一个具有完全四翼形式的四维混沌

本文通过引进一个非线性状态反馈控制器, 提出了一个新的四维混沌系统, 该混沌吸引子能在任何方向上都表现出四翼形式. 由于存在一个大的正李雅普诺夫指数, 混沌系统具有一些非常有趣和复杂的动力学行为. 对系统的一些基本动力学特性进行了数值模拟和理论分析, 如平衡点、耗散性、Poincaré映射、频谱、时域谱和混沌行为等. 通过对Lyapunov指数谱和分岔图的分析, 进一步研究了混沌行为的系统参数敏感性. 最后, 设计了一个实现四翼混沌系统的振荡电路, EWB观察结果与数值模拟结果具有良好的一致性.     

A four-wing and double-wing 3D chaotic system based on sign function

Many four-wing chaotic systems have been developed based on cross product or quadratic operations. Differently, we construct a three-dimensional chaotic system generating four-wing or double-wing attractors by virtue of sign function. Dynamical properties such as equilibrium points, Poincaré map, Lyapunov exponent spectra, Hopf bifurcations and bifurcation diagrams of the system are theoretically and numerically analyzed. Results of mathematical analyses and simulation tests indicate that the proposed chaotic system can keep chaotic to generate four-wing or double-wing attractors within a large scope of parameters. The system also shows hyperchaotic behaviors in some parameter range. Besides, circuit implementation of the chaotic system is studied. That proves the system is physically realizable.     

A novel four-wing non-equilibrium chaotic system and its circuit implementation

In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps. There is little difference between this chaotic system without equilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincaré maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.     

Novel four-dimensional autonomous chaotic system generating one-, two-, three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincaré map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations are in good agreement with the numerical simulation results.     

一个新的三维二次混沌系统及其电路实现

为了产生复杂的混沌吸引子，构造了一个新的三维二次自治混沌系统.该系统含有三个参数，每一个方程含有一个非线性乘积项.利用理论推导、数值仿真、Lyapunov指数谱和分岔图对系统的基本动力学特性进行了分析.结果表明，该系统具有五个平衡点，因而与Lorenz，Rsslor，Chen、Lü等混沌系统是非拓扑等价的；当其参数满足一定条件时，系统是混沌的.与Lorenz等混沌系统相比，该系统具有更大的正Lyapunov指数，能够产生复杂的混沌吸引子和一些有趣的动力学行为.最后，设计了实现该系统的混沌电路，电路实验结 关键词： 三维二次自治系统 混沌 混沌吸引子 电路实现     

一种新型的四维多翼超混沌吸引子及其在图像加密中的研究

提出了一种新的能产生多翼混沌吸引子的四维混沌系统,该系统在不同的参数条件下能产生混沌、超混沌吸引子.然后对此混沌系统的一些基本的动力学特性进行了理论分析和数值仿真,如平衡点、Poincaré映射、耗散性、功率谱、Lyapunov指数谱、分岔图等.同时设计了一个模拟振荡电路实现四翼超混沌吸引子,硬件电路模拟实验结果与数值仿真结果相一致.最后将此四维多翼超混沌系统用于物理混沌加密和高级加密标准加密级联的混合图像加密算法,这种利用物理混沌不可预测性的混合加密系统,不存在确定的明文密文映射关系,且密文统计特性也比其他加密系统要好.     

一个新的三维混沌系统的分析、电路实现及同步

中心流形理论提供了一个将高维系统降维研究的方法，应用该理论研究了一个新的混沌系统的基本特性，给出中心流形上流方程，分析这个新的混沌系统的叉式分岔.通过构建电路实现了该混沌系统,从而验证了系统的混沌行为，证实了混沌吸引子的存在.同时说明了由于电路信号频率与数值信号频率的不同所带来的数值仿真与物理实现之间在应用上有着重要区别.最后利用单变量反馈控制方法实现了新系统的同步控制，并给出了完整的同步实现电路. 关键词： 三维混沌系统 中心流形 电路实现 同步     

一个大范围超混沌系统的生成和电路实现

在对一些已有的超混沌系统研究和分析的基础上,提出了一个新的四维自治的超混沌系统,这个超混沌系统是通过引入一个状态变量到一个三维自治混沌系统而生成的,它较已有的超混沌系统而言,不仅最大的Lyapunov指数要大一些,而且在参数变化时,呈现超混沌的参数范围也很大.在对该系统进行数值仿真和分形分析的同时,也通过模拟电路对其进行了验证,电路实验结果表明,在电路中分别呈现的周期、伪周期、混沌、超混沌特性与数值仿真中获得的结果是一致的. 关键词： 超混沌 分形分析 超混沌电路 Lyapunov指数     

Hopf bifurcation analysis and circuit implementation for a novelfour-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare′ maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.     

