一类多翼蝴蝶混沌吸引子及其电路实现

提出了一种构造多翼蝴蝶混沌吸引子的新方法,在Liu混沌系统的基础上,通过设计一种新的分段线性函数,构造了一个产生多翼蝴蝶混沌吸引子的混沌系统,对系统的平衡点、Lyapunov指数谱、分岔图、相图、频谱和Poincare截面进行了分析。最后,设计了相应的硬件电路,电路实验结果与数值仿真结果一致,验证了该方法的可行性和有效性。     

一类多折叠环面多涡卷混沌吸引子的仿真研究

通过构造一个新的非线性函数，研究一类多折叠环面多涡卷混沌产生器.这种混沌产生器的主要特征是随着自然数n的增加，能产生大小相等、均匀分布的2n个折叠环面混沌吸引子和大小相等、均匀分布的2n+1个涡卷混沌吸引子，且折叠环面混沌吸引子与涡卷混沌吸引子之间相互间置，最左边和最右边均为涡卷混沌吸引子.基于虚拟电子实验室EWB软件，设计具体的电路，进行了仿真实验验证. 关键词： 折叠环面混沌吸引子 多涡卷混沌吸引子 涡卷     

A Simple Parallel Chaotic Circuit Based on Memristor

This paper reports a simple parallel chaotic circuit with only four circuit elements: a capacitor, an inductor, a thermistor, and a linear negative resistor. The proposed system was analyzed with MATLAB R2018 through some numerical methods, such as largest Lyapunov exponent spectrum (LLE), phase diagram, Poincaré map, dynamic map, and time-domain waveform. The results revealed 11 kinds of chaotic attractors, 4 kinds of periodic attractors, and some attractive characteristics (such as coexistence attractors and transient transition behaviors). In addition, spectral entropy and sample entropy are adopted to analyze the phenomenon of coexisting attractors. The theoretical analysis and numerical simulation demonstrate that the system has rich dynamic characteristics.     

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.     

Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity,coexisting multiple attractors,and offset boosting

Analyzing chaotic systems with coexisting and hidden attractors has been receiving much attention recently. In this article, we analyze a four dimensional chaotic system which has a plane as the equilibrium points. Also this system is of the group of systems that have coexisting attractors. First, the system is introduced and then stability analysis, bifurcation diagram and Largest Lyapunov exponent of this system are presented as methods to analyze the multistability of the system. These methods reveal that in some ranges of the parameter, this chaotic system has three different types of coexisting attractors, chaotic, stable node and limit cycle. Some interesting dynamics properties such as reversals of period doubling bifurcation and offset boosting are also presented.     

Generating multi-layer nested chaotic attractor and its FPGA implementation

Complex chaotic sequences are widely employed in real world, so obtaining more complex sequences have received highly interest. For enhancing the complexity of chaotic sequences, a common approach is increasing the scroll-number of attractors. In this paper, a novel method to control system for generating multi-layer nested chaotic attractors is proposed.At first, a piecewise(PW) function, namely quadratic staircase function, is designed. Unlike pulse signals, each level-logic of this function is square constant, and it is easy to realize. Then, by introducing the PW functions to a modified Chua's system with cubic nonlinear terms, the system can generate multi-layer nested Chua's attractors. The dynamical properties of the system are numerically investigated. Finally, the hardware implementation of the chaotic system is used FPGA chip.Experimental results show that theoretical analysis and numerical simulation are right. This chaotic oscillator consuming low power and utilization less resources is suitable for real applications.     

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional(5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.     

Dynamical behaviors of a chaotic system with no equilibria

Based on Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.     

