Discrete chaos in fractional sine and standard maps

Fractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively.     

Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family (, x) of maps at the interval [0, 1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent of these maps analytically, where the results thus obtained have been approved with the numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor for certain values of the parameter, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at those values of the parameter whose Lyapunov characteristic exponent begins to be positive.     

双降压式全桥逆变器非线性现象的研究

双降压式全桥逆变器具有无桥臂直通、输入直流电压利用率高、效率高、续流二极管可优化选取等优点,因而在高压输出场合得到广泛的应用.本文研究了双降压式全桥逆变器的分岔和混沌现象,建立了电流闭环比例控制下的二阶离散模型,得到了不同时间段内的频闪映射模型;通过折叠图和分岔图分析了不同比例系数k对于系统稳定性的影响,并搭建了Matlab/Simulink仿真模型,得到了电流闭环比例控制时电流iL的时域波形和相图轨迹,并在频域下分析了分岔和混沌对系统频谱的影响.同时,利用分岔图的方法分析了输入电压E、滤波电感L和开关周期T等外部参数变化时系统的非线性行为.研究结果表明,正确选择双降压式全桥逆变器的电路参数对于其稳定运行具有重要意义.     

单相三电平H桥逆变器分岔现象的研究

三电平逆变器相较于传统的两电平逆变器具有输出电压谐波畸变率小、开关管电压应力小等优点,因而在大功率场合受到了越来越多的关注. 本文针对一种复合式单相三电平逆变器,对其中的分岔和混沌现象进行了深入的研究,建立了电流闭环比例控制下的一阶离散模型,得到了不同时间段内的频闪映射模型. 以比例系数k,负载电阻R,负载电感L及输入电压E为变化参数,研究了三电平逆变器的分岔现象：通过分岔图和李雅普诺夫指数谱分析了慢变尺度下比例系数、负载电感、负载电阻和输入电压对系统动态性能的影响;通过折叠图直观地观测到了快变尺度下不同比例系数、负载电感所导致的分岔过程. 最后搭建了Matlab/Simulink仿真模型,得到了电流闭环比例控制时电流i的时域波形,仿真结果与理论分析相一致. 研究表明,正确选择单相三电平逆变器的电路参数对于其稳定运行具有重要意义. 关键词： 三电平逆变器 频闪映射 分岔 混沌     

Linear and nonlinear response of discrete dynamical systems II: Chaotic attractors

We investigate the average response to small external perturbations for discrete dynamical systems with chaotic attractors. The average linear response satisfies a fluctuation theorem, and in general diverges exponentially in the long-time limitt. It vanishes identically for allt>0 only in a number of special cases including the logistic model with bifurcation parameter =4. The nonlinear response turns out to be crucial. Its average is analyzed for a time-localized (pulse) perturbation. Near the onset of chaos it exhibits universal scaling behaviour expressed by two critical exponents. For static perturbations the resulting dynamics is extremely sensitive to the perturbation strength.Work supported by the Swiss National Science Foundation     

Theoretical and experimental study of discrete behavior of Shilnikov chaos in a CO 2 laser

The discrete distribution of homoclinic orbits has been investigated numerically and experimentally in a CO₂ laser with feedback. The narrow chaotic ranges appear consequently when a laser parameter (bias voltage or feedback gain) changes exponentially. Up to six consecutive chaotic windows have been observed in the numerical simulation as well as in the experiments. Every subsequent increase in the number of loops in the upward spiral around the saddle focus is accompanied by the appearance of the corresponding chaotic window. The discrete character of homoclinic chaos is also demonstrated through bifurcation diagrams, eigenvalues of the fixed point, return maps, and return times of the return maps. Received 28 September 2000 and 27 October 2000     

logistic模型的倍周期分岔控制

研究了logistic模型的倍周期分岔的控制问题，设计了各种线性控制器，得到了系统在控制前和控制后的分岔图，改变了动力系统的分岔特性.根据实际的应用目的可以设计不同的控制器，使倍周期分岔延迟或提前出现，甚至消失.适当选择控制器增益可以使分岔控制的效果更好. 关键词： logistic模型 倍周期分岔 分岔控制 控制器     

Hidden attractors in a new fractional–order discrete system: Chaos,complexity, entropy, and control

This paper studies the dynamics of a new fractional-order discrete system based on the Caputo-like difference operator.This is the first study to explore a three-dimensional fractional-order discrete chaotic system without equilibrium. Through phase portrait, bifurcation diagrams, and largest Lyapunov exponents, it is shown that the proposed fractional-order discrete system exhibits a range of different dynamical behaviors. Also, different tests are used to confirm the existence of chaos,such as 0–1 test and C0 complexity. In addition, the quantification of the level of chaos in the new fractional-order discrete system is measured by the approximate entropy technique. Furthermore, based on the fractional linearization method, a one-dimensional controller to stabilize the new system is proposed. Numerical results are presented to validate the findings of the paper.     

