An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.     

一种最简的并行忆阻器混沌系统

在提出的一种压控忆阻器的基础上, 构造了最简的并联忆阻器混沌系统, 分析其动力学特性, 得到了该系统的Lyapunov指数和Lyapunov维数, 给出了时域波形、相图、Lyapunov指数谱、分岔图、Poincaré映射等. 利用EWB软件设计了该新混沌系统的振荡电路并进行了仿真实验. 研究结果表明, 忆阻器的i-v特性在参数的变化时, 并不保持斜“8”字形, 会变为带尾巴的扇形. 该混沌系统与磁控忆阻器混沌系统不同, 系统只有一个平衡点, 初始条件在系统能振荡的情况下不影响系统状态. 电路实验仿真结果和数值仿真具有很好的一致性, 证实了该系统的存在性和物理上可实现性. 关键词： 忆阻器 混沌电路 并联 动力学行为     

Finite-space Lyapunov exponents and pseudochaos

We propose a definition of finite-space Lyapunov exponent. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by showing that, for large classes of chaotic maps, the corresponding finite-space Lyapunov exponent approaches the Lyapunov exponent of a chaotic map when M-->infinity, where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has pseudochaos if its finite-space Lyapunov exponent tends to a positive number (or to +infinity), when M-->infinity.     

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.     

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.     

利用小波分析计算离散动力系统的最大Lyapunov指数

最大Lyapunov指数是诊断和描述动力系统混沌的重要参数.在仿真计算的基础上,发现小尺度的小波变换模数的最大Lyapunov指数与离散动力系统本身是一致的.同时仿真计算还表明,由于小尺度小波变换的高通滤波性质,利用小波变换模数计算最大Lyapunov指数可有效克服极强的大尺度噪声的干扰. 关键词： 混沌 Lyapunvo指数 小波分析     

一种新的三维自治混沌系统

通过对蔡氏电路的研究,提出了一种新的混沌系统,并对该系统的基本动力学特性进行了深入研究,得到该系统的Lyapunov指数和Lyapunov维数,给出了相图、Lyapunov指数谱、分岔图、Poincaré映射以及功率谱等.利用OrCAD-PSpice软件设计了该新混沌系统的振荡电路并进行了仿真实验.研究结果表明,该系统与蔡氏电路产生的混沌吸引子并不拓扑等价,且该系统的参数变化范围较大,最大Lyapunov指数接近1,数值仿真和电路系统实验仿真具有很好的一致性,证实了该系统的存在性和物理上可实现性. 关键词： 混沌系统 Lyapunov指数谱 分岔图 电路实现     

非周期力在混沌控制中的双重功效

探讨了非周期力（有界噪声或混沌驱动力）在非线性动力系统混沌控制中的影响.以一类典型的含有五次非线性项的Duffing-van der Pol系统为范例,通过对系统的轨道、最大Lyapunov指数、功率谱幅值及Poincar截面的分析,发现适当幅值的有界噪声或混沌信号,一方面可以消除系统对初始条件的敏感依赖性,抑制系统的混沌行为,将系统的混沌吸引子转化为奇怪非混沌吸引子;另一方面也可以诱导系统的混沌行为,将系统的周期吸引子转化为混沌吸引子.从而揭示了非周期力在混沌控制中的双重功效：抑制混沌和诱导混沌. 关键词： 混沌控制 有界噪声 混沌驱动力     

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     

推广恒Lyapunov指数谱混沌系统及其演变研究

基于恒Lyapunov指数谱改进系统,通过在系统方程中添加线性项与常数项,实现了恒Lyapunov指数谱混沌系统的推广.首先结合Lyapunov指数谱、分岔图和状态变量幅值演变的数值仿真,揭示了该系统的动力学行为;接着通过组合不同的线性项,从推广系统演变得到一族性质类似而又相轨不同的子系统,并分析了各个子系统的平衡点、特征值与Lyapunov指数等动力学特征;最后,指出该系统在混沌雷达、保密通信和其他信息处理系统中具有广阔的应用前景. 关键词： 推广混沌系统 Lyapunov指数谱 演变 子系统     

