Synchronizing chaotic systems with positive conditional Lyapunov exponents by using convex combinations of the drive and response systems

We introduce a new method for synchronizing chaotic systems with positive conditional Lyapunov exponents, i.e., systems that do not synchronize in the Pecora-Carroll sense. This method works by considering a convex combination of the drive and response systems as a new driving signal. In this combination, the compoent associated with the response system acts as a chaos suppression method stabilizing the dynamics of the response system. This allows the chaotic component from the drive signal to synchronize both systems. The method is applied to synchronize some connections of the Rössler, Lorenz and van der Pol-Duffing systems that do not synchronize using the Pecora-Carroll scheme.     

Chaotic Synchronization of the Master Slave Chaotic Systems with Different Structures Based on BANG-BANG Control Principle

We propose a Bang-Bang control scheme that can synchronize master-slave chaotic systems. The chaotic systems considered here are structurally different from each other. Different from some control strategies reported previously, the scheme proposed here can be taken as a general one that is independent of the chaotic system itself.     

Chaotic synchronization of coupled ergodic maps

With few exceptions, studies of chaotic synchronization have focused on dissipative chaos. Though less well known, chaotic systems that lack dissipation may also synchronize. Motivated by an application in communication systems, we couple a family of ergodic maps on the N-torus and study the global stability of the synchronous state. While most trajectories synchronize at some time, there is a measure zero set that never synchronizes. We give explicit examples of these asynchronous orbits in dimensions two and four. On more typical trajectories, the synchronization error reaches arbitrarily small values and, in practice, converges. In dimension two we derive bounds on the average synchronization time for trajectories resulting from randomly chosen initial conditions. Numerical experiments suggest similar bounds exist in higher dimensions as well. Adding noise to the coupling signal destroys the invariance of the synchronous state and causes typical trajectories to desynchronize. We propose a modification of the standard coupling scheme that corrects this problem resulting in robust synchronization in the presence of noise.     

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.     

阶次不等的分数阶混沌系统同步

针对阶次不等的分数阶混沌系统同步问题,提出了一种将不等阶分数阶系统转化为等阶系统的方法,将不等阶分数阶系统的同步问题转化为等阶的异结构同步问题.利用该方法实现了阶次不等的分数阶Lorenz混沌系统的同步,数值仿真结果验证了该理论的正确性. 关键词： 不等阶 分数阶混沌系统 同步 异结构     

Synchronization and emergence in complex systems

We show how novel behaviour can emerge in complex systems at the global level through synchronization of the activities of their constituent units. Two mechanisms are suggested for the emergence, namely non-diffusive coupling and time delays. In this way, simple units can synchronize to display complex dynamics, or conversely, simple dynamics may arise from complex constituents.     

基于收缩映射的奇异非混沌系统同步

提出一种基于收缩映射的奇异非混沌系统同步方案．通过利用一种混沌系统驱动另一种混沌系统产生出奇异非混沌吸引子,由于奇异非混沌吸引子的Ｌｙａｐｕｎｏｖ指数为负值,因而可有效抑制混沌系统对初始状态的敏感程度．为实现两个奇异非混沌吸引子的同步,文中采用收缩映射实现混沌驱动系统的快速同步．研究表明,该方案能够快速实现同步,并且有较强的鲁棒性,易于实现,可用于混沌保密通信 关键词：     

激光混沌并联同步及其在全光逻辑门中的应用研究

提出多量子阱激光器混沌“主-从-响应”式结构同步系统,研究其并联同步在光学逻辑门中的应用. 利用一个注入多量子阱激光器混沌系统注入驱动实现了两个响应多量子阱激光系统的混沌并联同步,同时还获得了“主-从”式结构的混沌同步. 基于响应子系统的混沌并联同步思想,提出了全光逻辑门的基本理论模型并定义了计算原则与方法. 利用光的外部调制方法对两个驱动光进行调制与控制,让两个响应子系统实现同步与非同步,使系统获得了并具有全光逻辑门函数功能与特点,并成功地进行了数字逻辑计算. 具体提出了全光XNOR、NOR、NOT等逻辑门及逻辑计算方法,数值模拟结果证明了系统方案的可行性. 关键词： 混沌 同步 逻辑门 多量子激光器     

Unraveling the primary mechanisms leading to synchronization response in dissimilar oscillators

We study how the phenomenon of response to synchronization arises in sets of pulse-coupled dissimilar oscillators. One of the sets is constituted by oscillators that can easily synchronize. Conversely, the oscillators of the other set do not synchronize. When the elements of the first set are not synchronized, they induce oscillation death in the constituents of the second set. By contrast, when synchronization is achieved in oscillators of the first set, those of the second set recover their oscillatory behavior and thus, responding to synchronization. Additionally, we found another interesting phenomenon in this type of systems, namely, a new control of simultaneous firings in a population of similar oscillators attained by means of the action of a dissimilar oscillator.     