一个新的恒Lyapunov指数谱混沌吸引子与电路实现

通过对改进恒Lyapunov指数谱混沌系统进行进一步演变,并引入新的绝对值项,发现了一种新的混沌吸引子.首先,通过相图、Poincar映射、Lyapunov指数以及功率谱,证明该混沌吸引子的存在性.接着,分析研究了这种新型混沌吸引子的基本动力学行为.Lyapunov指数谱、分岔图和状态变量幅值演变的数值仿真说明,该系统存在全局线性调幅参数,在该参数的调整下,系统输出三维信号的幅度皆能得到线性调整,而系统保持相同的混沌吸引子与Lyapunov指数谱.最后,通过构建电路实现了该混沌系统,观察到相应的混沌吸引子,也验证了全局线性调幅参数的调幅作用,数值仿真与电路实现有很好的一致性.     

基于忆阻器的多涡卷混沌系统及其脉冲同步控制

忆阻器是一种具有记忆功能和纳米级尺寸的非线性元件,作为混沌系统的非线性部分,能够提高混沌系统的信号随机性和复杂度.本文基于增广Lü系统设计了一个三维忆阻混沌系统.仅仅通过改变系统的一个参数,该系统能产生单涡巻、双涡卷和四涡巻的混沌吸引子,说明该系统具有丰富的混沌特性.首先对该忆阻混沌系统的基本动力学行为进行了理论分析和数值仿真,如平衡点稳定性、对称性,Lyapunov指数和维数,分岔图和Poincare截面等.同时,建立了模拟该忆阻混沌系统的SPICE(simulation program with integrated circuit emphasis)电路,给出了不同参数下的电路实验相图,其仿真结果与数值分析相符,从而验证了该忆阻混沌系统的混沌产生能力.由于脉冲同步只在离散时刻传递信息,能量消耗小,同步速度快,易于实现单信道传输,因而在混沌保密通信中更具有实用性.因此,本文从最大Lyapunov指数的角度实现了该忆阻混沌系统的脉冲混沌同步,数值仿真证实了忆阻混沌系统的存在性以及脉冲同步控制的可行性,为进一步研究该忆阻混沌系统在语音保密通信和信息处理中的应用提供了实验基础.     

A novel one equilibrium hyper-chaotic system generated upon Lü attractor

By introducing an additional state feedback into a three-dimensional autonomous chaotic attractor Lü system, this paper presents a novel four-dimensional continuous autonomous hyper-chaotic system which has only one equilibrium. There are only 8 terms in all four equations of the new hyper-chaotic system, which may be less than any other four-dimensional continuous autonomous hyper-chaotic systems generated by three-dimensional (3D) continuous autonomous chaotic systems. The hyper-chaotic system undergoes Hopf bifurcation when parameter c varies, and becomes the 3D modified Lü system when parameter k varies. Although the hyper-chaotic system does not undergo Hopf bifurcation when parameter k varies, many dynamic behaviours such as periodic attractor, quasi periodic attractor, chaotic attractor and hyper-chaotic attractor can be observed. A circuit is also designed when parameter k varies and the results of the circuit experiment are in good agreement with those of simulation.     

基于参数切换算法的混沌系统吸引子近似及其电路设计

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.     

One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics

A novel 3D fractional-order chaotic system is proposed in this paper. And the system equations consist of nine terms including four nonlinearities. It's interesting to see that this new fractional-order chaotic system can generate one-wing, two-wing, three-wing and four-wing attractors by merely varying a single parameter. Moreover, various coexisting attractors with respect to same system parameters and different initial values and the phenomenon of transient chaos are observed in this new system. The complex dynamical properties of the presented fractional-order systems are investigated by means of theoretical analysis and numerical simulations including phase portraits, equilibrium stability, bifurcation diagram and Lyapunov exponents, chaos diagram, and so on. Furthermore, the corresponding implementation circuit is designed. The Multisim simulations and the hardware experimental results are well in accordance with numerical simulations of the same system on the Matlab platform, which verifies the correctness and feasibility of this new fractional-order chaotic system.     