The evolution of chaotic dynamics for fractional unified system

Based on reliable numerical approach, this Letter studies the chaotic behavior of the fractional unified system. The lowest orders for this system to have a complete chaotic attractor (the attractor covers the three equilibrium points of the classical unified system) at different parameter values are obtained. A striking finding is that with the increase of the parameter α of the fractional unified system from 0 to 1, the lowest order for this system to have a complete chaotic attractor monotonically decreases from 2.97 to 2.07. Because of the inherent attribute (memory effects) of fractional derivatives, this finding reveals that the chaotic behavior of fractional (classical) unified system becomes stronger and stronger when α increases from 0 to 1. Furthermore, this Letter introduces a novel measure to characterize the chaos intensity of fractional (classical) differential system.     

Period matching in modulated maps

We study discrete nonlinear maps in which the control parameter is itself “modulated” by another discrete nonlinear map. We show that for a certain class of such maps, which includes for example the logistic map, the periodicity of the modulated signal is either one, independent of the periodicity of the modulating signal, or its periodicity is an integral multiple of the periodicity of the modulating signal or it is chaotic.     

Globally coupled maps with sequential updating

A system of globally coupled logistic maps with sequential updating is analyzed numerically. It is found that deterministic asynchronous updating schemes may have dramatic influences on the dynamical behaviors of globally coupled systems. Transitions from spatio–temporal chaos to spatially organized states are observed as the coupling parameter varies. It is shown that the model system may exhibit a variety of collective properties such as the clustering, traveling wave patterns, and spatial bifurcation cascades.     

单参数Lorenz混沌系统的电路设计与实现

为了研究混沌系统的性质及其应用,采用分立元件设计并实现了单参数Lorenz混沌系统,系统参数与电路元件参数一一对应.通过调节电路中的可变电阻,观察到了该单参数系统的极限环、叉式分岔、倍周期分岔和混沌等动力学现象,以及该系统由倍周期分岔进入混沌的过程.研究了分数阶单参数Lorenz系统存在混沌的必要条件,找出了分数阶单参数Lorenz系统出现混沌的最低阶数以及最低阶数随系统参数变化的一般规律.电路仿真与电路实现研究表明,单参数Lorenz系统具有物理可实现性、丰富的动力学特性以及理论分析与实验结果的一致性.     

Dynamics of the chain of forced oscillators with long-range interaction: from synchronization to chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0相似文献   

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

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.     

空间单位区域双二次有理贝赛尔曲面混沌特性研究

研究空间单位区域内两个二次曲面映射构成的函数的混沌特性, 发现了一种构造混沌的方法. 当一个曲面是单位区域内标准曲面, 另一个曲面随机生成时, 此函数是混沌的概率可以大于十分之一, 说明在满足一定条件时, 混沌是极其普遍的. 通过计算Lyapunov指数以及绘制分岔图等对该类函数的混沌特性进行分析, 根据参数变化的分岔图以及混沌曲面控制点的区域分布特性等寻找混沌映射函数, 得到了大量的二维混沌吸引子图形, 并对其中三个进行了详细研究. 另外, 把灰度图像作为离散二维函数, 首次研究了图像作为迭代表达式时表现出的一些混沌特性. 研究发现, 相同的或者相近的图像易于收敛到周期点上, 这个结果可以用于图像识别等研究领域. 关键词： 混沌 迭代 图像     

Generalized complexity measures and chaotic maps

The logistic and Tinkerbell maps are studied with the recently introduced generalized complexity measure. The generalized complexity detects periodic windows. Moreover, it recognizes the intersection of periodic branches of the bifurcation diagram. It also reflects the fractal character of the chaotic dynamics. There are cases where the complexity plot shows changes that cannot be seen in the bifurcation diagram.     

Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincar\'{e} maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.     

The dynamics of a memristor-based Rulkov neuron with fractional-order difference

The exploration of the memristor model in the discrete domain is a fascinating hotspot. The electromagnetic induction on neurons has also begun to be simulated by some discrete memristors. However, most of the current investigations are based on the integer-order discrete memristor, and there are relatively few studies on the form of fractional order. In this paper, a new fractional-order discrete memristor model with prominent nonlinearity is constructed based on the Caputo fractional-order difference operator. Furthermore, the dynamical behaviors of the Rulkov neuron under electromagnetic radiation are simulated by introducing the proposed discrete memristor. The integer-order and fractional-order peculiarities of the system are analyzed through the bifurcation graph, the Lyapunov exponential spectrum, and the iterative graph. The results demonstrate that the fractional-order system has more abundant dynamics than the integer one, such as hyper-chaos, multi-stable and transient chaos. In addition, the complexity of the system in the fractional form is evaluated by the means of the spectral entropy complexity algorithm and consequences show that it is affected by the order of the fractional system. The feature of fractional difference lays the foundation for further research and application of the discrete memristor and the neuron map in the future.