Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems)

Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.     

对称非线性耦合混沌系统的同步

研究两个对称非线性耦合混沌系统的同步问题.通过对系统线性项与非线性项的适当分离， 构造一个特殊的非线性耦合项，发现在耦合强度α＝05附近的某一区域里存在稳定的 混沌同步现象.提供判断同步误差稳定性的方程，利用线性系统的稳定性分析准则和条件Lya punov指数来检验同步状态的稳定性.新方法适用于连续时间系统的混沌同步，也适用于具有 两个（或多于两个）正Lyapunov指数的超混沌系统.以Lorenz系统，超混沌Rssler 系统作 为算例，数值模拟结果证实所提新方法的有效性. 关键词： 混沌 同步 非线性耦合 稳定性准则 超混沌     

基于弹性控制器的混沌同步

利用线性输出反馈，实现了混沌系统的弹性同步，削弱了控制器本身不确定性的影响.结合linear matrix inequality工具箱，方便地得到了控制器的增益形式.仿真结果表明了研究的 意义及控制器的有效性. 关键词： 混沌系统 同步 弹性控制     

一类具有可变系数的混沌系统的同步

本文基于系统传递函数矩阵的严格正实性, 针对一类具有可变系数的混沌 (或超混沌) 系统的自同步与异结构同步问题提出了解决方法. 通过在响应系统中加入同步控制器, 并将待同步系统导出的误差系统中的非线性部分作为误差系统输入, 将误差状态变量作为误差系统输出, 使误差系统的传递函数矩阵成为严格正实的, 这样可使误差系统的原点是渐近稳定的, 即两系统达到稳定的混沌 (或超混沌) 同步. 所设计的同步控制器参数选取范围明确, 均为线性的, 且对于待同步系统的系数变化具有一定的鲁棒性. 文中给出了同步控制器的具体设计过程和同步结果, 并结合数值仿真验证了该方法的可行性与有效性. 关键词： 严格正实 可变系数 混沌同步     

一类混沌系统的同步方法

针对一类混沌系统的特点,任意构造满足一定条件的响应系统,根据滑模控制策略得到实现两者同步的反馈控制器.在可测同步误差基础上,利用扩张状态观测器来估计驱动系统与“构造”的响应系统之间的差异,把上述反馈控制策略转变为物理可实现的控制器并实现两者的同步.以Duffing和Chua’s电路为例的仿真表明效果良好 关键词： 混沌同步 滑模控制 扩张状态观测器     

输入受限的混沌系统同步控制

在混沌系统的同步控制中, 由于混沌系统对初始状态的敏感性, 一旦两个混沌系统的状态初值偏差大, 其状态同步往往需要高幅值的控制律来达到, 这给同步控制实现带来了困难, 并且在同步控制中, 两个混沌系统的初始值通常是未知的. 本文考虑控制输入受限情况下的混沌同步控制问题, 基于符号函数的近似表示式, 将受限的控制输入建模为连续可微的光滑函数, 在每一个采样点将同步控制误差系统近似为局部最优线性模型并设计连续型线性二次型调节器(LQR)最优控制律. 为降低混沌同步控制律的幅值和维持同步系统采样时刻之间的动态, 设计了等价的离散最优控制律, 并通过调整LQR性能加权矩阵值, 确保同步控制信号不会超出其受限的上界. 最后对统一混沌模型下的三种不同混沌系统同步控制进行了仿真研究. 仿真结果验证了方法的有效性. 关键词： 统一混沌模型 符号函数 输入受限 同步控制     

Synchronization of chaotic systems with different orders

A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state variables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.     

Synchronization of chaotic systems with coexisting attractors

Synchronization of coupled oscillators exhibiting the coexistence of chaotic attractors is investigated, both numerically and experimentally. The route from the asynchronous motion to a completely synchronized state is characterized by the sequence of type-I and on-off intermittencies, intermittent phase synchronization, anticipated synchronization, and period-doubling phase synchronization.     

Synchronization in chaotic Hamiltonian systems and a geophysical application

This paper addresses the question of the rate of synchronization of two identical systems as a function of the inserting time interval Delta t between inserted variables of the driving system in the role of the same variables of the driven system in a simplified Hamiltonian system and its application to a simplified geophysical model. We start by analyzing the synchronization in a simplified two-degree Hamiltonian system. The synchronization rate turns out to be a decreasing function of the inserting time interval Delta t up to a certain limit Delta t(o) where the process reverses and the synchronization rate becomes slower as the inserting frequency decreases. The key point of the analysis uses a second-order Taylor expansion of the system resolvent which indicates that synchronization rate is basically of order O(Delta t(2)) for small Delta t. The study is then extended to include a simplified geophysical system. A nonlinear one-dimensional shallow-water model on a periodic domain meant to represent a latitudinal circle around 45 degrees N is used. It is found that when the zonal wind is inserted, the maximum synchronization rate is obtained when the inserting time interval is approximately 4 h. When the meridional wind is inserted, it is obtained at slightly less than 4 h. It is shown, in particular, that the synchronization rate depends on the latitude (or the Coriolis parameter). A low-order simplified dynamical system derived from the one-dimensional shallow-water model is used to show that this optimum time interval Delta t(o) when the zonal wind and the geopotential, for example, are inserted varies approximately as square root of [2]/2 Omega sin phi to accuracy O(Delta t(3)). Analyses performed with a linear version of the shallow-water model reveal that this latter can be used to explain the observed convergence behavior in the nonlinear model. The only point is the choice of the stationary state for linearization purposes. It is then suggested that in more complicated geophysical systems, the closest stationary state to the climatology can be used to estimate the crossover point Delta t(o).     

