The open-plus-closed loop （OPCL） method for chaotic systems with multiple strange attractors

Based on the open-plus-closed-loop (OPCL) control method a systematic and comprehensive controller is presented in this paper for a chaotic system, that is, the Newton--Leipnik equation with two strange attractors: the upper attractor (UA) and the lower attractor (LA). Results show that the final structure of the suggested controller for stabilization has a simple linear feedback form. To keep the integrity of the suggested approach, the globality proof of the basins of entrainment is also provided. In virtue of the OPCL technique, three different kinds of chaotic controls of the system are investigated, separately: the original control forcing the chaotic motion to settle down to the origin from an arbitrary position of the phase space; the chaotic intra-attractor control for stabilizing the equilibrium points only belonging to the upper chaotic attractor or the lower chaotic one; and the inter-attractor control for compelling the chaotic oscillation from one basin to another one. Both theoretical analysis and simulation results verify the validity of the proposed means.     

Chaos detection and control in a typical power system

In this paper, a new chaotic system is introduced. The proposed system is a conventional power network that demonstrates a chaotic behavior under special operating conditions. Some features such as Lyapunov exponents and a strange attractor show the chaotic behavior of the system, which decreases the system performance. Two different controllers are proposed to control the chaotic system. The first one is a nonlinear conventional controller that is simple and easy to construct, but the second one is developed based on the finite time control theory and optimized for faster control. A MATLAB-based simulation verifies the results.     

A new 5D hyperchaotic system with stable equilibrium point,transient chaotic behaviour and its fractional-order form

In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.     

一个四翼混沌吸引子

在新的四维混沌系统中数值观察到四翼混沌吸引子，然而，通过进一步分析发现，该四翼吸引子并非真实的，实际上它是上、下两个共存的双翼混沌吸引子，他们各自有独立的混沌吸引域，由于其位置靠得太近和数值误差产生的一种假象.通过引入一个线性状态反馈控制项，系统的一些相似性被破坏，受控系统能产生穿越上下吸引域界限的对角双翼混沌吸引子，进一步，随着动力学模态的演化，上下混沌吸引子与对角混沌吸引子融合成一个真正的四翼混沌吸引子.最后，通过比较该四翼混沌吸引子的系统、Lorenz系统、Chua氏电路等混沌信号的频谱发现，四翼混沌吸引子的系统信号具有极宽的频谱带宽，该特性在通讯加密等工程应用中具有重要价值. 关键词： 四维混沌系统 双翼吸引子 四翼吸引子 频谱分析     

三维四翼广义增广Lü系统

增广Lü系统是仅含有两个控制参数的三维混沌系统,可产生一个伪四翼吸引子.利用混沌反控制方法,通过在增广Lü系统中引入一个简单的线性状态控制器,提出了一个新三维混沌系统——广义增广Lü系统.新系统实现了一个真正的具有复合结构的四翼吸引子,它包含了两个镜像对称的子系统.理论分析、数值仿真和电路实验证明了广义增广Lü系统四翼吸引子的存在性,也阐述了广义增广Lü系统有着较增广Lü系统更加丰富的动力学特性. 关键词： 广义增广Lü系统 四翼吸引子 数值仿真 电路实验     

一种基于washout滤波器技术的参数受扰混沌系统的控制

基于washout滤波器技术的混沌控制方法不仅能用于参数不变的混沌系统的控制，还可用于参数受扰动时的混沌系统的控制.给出了参数受扰动情况下基于washout滤波器技术的控制器设计的一个充分条件.数值仿真表明，该控制方法易于实现、控制代价小、能使原混沌系统的不稳定不动点稳定，而且具有一定的鲁棒性，很适合于工程应用. 关键词： 混沌控制 参数受扰动 washout滤波器     

Stabilization and Synchronization of a Complex Hidden Attractor Chaotic System by Backstepping Technique

In this paper, the stabilization and synchronization of a complex hidden chaotic attractor is shown. This article begins with the dynamic analysis of a complex Lorenz chaotic system considering the vector field properties of the analyzed system in the Cn domain. Then, considering first the original domain of attraction of the complex Lorenz chaotic system in the equilibrium point, by using the required set topology of this domain of attraction, one hidden chaotic attractor is found by finding the intersection of two sets in which two of the parameters, r and b, can be varied in order to find hidden chaotic attractors. Then, a backstepping controller is derived by selecting extra state variables and establishing the required Lyapunov functionals in a recursive methodology. For the control synchronization law, a similar procedure is implemented, but this time, taking into consideration the error variable which comprise the difference of the response system and drive system, to synchronize the response system with the original drive system which is the original complex Lorenz system.     

