The design and implementation of a multi‐wing chaotic attractor based on a five‐term three‐dimension system

The last two decades have seen great progress about the generation and circuit realization of multi‐wing chaotic attractor. In this paper, several multi‐scroll chaotic attractors are generated from a five‐term system by adding a piecewise linear function. Moreover, some basic properties in terms of symmetry and dissipation, equilibrium points, eigenvalues of the Jacobian matrices, Lyapunov exponent spectrum, bifurcation diagram, and Poincaré map are studied. In particular, an analog circuit is designed to implement the proposed multi‐scroll attractors, which are different from the traditional attractors. Furthermore, an integrated circuit diagram is designed to realize the fractional‐order multi‐scroll attractors. Finally, the performed experimental results confirm the theoretical analysis, and our work lends itself to many potential applications in engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

Creation and circuit implementation of two-to-eight–wing chaotic attractors using a 3D memristor-based system

This paper constructs a three-dimensional (3D) memristor-based system for creating multiwing chaotic attractors. A second-degree polynomial memristance function and a sixth-order exponent internal state memristor function with one parameter are employed, and the complexity of attractors is increased. A detailed analysis on dynamical behaviors of the proposed system are described, such as the bifurcation diagrams, finite-time local Lyapunov exponents, time series, phase portraits, and Poincaré maps. By adjusting the design parameters, the system displays two-to-eight–wing chaotic attractors, especially the five-wing and seven-wing attractors, which have never been found in the known systems. Further, we provide the calculation formula of the number of wings in the system, discuss the distribution of the involving inner holes on the plane, and design an electronic circuit to realize the proposed system. The experimental results of the circuit implementation agree with the numerical simulations on Matlab well. It indicates the potential engineering applications for various chaos-based information systems.     

超磁致伸缩作动器的分岔和混沌行为

混沌和拟周期运动是两种不稳定的响应,为研究超磁致伸缩作动器的这些不稳定行为,建立了其非线性动态力学模型,用数值方法得到了反映系统非线性响应特性的分岔图、相图、Poincaré截面图、幅值谱等。分析结果表明：阻尼系数的增加,有助于系统输出的稳定;系统刚度过低,会导致混沌现象的出现;激励电流的角频率对系统的响应特性有很大的影响。     

Classical and quantum mechanical transport simulations in open quantum dots

Simulations of open dot arrays are performed where the electron motion is considered in a harmonic potential and in a static magnetic field. Results concern electron trajectories and Poincaré sections from the classical calculation. The analysis of the Poincaré sections reveals a strong influence of the length of the constriction on the chaotic regions in phase space. Density probabilities from the quantum mechanical calculation show a significant correspondence to classically calculated trajectories.     

Novel chaotic behavior in the Muthuswamy–Chua system using Chebyshev Polynomials

In this paper, three kinds of memristors with memristance functions obtained from the Chebyshev polynomials are used in the Muthuswamy–Chua system, which has only three circuit elements: a linear passive inductor, a linear passive capacitor and a nonlinear active memristor. We use multivariable second‐order polynomial functions of current and memristor state for the internal state function of the memristor. This enables our system to generate not only double‐scroll but also four‐scroll attractors. Systematic studies of chaotic behavior in these systems are performed using phase portraits, bifurcation diagrams and Lyapunov exponents. Simulation results show that all these systems exhibit chaotic behavior over a range of control parameters. Copyright © 2014 John Wiley & Sons, Ltd.     

基于分岔理论的铁磁谐振研究

电力系统的铁磁谐振所产生的过电压严重威胁系统的安全稳定运行。为了了解系统响应性质随系统参数变化的情况,基于分岔理论,对一典型的110 kV变电站系统进行了数值仿真研究,绘制了以电源电压为分岔参数的分岔图,并结合系统的时域、频域、相图和庞加莱映射,分析了该系统所表现的基频、分频和混沌谐振。研究结果表明,分岔图能够很好的显示系统在某一参数范围内的全局性质,有利于为提出抑制谐振措施拓宽思路。     

Multi‐scroll chaotic attractors from a generalized time‐delay sampled‐data system

In this paper, the first generalization for time‐delay sampled‐data chaotic system in order to generate multi‐scroll attractor is introduced with its circuit implementation. An efficient delay‐line with binary priority encoding, parallel shifting, and binary decoding is also suggested and implemented to overcome the delay line realization drawback in such systems. The proposed system enhances the complexity of chaotic behavior by means of multi‐scroll feature and exemplifies the simplification of chaotic systems for better realizations. Copyright © 2015 John Wiley & Sons, Ltd.     