基于区间系统理论的分数阶混沌系统同步

通过设计一个非线性反馈控制器,实现了分数阶混沌系统的同步.与其他的分数阶混沌系统同步方法相比,提出的控制器设计方法保留了部分误差系统中的非线性项,而没有完全抵消同步误差系统的非线性项,有效改善了误差系统的控制性能.同时,应用区间分数阶线性时不变系统稳定性原理和线性矩阵不等式技术,得到了一个新的分数阶混沌系统同步的充分条件,进而获得的控制器保证了混沌系统同步.仿真结果验证了提出方法的有效性. 关键词： 区间分数阶时不变系统 分数阶混沌系统 混沌同步     

分数阶混沌系统的异结构同步

基于分数阶线性系统稳定性理论,结合反馈控制和主动控制方法,提出了一种分数阶混沌系统异结构同步方法,给出了同步控制器解析式. 以分数阶Chen混沌系统和分数阶Liu混沌系统、分数阶新超混沌系统和分数阶超混沌Rssler系统的异结构同步为例, 进行了数值模拟,证实了该方法的有效性和可行性. 关键词： 分数阶混沌 异结构混沌同步 新超混沌系统 超混沌Rssler系统     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Synchronization of a network coupled with complex-variable chaotic systems

In this paper, synchronization of a network coupled with complex-variable chaotic systems is investigated. Adaptive feedback control and intermittent control schemes are adopted for achieving adaptive synchronization and exponential synchronization, respectively. Several synchronization criteria are established. In these schemes, the outer coupling matrix is not necessarily assumed to be symmetric or irreducible. Further, for a class of networks with an irreducible and balanced outer coupling matrix, a pinning control scheme is adopted for achieving synchronization. Numerical simulations are demonstrated to verify the effectiveness of the theoretical results.     

Synchronization of chaotic systems based on adaptive observer design

The synchronizing problem of a chaotic system is investigated based on the observer design. The nonlinear section is assumed to satisfy the Lipschitz condition. Firstly, the normal observer is designed based on the known Lipschitz constant and the results are given in linear matrix inequality (LMI) form. Then a fairly simple adaptive observer is designed with the Lipschitz constant unknown. Simulations on synchronizing the Lorenz system are investigated and the results show the validity and feasibility of our main results.     

Synchronization between two different chaotic systems with noise perturbation

This paper investigates the chaotic synchronization between the noise-perturbed Lorenz system and one of the noise-perturbed Chen and Lu¨ systems.Based on the active control method and the Lyapunov theory in stochastic differential equations,sufficient conditions for the stability of the error dynamics are derived.Numerical simulations are also shown to demonstrate the effectiveness of these theoretic results.     

一类不确定混沌系统的观测器同步

基于降维观测器的方法实现了一类具有外部扰动的混沌系统的同步. 无需知道系统外部扰动项的任何信息，就可对驱动系统设计基于降维观测器的响应系统，从而实现了混沌系统的同步. 数值仿真表明该方法是有效的. 关键词： 混沌同步 混沌系统 观测器     

Synchronization between two different noise-perturbed chaotic systems with unknown parameters

In this paper, a general method of synchronizing noise-perturbed chaotic systems with unknown parameters is proposed. Based on the LaSalle-type invariance principle for stochastic differential equations and by employing a combination of feedback control and adaptive control, some sufficient conditions of chaos synchronization between these noise-perturbed systems with unknown parameters are established. The model used in the research is the chaotic system, but the method is also applicable to the hyperchaotic systems. Unified system and noise-perturbed RSssler system, hyperchaotic Chen system and nolse-perturbed hyperchaotic RSssler system are taken for illustrative examples to demonstrate this technique.[第一段]     

Synchronization of chaotic fractional-order systems via active sliding mode controller

In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master-slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lü-Lü systems, synchronizing fractional order Chen-Chen systems and synchronizing fractional-order Lü-Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations.     

利用标量控制器实现一类混沌系统同步

研究了一类混沌系统的混沌同步，对此类混沌系统，通过设计一个合适标量控制器，可以实现系统的混沌同步.给出了该标量控制器设计的一般方法，并从理论上得到了混沌同步的充分和必要条件，且此充分和必要条件与混沌系统的性质无关. 关键词： 混沌系统 标量控制器 混沌同步