Synchronizing spatiotemporal chaos

We show analytically and numerically that a pair of uni-directionally coupled spatially extended systems can synchronize. For the case of partial differential equations the synchronization can be achieved by applying the scalar driving signals only at finite number of space points. Our approach is very general and can be useful for practical applications since the synchronization is achieved via feeding in the response system only the information from certain (discrete) spatial locations of the drive system. We also stress some open problems in the field of synchronization of spatiotemporal chaos. (c) 1997 American Institute of Physics.     

General form of coupling leading to synchronization of oscillating dynamical systems

There are numerous examples in nature where oscillating or pulsing physical and biological systems automatically synchronize. Here we pose the question: Is there a general type of coupling that leads to such synchronization? It is demonstrated on numerous examples that dissipative coupling results in synchronization for a variety of systems.     

Adaptive synchronization of a class of continuous chaotic systems with uncertain parameters

This Letter investigates the problem of synchronization between two different chaotic systems. A generic adaptive synchronization scheme is proposed for a class of chaotic systems with uncertain parameters. Based on the Lyapunov stability theorem, an adaptive controller and the corresponding parameters update laws are designed to synchronize two chaotic systems. Numerical simulations are also given to show the effectiveness of the proposed method.     

Synchronization of noisy systems by stochastic signals

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.     

Projective synchronization of chaotic systems with bidirectional nonlinear coupling

This paper presents a new scheme for constructing bidirectional nonlinear coupled chaotic systems which synchronize projectively. Conditions necessary for projective synchronization (PS) of two bidirectionally coupled chaotic systems are derived using Lyapunov stability theory. The proposed PS scheme is discussed by taking as examples the so-called unified chaotic model, the Lorenz–Stenflo system and the nonautonomous chaotic Van der Pol oscillator. Numerical simulation results are presented to show the efficiency of the proposed synchronization scheme.     

Detection of Mechanism of Noise-Induced Synchronization between Two Identical Uncoupled Neurons

We investigate the noise-induced synchronization between two identical uncoupled Hodgkin-Huxley neurons with sinusoidal stimulations. The numerical results confirm that the value of critical noise intensity for synchronizing two systems is much less than the magnitude of mean size of the attractor in the original system, and the deterministic feature of the attractor in the original system remains unchanged. This finding is significantly different from the previous work [Phys. Rev. E 67 （2003） 027201] in which the value of the critical noise intensity for synchronizing two systems was found to be roughly equal to the magnitude of mean size of the attractor in the original system, and at this intensity, the noise swamps the qualitative structure of the attractor in the original deterministic systems to synchronize to their stochastic dynamics. Further investigation shows that the critical noise intensity for synchronizing two neurons induced by noise may be related to the structure of interspike intervals of the original systems.     

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.     

Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller

This paper provides a novel method to synchronize uncertain fractional-order chaotic systems with external disturbance via fractional terminal sliding mode control. Based on Lyapunov stability theory, a new fractional-order switching manifold is proposed, and in order to ensure the occurrence of sliding motion in finite time, a corresponding sliding mode control law is designed. The proposed control scheme is applied to synchronize the fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with uncertainty and external disturbance parameters. The simulation results show the applicability and efficiency of the proposed scheme.     

On the geometry of master-slave synchronization

In 1990, Pecora and Carroll reported the observation that one can synchronize the orbits of two identical dynamical systems, which may be chaotic, by feeding state variables of one of them to the other one with no feedback, a phenomenon often called master-slave synchronization. We report here some results on the theory of master-slave synchronization for maps and flows, which are all inspired by a similar geometric and coordinate independent point of view to the one introduced in master-slave synchronization by Tresser, Worfolk, and Bass. Our results are variations on the theme that projection often can compensate for expansion.(c) 2002 American Institute of Physics.     

一种新的分段混沌系统的同步控制

利用状态观测器思想构造驱动和响应系统，实现了一个新近提出的分段线性混沌系统的同步控制，并且推导了两个不同初值的系统实现同步的充分条件.理论分析说明该方法对原系统结构没有特殊要求，无需求解Lyapunov函数，适合应用于分段混沌系统.仿真实验的良好结果验证了该算法的有效性，对于不同初值的分段系统均能实现快速同步.同时该算法适用于一类分段混沌系统的同步控制. 关键词： 分段线性混沌系统 状态观测器 同步控制     

Synchronization of Chaotic Optical Systems by Remote Feedback Controlling

A dynamical variable feedback scheme is proposed to synchronize chaotic motion of two identical and nearly identical laser systems with an injected signal. The feedback synchronization method works well while the conventional variable-replacement synchronization methods are not realizable. Various practical situations for synchronizing two chaotic laser systems located distance away from each other are taken into account. The application of this synchronization approach to wide range of practical systems is emphasized.