Qualitative analysis of an interrupted electric circuit with spike noise

Switching non‐ideality and its effects have been reported in DC‐DC converters. In this paper, we examine the qualitative property of an interrupted electric circuit with spike noise. First, we show the circuit model that have the switch interrupted by its own state and a periodic interval. Here, we artificially add spike noise via every switching action. Then, we explain its dynamics and derive the Poincaré map for the rigorous analysis in a circuit with ideal switching and a circuit with spike noise, respectively. Finally, we discuss the dynamical effects of spike noise from experimental and analytical viewpoints based on the Poincaré map and bifurcation diagrams. As a result, some dynamical effects of spike noise are clarified in terms of the invariant set, bifurcations, and existence regions of coexisting attractor. Copyright © 2010 John Wiley & Sons, Ltd.     

A multi‐wing butterfly chaotic system and its implementation

It has been a common brief that the principle for generating multi‐wing attractor from a Lorenz‐like chaotic system is to construct an even symmetric function for the system. In this pursuit, research effort on exploring the possibility of using non‐even symmetric function is scant, if any. This paper attempts to reveal the intrinsic relation between such a system function and its generated multi‐wing attractor in a chaotic system. It will be shown that a rather simple asymmetric function applying to the Shimizu–Morioka system can indeed generate a multi‐wing butterfly chaotic attractor. Then, some basic properties, compound structure, bifurcation diagram, Lyapunov exponent spectrum, and dynamical analysis of the new chaotic system are investigated in detail. Finally, an electronic circuit is designed and its experimental results are presented, which well match the numerical simulation results, thereby verifying the feasibility of the new method. Copyright © 2017 John Wiley & Sons, Ltd.     

Spiral periodic structure inside chaotic region in parameter‐space of a Chua circuit

In this letter we investigate, via numerical simulations, the parameter‐space of the set of autonomous first‐order differential equations of a Chua circuit. We show that this parameter‐space presents self‐organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period‐adding cascades, along specific directions that point towards this same focal point. Copyright © 2010 John Wiley & Sons, Ltd.     

Lyapunov exponent and bifurcation characteristics in synchronous machines with an internal fault

In this paper, first, turn‐to‐turn fault in a synchronous machine stator winding is investigated, and its inductances are derived with the winding function method. Then, using the nonlinear control theory, the first‐order Poincaré map of the synchronous machine is computed, and a new extended Poincaré map of the machine is established. Also, the characteristic multipliers of the synchronous machine are calculated by the Poincaré map. This new map is capable of analyzing electrical machines in the nonlinear and unbalanced cases for evaluation of stability, bifurcation, and chaos phenomena. The results show that the new map is an effective technique for modeling and analyzing any AC electrical machines. The characteristic multipliers and the Lyapunov exponents of the extended map indicate the system's condition. Chaotic phenomena are observed in some machines with internal faults. Since chaotic systems have a continuous spectrum and various kinds of frequencies in their spectrum, identification of the internal fault location is not easy with harmonic analyses such as Fourier spectrum method on the excitation current in synchronous machines. © 2013 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

混沌振荡与真空断路器动作特性之间相互作用的研究

本文主要研究了真空断路器的动作特性与混沌振荡之间的相互作用.以一个简化的电力系统模型为例,从系统分歧的观点出发,研究了在不同参数条件下,断路器的断开所引起的系统动态演变过程,并探讨了在混沌振荡制式条件下断路器的工作状况.     

基于频闪映射的Boost PFC变换器中的间歇性分岔和混沌现象分析

该文推导出了一组频闪映射方程来精确描述峰值电流控制型Boost PFC变换器的非线性动力学特性。数值模拟发现了系统电流在输入电压周期内会出现间歇性分岔和混沌现象，并对这种间歇性分岔和混沌的周期性进行了分析。通过对系统占空比的详细研究，发现系统并没有经过倍周期分岔的充分发展到达混沌，而是在倍周期分岔的过程中，发生边界碰撞分岔后以非光滑的方式直接进入混沌的。最后，给出了在输入电压幅值、参考电压和负载变化下系统的稳定域。这些结果对实际电路的设计具有重要的参考价值。     

Bifurcation analysis of a piecewise smooth system with non‐linear characteristics

In previous works, there are no results about the bifurcation analysis for a piecewise smooth system with non‐linear characteristics. The main purpose of this study is to calculate the bifurcation sets for a piecewise smooth system with non‐linear characteristics. We first propose a new method to track the bifurcation sets in the system. This method derives the composite discrete mapping, Poincaré mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. As an illustrated example, we then apply this general methodology to the Rayleigh‐type oscillator containing a state‐ period‐dependent switch. In the circuit, we can find many subharmonic bifurcation sets including global bifurcations. We also show the bifurcation sets for the border‐collision bifurcations. Some theoretical results are verified by laboratory experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Security analysis of image encryption based on twodimensional chaotic maps and improved algorithm