四维系统中多涡卷混沌吸引子的仿真研究

通过构造一个新的非线性函数，研究一种新型四维系统多涡卷混沌信号发生器，这种多涡卷混沌信号发生器的主要特征是随着自然数n的增加，能产生2n+2个多涡卷混沌吸引子，通过改变控制参数k可以改变多涡卷混沌吸引子的混沌边界.并在EWB平台上设计了具体的电路，进行仿真实验验证. 关键词： 四维混沌系统 多涡卷混沌吸引子 非线性函数     

忆阻器混沌电路产生的共存吸引子与Hopf分岔

利用荷控忆阻器和一个电感串联设计一种新型浮地忆阻混沌电路.用常规动力学分析方法研究该系统的基本动力学特性,发现系统可以产生一对关于原点对称的"心"型吸引子.将观察混沌吸引子时关注的电压、电流推广到功率和能量信号,观察到蝴蝶结型奇怪吸引子的产生.理论分析Hopf分岔行为并通过数值仿真进行验证,结果表明系统随电路参数变化能产生Hopf分岔、反倍周期分岔两种分岔行为.相对于其它忆阻混沌电路该电路采用的是一个浮地型忆阻器,并且在初始状态改变时,能产生共存吸引子和混沌吸引子与周期极限环共存现象.     

Liu混沌系统的混沌分析及电路实验的研究

研究了一种新型混沌系统——Liu混沌系统的基本动力学行为以及电路实现的问题，给出了相图、庞卡莱映射、功率谱以及李雅普诺夫指数，基于李雅普诺夫指数谱和分叉图分析了系统参数对Liu混沌系统的影响.最后设计硬件电路证实了Liu混沌系统以及Liu混沌系统随系统参数变化时的各种状态的存在.给出数值仿真和电路实验的结果. 关键词： Liu混沌系统 分岔 电路实验     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

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.     

Generation and circuit implementation of multi-block multidirectional grid multi-scroll chaotic attractors

Due to the dynamic characteristics of the Chua's system, multi-scroll chaotic attractors are still confined in a single block and fail to break the limit. This paper proposes an approach for generating novel multi-block multidirectional grid multi-scroll chaotic attractors that can break the limit via novel nonlinear modulating functions. According to this method, the recursion rules used to generate multi-block multidirectional grid multi-scroll attractors are mathematically obtained. The new system is autonomous; the effectiveness of this method has been verified by theoretical analysis, numerical simulation, and circuit implementation.     

一类多折叠环面混沌吸引子

在双折叠环面混沌吸引子方程中构造一种具有多个分段线性化的奇函数，基于递归算法，导出可产生一类多折叠环面混沌吸引子的递归公式.通过适当选取各个分段线性区间的斜率值，利用所得的递归公式计算出分段线性化函数中各个平衡点和转折点之值，最终可产生一类多折叠环面混沌吸引子.给出了产生这类混沌吸引子的计算机数值模拟结果. 关键词： 双折叠环面混沌吸引子 多折叠环面混沌吸引子 递归算法     

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.     

Generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems

Based on our previous works and Lyapunov stability theory, this paper studies the generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems. A fourth-order circuit, by introducing additional breakpoints in the modified Chua oscillator, is implemented for the study of generation and synchronization of N-scroll chaotic attractors.This confirms the consistency of theoretical calculation, numerical simulation and circuit experiment.Furthermore,we give a refined and extended study of generating and synchronizing N-scroll hyperchaotic attractors in the fourth-order MCK system and report the new theoretical result, which is verified by computer simulations.     

一类多涡卷混沌吸引子及电路设计

提出了一种产生多涡卷混沌吸引子的方法并给出了相应的电路设计.该方法由若干个分段线性函数构成混沌系统的非线性函数,构造不同的分段数可以获得不同数量的涡卷混沌吸引子.数值仿真和EWB仿真结果表明了该方法的有效性. 关键词： 多涡卷混沌吸引子 分段线性函数 电路设计 电路实验结果     

n-scroll chaotic attractors from a first-order time-delay differential equation

In this paper, a novel first-order delay differential equation capable of generating n-scroll chaotic attractor is presented. Hopf bifurcation of the introduced n-scroll chaotic system is analytically and numerically determined. The bifurcation diagram and Lyapunov spectrum of the system are calculated and the results show that the system has a chaotic regime in a wider parameter range. Furthermore, period-3 behavior has been observed on the system. Circuit realizations of two-, three-, four-, and five-scroll chaotic attractors are also presented.     

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially, this work can be used for some real applications in secure communication, such as data and image encryptions.