Embedding any desired number of coexisting attractors in memristive system

A simple variable-boostable system is selected as the structure for hosting an arbitrarily defined memristor for chaos producing. The derived three-dimensional (3-D) memristive chaotic system shows its distinct property of offset, amplitude and frequency control. Owing its merits any desired number of coexisting attractors are embedded by means of attractor doubling and self-reproducing based on function-oriented offset boosting. In this circumstance two classes of control gates are found:one determines the number of coexisting attractors resorting to the independent offset controller while the other is the initial condition selecting any one of them. Circuit simulation gives a consistent output with theoretically predicted embedded attractors.     

基于指数加权-核在线序列极限学习机的混沌系统动态重构研究

利用指数加权在线核序列极限学习机(exponential weighted online sequential extreme learning machine with kernel, EW-KOSELM)辨识算法,开展了针对混沌动力学系统的动态重构研究. EW-KOSELM算法将核递归最小二乘(kernel recursive least squares, KRLS)算法直接延伸至在线ELM (extreme learning machine)框架中,通过引入遗忘因子削弱了旧数据的影响,并基于"固定预算(fixed-budget, FB)"内存技术,应对在线核学习算法所固有的规模不断增长的计算困难.将所提辨识算法应用于Duffing-Ueda振子的混沌动力学系统数值仿真实例中,对基于FB-EW-KOSELM的辨识模型与原系统的动态性能进行了定性与定量的分析校验,定性校验准则是基于对比辨识模型与原系统吸引子(轨迹嵌入)、庞加莱映射、分岔图、极限环完成的,定量校验准则包括对比辨识模型与原系统的李雅普诺夫指数与关联维.进一步将其分别应用于来自测量蔡氏电路产生双涡卷吸引子与螺旋吸引子的实测数据实验及某一实际混沌电路所产生的时间序列中,对于具有低信噪比的实测电压或电流数据还需进行了小波降噪预处理.通过分析辨识模型重构吸引子,实验结果表明,FB-EW-KOSELM算法具有良好的动态重构性能,能精确地再生出展示混沌动态行为的过程非线性模型,且具有与原混沌系统非常接近的动态不变性指标.     

The stability control of fractional order unified chaotic system with sliding mode control theory

This paper studies the stability of the fractional order unified chaotic system with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for the fractional order unified chaotic system. By the existing proof of sliding manifold, the sliding mode controller is designed. To improve the convergence rate, the equivalent controller includes two parts: the continuous part and switching part. With Gronwall’s inequality and the boundness of chaotic attractor, the finite stabilization of the fractional order unified chaotic system is proved, and the controlling parameters can be obtained. Simulation results are made to verify the effectiveness of this method.     

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.     

用截面映象非线性延时反馈控制混沌

提出了利用相空间Poincare截面上的非线性延时反馈稳定微分动力系统混沌吸引子中不稳定周期轨道的新方法.以Henon映象和带注入信号的双光子激光系统为例，给出了理论分析和数值模拟结果．这种方法的主要特点是不需要获取混沌系统中不稳定周期轨道的任何信息，并且可以在混沌态的任意时刻施加控制作用．     

Bifurcation control and chaos in a linear impulsive system

Bifurcation control and the existence of chaos in a class of linear impulsive systems are discussed by means of both theoretical and numerical ways. Chaotic behaviour in the sense of Marotto's definition is rigorously proven. A linear impulsive controller, which does not result in any change in one period-1 solution of the original system, is proposed to control and anti-control chaos. The numerical results for chaotic attractor, route leading to chaos, chaos control, and chaos anti-control, which are illustrated with two examples, are in good agreement with the theoretical analysis.     

