)计算结果对比,进一步说明C0算法分析混沌系统复杂度的有效性.系统复杂度随时间演化的特性分析表明,系统复杂度在一定范围内波动,即系统具有演化稳定性,两连续系统中y序列复杂度最大.为混沌系统应用于信息加密、保密通信领域提供了理论与实验依据.

混沌系统的自适应函数投影同步与参数辨识

为了实现两未知参数混沌系统的同步控制与参数辨识,采用自适应函数投影同步控制策略,基于李亚普诺夫稳定性原理,设计了实现参数未知、不同初值的两同构或异构混沌系统同步的控制器和参数自适应控制律,给出了实现同步的控制参数的取值范围,分析了控制参数对同步系统性能的影响规律.以最新提出的单参数简化洛仑兹混沌系统模型为研究对象,采用Matlab/Simulink进行动态仿真研究,表明了理论分析的正确性和同步控制与参数辨识方法的有效性.

混沌伪随机序列发生器的FPGA设计与实现

基于简化Lorenz系统,提出混沌伪随机序列发生器的一种设计方法。根据IEEE-754浮点运算标准,按照模块化设计理念,设计混沌方程所需的浮点运算模块,并在FPGA(Field Programmable Gate Array)上实现了简化Lorenz混沌系统。设计混沌伪随机序列量化算法,对生成的混沌伪随机序列进行复杂度分析。分析结果表明,量化算法显著提高了序列复杂度。使用NIST标准进行伪随机序列性能测试,测试结果表明,序列具有良好的随机特性,可直接用于实际加密应用。为连续混沌系统FPGA实现和混沌伪随机序列在信息安全中的应用奠定了基础。

基于线性反馈控制的超混沌系统设计与电路实现

基于最新提出的简化Lorenz混沌系统,采用线性控制策略,设计了一个超混沌系统。利用相图、Lyapunov指数谱、分岔图以及Poincaré截面分析方法,详细地研究了系统参数变化时简化Lorenz超混沌系统的动力学行为,数值仿真结果显示,Lyapunov指数谱与分岔图分析一致。采用分立元件,设计并实现了该系统的模拟电子电路,通过调节电路中的可变电阻,观察到了该系统的极限环、叉式分岔、倍周期分岔、混沌和超混沌,以及由倍周期分岔而进入混沌的道路等动力学现象。数值仿真和电路实验结果表明,该系统具有丰富的动力学特性,且电路实验结果与数值仿真相吻合。

简化Lorenz多涡卷混沌吸引子的设计与应用

将简化Lorenz系统线性化成两个线性系统,采用控制方法得到两涡卷混沌系统,通过扩展两涡卷混沌系统的指标2鞍焦点,设计了多涡卷混沌吸引子.利用相图、分岔图、Poincaré截面和最大Lyapunov指数等方法,分析了该多涡卷混沌系统的动力学特性.设计了多涡卷混沌吸引子的模拟电路,并进行了仿真,数值仿真与电路仿真相一致.将多涡卷混沌系统应用于图像加密,设计了多涡卷混沌与高级加密标准(AES)的改进混合加密算法,并分析了其加密性能.结果表明,基于多涡卷混沌系统的改进混合加密算法具有更高的安全性.

分数阶混沌系统的Adomian分解法求解及其复杂性分析

根据分数阶微分定义,采用Adomian分解算法,研究了分数阶简化Lorenz系统的数值解.研究发现,该算法与预估-校正算法相比,求解结果更准确,所耗计算资源和内存资源更少,求解整数阶系统时较Runge-Kutta算法更准确;利用Adomian算法得到的分数阶简化Lorenz系统出现混沌的最小阶数为1.35,比利用预估-校正算法得到的最小阶2.79更小.采用相图、分岔图分析了该系统的动力学特性,基于谱熵算法(SE)和C0算法分析了该系统的复杂度.结果表明,复杂度结果和分岔图一致,说明系统的复杂度同样能反映出系统动力学特性;复杂度随阶数q的增加呈总体减小的趋势,而混沌态时系统参数c变化对系统复杂度影响不大.为分数阶混沌系统应用于信息加密、保密通信领域提供了理论与实验依据.

简化洛伦兹混沌系统的追踪同步控制

基于李亚普诺夫稳定性原理,推导出简化洛伦兹系统的追踪同步控制定理,并从理论上证明设计的追踪控制器可以使系统追踪任意可微参考信号.采用设计的追踪同步控制器,实现简化洛伦兹系统追踪直流信号、正弦信号和超混沌信号的情况.分析控制参数的大小与混沌追踪同步性能之间的关系,发现采用较大的参数能缩短同步时间.数值仿真结果表明该方法的可靠性和有效性.为实现简化洛伦兹混沌系统保密通信提供了新的同步方案.