The article proposes a new algorithm to improve the security of image encryption based on twodimensional chaotic maps. Chaotic maps are often used in encrypting images. However, the encryption has periodicity, no diffusion, and at the same time, the real keys space of encryption are fewer than the theoretical keys space, which consequently results in potential security problems. Thus, this article puts forward several ways to solve the problems including adding diffusion mechanism, changing the design of keys and developing a composite encryption system. It designs an algorithm for the version B of the discretized baker map, which is one of the most prevalent chaotic maps, based on which a new image encryption is proposed to avoid the above problems. The simulation results show that the new encryption algorithm is valid and the result can be applied to other two-dimensional chaotic maps, such as the cat map. __________ Translated from Journal of Harbin Institute of Technology, 2007, 39(9): 1411–1414 [译自: 哈尔滨工业大学学报]     

Hyperchaos evolved from the generalized Lorenz equation

In this letter, a new hyperchaotic system is formulated by introducing an additional state into the third‐order generalized Lorenz equation. The existence of the hyperchaos is verified with bifurcation analysis, and the bifurcation routes from periodic, quasi‐periodic, chaotic and hyperchaotic evolutions are observed. Various attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit. Copyright © 2005 John Wiley & Sons, Ltd.     

双馈风机系统的切换型混沌振荡及其非光滑分岔特性

新能源电力系统含限幅或控制切换等非线性环节,本质上为非光滑系统,其稳定性和失稳形式复杂。对此,研究了大扰动下双馈风机并网系统中由限幅饱和导致的混沌振荡失稳机理,在数学上对应着非光滑系统轨迹发生从收敛至平衡点到混沌的非光滑分岔。首先,介绍了切换动力系统(非光滑系统)的非光滑分岔和切换型混沌;其次,分析了交流故障后双馈风机系统发生的复杂振荡现象,利用功率谱和Lyapunov指数确认其为混沌振荡,并分析了其拟周期的频率漂移特性;然后,分析了混沌振荡与限幅饱和间的关系,揭示了该混沌是由多个限幅饱和(非光滑系统切换)引起的,本质上为切换型混沌振荡;最后,从短路比、故障条件和风速等角度分析了大扰动后的切换型混沌振荡的非光滑分岔特性,结果表明在较低短路比、较大故障冲击下系统更易发生切换型混沌振荡。     

Adaptive actuator failure compensation design for unknown chaotic multi‐input systems

In this paper, an adaptive control approach is designed for compensating the faults in the actuators of chaotic systems and maintaining the acceptable system stability. We propose a state‐feedback model reference adaptive control scheme for unknown chaotic multi‐input systems. Only the dimensions of the chaotic systems are required to be known. Based on Lyapunov stability theory, new adaptive control laws are synthesized to accommodate actuator failures and system nonlinearities. An illustrative example is studied. The simulation results show the effectiveness of the design method. Copyright © 2014 John Wiley & Sons, Ltd.     

Elimination of chaotic ferroresonance in power transformer by ISFCL

Ferroresonance occurrences as one of the complex phenomena in power system not only lead to unwanted thermal and insulation problems, but also can cause chaotic oscillations. In this paper, a new method is proposed to eliminate and damp ferroresonant oscillations. Hence, a kind of inductive superconducting fault current limiters (ISFCL) is applied to eliminate chaotic ferroresonance oscillations alongside utilizing the current limiting ability of ISFCL. With regard to that ferroresonance is classified as nonlinear dynamics, chaos theory is applied. By using this theory, changes in system parameters which occur chaotic ferroresonant oscillations, can be better investigate and analyze. Behavior of the system, with and without ISFCL, is shown in bifurcation and phase plane diagrams. Behavioral changes of the system can be easily seen in these diagrams. Obtained results strongly show effectiveness of using this method for eliminating and damping ferroresonant oscillations.     

交错并联开关电感Boost变换器的失稳分析与控制

为提高输出电压增益，减小开关管电压应力，将开关电感结构代替传统电感嵌入到交错并联Boost变换器中。利用频闪映射法得到交错并联开关电感Boost变换器在峰值电流控制下的离散数学模型，并通过分岔图、相轨图、Jacobian矩阵特征值研究其失稳机理，结合参数共振微扰与电荷控制的优点，设计了补偿方案对失稳系统进行混沌控制。研究结果表明，开关电感的嵌入有效拓宽了交错并联Boost变换器的稳定范围，并通过补偿方案使得混沌系统重回周期-1稳态，提高了变换器的工作性能和稳定性。