一种恒Lyapunov指数谱混沌吸引子及其Jerk电路实现

基于Colpitts方程,提出了一种新的三维混沌吸引子.该混沌吸引子在系统变幅参数改变时,输出混沌信号中的两维信号的幅值随着参数作线性变化,第三维信号的幅值保持在同样的数值区间,而系统的Lyapunov指数谱却保持恒定.该混沌系统通过改造Colpitts混沌系统归一化方程中的指数项为绝对值项而得到.通过相图、庞加莱映射、功率谱以及Lyapunov指数,证明了该混沌吸引子的存在性.对这种新型混沌吸引子的基本动力学行为予以分析,基于Lyapunov指数谱阐述并论证了该系统能够呈现周期态和混沌态.最后,给出该特 关键词： Colpitts系统 恒定Lyapunov指数谱 混沌吸引子 分岔图     

Transition to chaos in continuous-time random dynamical systems

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

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.     

Lyapunov exponent diagrams of a 4-dimensional Chua system

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.     

基于Lyapunov指数的摇摆条件下自然循环流动不稳定性混沌预测

运用基于最大Lyapunov指数的混沌预测方法对摇摆条件下自然循环系统的流量脉动进行了预测. 对不规则复合型脉动的流量脉动实验数据进行相空间重构, 计算关联维数、二阶Kolmogorov熵和最大Lyapunov指数等几何不变量, 在说明不规则复合型脉动是混沌运动的基础上, 根据最大Lyapunov指数对不规则复合型脉动进行了预测. 通过预测结果和实验结果对比发现: 对于复杂的两相自然循环流动不稳定性, 预测结果具有较高的精度, 说明预测方法的可行性. 同时, 确定了混沌系统可预测的尺度, 提出用动态预测的方式监测系统流量脉动. 本文的研究方法为两相流复杂的流动不稳定性研究提供了新的思路. 关键词： 混沌时间序列 实时预测 最大Lyapunov指数 两相流动不稳定性     

试用Lyapunov指数探讨气候突变及其前兆信号

Lyapunov 指数是标志一个系统做规则运动还是混沌运动的一个重要物理量.鉴于此, 本文利用Lyapunov指数研究系统的混沌特性, 研究气候的突变.计算结果表明, 定义法求得的Lyapunov指数是一种可靠的突变检测方法, 无论是理想序列还是实际存在突变的序列, 利用该方法都能准确地找出突变位置; 而利用非线性局部Lyapunov指数的可预报期限从理论上佐证了基于临界慢化现象的气候突变前兆信号的可靠性, 通过计算各个时间段的最大Lyapunov指数能够反映系统的内在性质、研究其混沌特性. 研究结果为该方法在实际观测资料中的广泛应用提供了理论基础. 关键词： Lyapunov指数 气候突变 前兆信号     

周期切换下Rayleigh振子的振荡行为及机理

本文研究了自治与非自治电路系统在周期切换连接下的动力学行为及机理.基于自治子系统平衡点和极限环的相应稳定性分析和切换系统李雅普诺夫指数的理论推导及数值计算.讨论了两子系统在不同参数下的稳态解在周期切换连接下的复合系统的各种周期振荡行为,进而给出了切换系统随参数变化下的最大李雅普诺夫指数图及相应的分岔图,得到了切换系统在不同参数下呈现出周期振荡,概周期振荡和混沌振荡相互交替出现的复杂动力学行为并分析了其振荡机理.给出了切换系统通过倍周期分岔,鞍结分岔以及环面分岔到达混沌的不同动力学演化过程.     

一种新型混沌系统的分析及电路实现

研究了新混沌动力系统的基本动力学行为,确定了新混沌系统的对称性、平衡点稳定性等相关问题.重点分析了系统参数对整个混沌系统的影响,给出了混沌系统关于随系统参数变化的Lyapunov指数谱、分岔图等.据此得出在系统参数的某一区间的新混沌系统状态.最后根据新混沌系统的数学模型设计具体的实际的电子电路,给出各种参数值的电路实验相图.     

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.