New robust chaotic system with exponential quadratic term

This paper proposes a new robust chaotic system of three-dimensional quadratic autonomous ordinary differential equations by introducing an exponential quadratic term. This system can display a double-scroll chaotic attractor with only two equilibria, and can be found to be robust chaotic in a very wide parameter domain with positive maximum Lyapunov exponent. Some basic dynamical properties and chaotic behaviour of novel attractor are studied. By numerical simulation, this paper verifies that the three-dimensional system can also evolve into periodic and chaotic behaviours by a constant controller.     

一种具有恒Lyapunov指数谱的混沌系统及其电路仿真

提出了一种新的具有恒Lyapunov指数谱的三维混沌系统,该系统含有六个参数,其中一个方程含有一个非线性乘积项,一个方程含有平方项.通过理论推导、数值仿真、Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图研究了系统的动力学特性,并分析了不同参数变化对系统动力学行为的影响,其中,平方项系统参数变化时,系统的Lyapunov指数谱保持恒定,输出信号中的两维信号的幅值与参数呈幂函数关系变化,其指数为－1/2,第三维信号的幅值保持在同样的数值区间.最后,设计了该混沌系统的硬件电路并运用Multisim软件对该电路进行仿真实验,证实了该混沌系统的可实现性. 关键词： 混沌系统 恒Lyapunov指数谱 Poincare截面图 混沌电路     

新三维混沌系统及其电路仿真

提出了一个新的不同于Lorenz系统和Chen系统的三维连续自治混沌系统.该系统含有五个参数,其中两个方程中各含有一个非线性乘积项.通过理论推导、数值仿真、Lyapunov指数谱、分岔图、Lyapunov维数、Poincare截面图研究了系统的基本动力学特性,并分析了改变不同参数时系统动力学行为的变化.最后设计了硬件电路并运用电子工作平台Multisim软件对该电路进行仿真实验,证实了该混沌系统的可实现性. 关键词： 混沌系统 Lyapunov指数 Poincare截面图 电路仿真     

新的切换四涡卷超混沌系统及其电路实现

构建了新的包含2个子系统的切换四涡卷超混沌系统,2个子系统之间既相互关联又相互切换.分析了2个子系统的分岔图、平衡点的稳定性、Lyapunov指数以及动力学行为的演化过程.设计了实现切换四涡卷超混沌系统的电路,系统以时间依赖和状态依赖两种切换律进行任意切换和自主切换,仅用一个电路就能实现2个子系统的切换功能.     

一类自治混沌系统的动力学分析与电路实现

分析讨论一类新的三维二次自治混沌系统.通过理论分析、数值仿真,计算系统的Lyapunov指数谱和Lyapunov维数,以及系统的分岔图等,对系统的基本动力学特性进行分析.设计实现该系统的混沌电路,建立系统参数和电路参数的对应关系,运用Pspice软件对电路进行仿真实验,证实分析的正确性.所讨论的系统不同于Lorenz系统、Rössler系统、Chen系统和Lü系统等,且电路实现很简单实用.     

一个改进恒Lyapunov指数谱混沌系统的电路实现与同步控制

基于提出的恒Lyapunov指数谱混沌系统,通过将系统中的参数进行剥离,得到一个改进型的恒Lyapunov指数谱混沌系统.该混沌系统存在三个重要的特性：双参数恒Lyapunov指数谱、存在全局线性调幅参数和倒相参数.通过Lyapunov指数谱与分岔图结合理论证明与推理,揭示了该新系统存在的上述动力学特征.构建实验电路,实现了改进混沌系统,物理实验验证了新系统的混沌行为.最后,利用单变量反馈控制方法实现了新系统的同步控制,通过物理实验验证了新系统同步控制的条件. 关键词： 改进恒Lyapunov指数谱混沌系统 电路实现 同步控制