参数不确定永磁同步电机混沌系统的有限时间稳定控制

针对含有参数不确定性的永磁同步电机混沌系统, 结合主动控制与有限时间稳定控制理论, 提出一种改进的主动有限时间稳定控制器. 该控制器不仅具有较强的鲁棒性, 而且可以通过调整终端吸引子比例系数, 有效地缩短系统的响应时间, 以提高系统的响应能力. 通过仿真实验, 验证了该控制器比传统的控制器的具有更强的鲁棒性和响应能力. 关键词： 永磁同步电机 混沌控制 有限时间稳定 主动控制     

A new four-dimensional hyperjerk system with stable equilibrium point,circuit implementation,and its synchronization by using an adaptive integrator backstepping control

This paper reports a new simple four-dimensional(4 D) hyperjerk chaotic system. The proposed system has only one stable equilibrium point. Hence, its strange attractor belongs to the category of hidden attractors. The proposed system exhibits various dynamical behaviors including chaotic, periodic, stable nature, and coexistence of various attractors. Numerous theoretical and numerical methods are used for the analyses of this system. The chaotic behavior of the new system is validated using circuit implementation. Further, the synchronization of the proposed systems is shown by designing an adaptive integrator backstepping controller. Numerical simulation validates the synchronization strategy.     

Chaos control via a simple fractional-order controller

In this Letter, we propose a fractional-order controller to stabilize the unstable fixed points of an unstable open-loop system. Also, we show that this controller has strong ability to eliminate chaotic oscillations or reduce them to regular oscillations in the chaotic systems. This controller has simple structure and is designed very easily. To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of uncertain chaotic systems.     

混沌背景下信号的盲分离

混沌信号与确定性小信号叠加生成的混合信号是一更高维的混沌信号,因而不能用一般的混沌信号噪声抑制的方法进行分离.提出了一种这类信号盲分离的方法.在重构未知的混沌信号的动力方程时,充分利用混沌吸引子的几何特性,并且限定动力映射为原混沌吸引子所在流形的内部映射,从而保证了重构的动力系统方程对应于原混沌信号,而不是同样具有混沌特性的混合信号.然后利用重构的动力方程,借用混沌信号中的噪声抑制思想,估计出原混沌信号对应的轨道,实现信号分离.通过对Lorenz系统中谐波信号、Henon映象中自回归过程,以及脑电信号中谐波信号进行提取的数值实验,验证了信号盲分离方法的有效性和可行性. 关键词： 混沌 非线性 信号处理 盲分离     

共轭Chen混沌系统的分岔分析及基于该系统的超混沌生成研究

分析了一个三维自治混沌系统的Hopf分岔现象,该系统的混沌吸引子属于共轭Chen混沌系统.通过引入一个控制器,基于该混沌系统构建了一个四维自治超混沌系统.该超混沌系统含有一个单参数,在一定的参数范围内呈现超混沌现象.通过Lyapunov指数和分岔分析,随着参数的变化该系统轨道呈现周期轨道、准周期轨道、混沌和超混沌的演化过程. 关键词： 混沌 超混沌生成 Hopf分岔 分岔分析     

Chaotic attractor transforming control of hybrid Lorenz-Chen system

Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization. According to the Jacobian matrix of the nonlinear system, the stabilization control region is gotten. The controller is designed to stabilize fast the minimum phase Lorenz-Chen chaotic system after equivalently transforming from chaotic system to passive system. The simulation results show that the system not only can be controlled at the different equilibria, but also can be transformed between the different chaotic attractors.     

Suppression of chaotic oscillations in a microchip laser by injection of a new orbit into the chaotic attractor

Suppression of chaotic instability arising in a Nd:YVO(4) microchip laser subject to frequency-shifted optical feedback is accomplished by injection of one of the periodic orbits into the bifurcation region of another chaotic system driven by pump modulation. Various periodic patterns, which do not exist in the original chaotic attractor, can be extracted from the chaotic oscillation by use of this nonfeedback chaos-control technique.