Boundedness and synchronization of y-coupled Lorenz systems with or without controllers

The purpose of this paper is to investigate the boundedness and synchronization of y-coupled Lorenz systems. When the coupling term is only added to the second variable, we call them y-coupled Lorenz systems. In this paper, we first prove the boundedness of y-coupled Lorenz systems, which ensures the existence and uniqueness of the solution when t→∞. Based on the boundedness, we prove that for y-coupled Lorenz systems, QUAD condition is satisfied. It should be pointed out that QUAD condition plays an essential role in the discussion of synchronization. Based on the boundedness and QUAD condition, we prove that if the coupling is strong enough, the y-coupled Lorenz systems can achieve the complete synchronization globally and exponentially.     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

基于线性分离的自治混沌系统的投影同步

分析了自治混沌系统的投影同步问题.基于线性系统的稳定判定准则，提出一种新的线性分离的同步方法，并采用该方法实现了Lorenz系统，Rssler系统和超混沌Chen系统的投影同步.数值模拟进一步验证了所提出方案的有效性. 关键词： 投影同步 自治混沌系统 线性分离     

Fuzzy modeling and synchronization of different memristor-based chaotic circuits

This Letter is concerned with the problem of fuzzy modeling and synchronization of memristor-based Lorenz circuits with memristor-based Chua?s circuits. In this Letter, a memristor-based Lorenz circuit is set up, and illustrated by phase portraits and Lyapunov exponents. Furthermore, a new fuzzy model of memristor-based Lorenz circuit is presented to simulate and synchronize with the memristor-based Chua?s circuit. Through this new fuzzy model, two main advantages can be obtained as: (1) only two linear subsystems are needed; (2) fuzzy synchronization of these two different chaotic circuits with different numbers of nonlinear terms can be achieved with only two sets of gain K. Finally, numerical simulations are used to illustrate the effectiveness of these obtained results.     

Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control

Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control is investigated.The chaotic systems are redesigned by using the generalized Hamiltonian systems and observer approach.Based on Lyapunov’s stability theory,linear and nonlinear feedback control of adaptive H∞ synchronization is established in order to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance on an H∞-norm constraint.Adaptive H∞ synchronization of chaotic systems via three kinds of control is investigated with applications to Lorenz and Chen systems.Numerical simulations are also given to identify theeffectiveness of the theoretical analysis.     

Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems

Scaling factor characterizes the synchronized dynamics of projective synchronization in partially linear chaotic systems but it is difficult to be estimated. To manipulate projective synchronization of chaotic systems in a favored way, a control algorithm is introduced to direct the scaling factor onto a desired value. The control approach is derived from the Lyapunov stability theory. It allows us to arbitrarily amplify or reduce the scale of the response of the slave system via a feedback control on the master system. In numerical experiments, we illustrate the application to the Lorenz system. (c) 2001 American Institute of Physics.     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

H∞ synchronization of chaotic systems using output feedback control design

This article investigates the H_∞ synchronization problem for a general class of chaotic systems. Based on Lyapunov theory, linear matrix inequality (LMI) and linear matrix equality (LME) formulation, the output feedback controller is established to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance to an H_∞-norm constraint. Two illustrative examples are provided to demonstrate the effectiveness of the developed theoretical results.     

Perfect synchronization of chaotic systems： a controllability perspective

This paper presents a synchronization method, motivated from the constructive controllability analysis, for two identical chaotic systems. This technique is applied to achieve perfect synchronization for Lorenz systems and coupled dynamo systems. It turns out that states of the drive system and the response system are synchronized within finite time, and the reaching time is independent of initial conditions, which can be specified in advance. In addition to the simultaneous synchronization, the response system is synchronized un-simultaneously to the drive system with different reaching time for each state. The performance of the resulting system is analytically quantified in the face of initial condition error, and with numerical experiments the proposed method is demonstrated to perform well.     

Stochastic linear generalized synchronization of chaotic systems via robust control

In this Letter, based on robust control, we provide a general theoretical result on stochastic linear generalized synchronization (GS) of chaotic systems. Given a driving system with noise perturbations and a linear synchronization function, a response system is developed easily according to the scheme derived here. By introducing the Lyapunov stability theory and linear matrix inequalities (LMIs), the condition for synchronization is proved to be effective. Finally, the Lorenz system is taken for illustration and verification.     

Synchronisation of fractional-order time delayed chaotic systems with ring connection

In this paper, synchronisation of fractional-order time delayed chaotic systems in ring networks is investigated. Based on Lyapunov stability theory, a new generic synchronisation criterion for N-coupled chaotic systems with time delay is proposed. The synchronisation scheme is applied to N-coupled fractional-order time delayed simplified Lorenz systems, and the Adomian decomposition method (ADM) is developed for solving these chaotic systems. Performance analysis of the synchronisation network is carried out. Numerical experiments demonstrate that synchronisation realises in both state variables and intermediate variables, which verifies the effectiveness of the proposed method.     

SC混沌比例投影同步方法在保密通信中的应用

利用基于线性稳定性准则的SC混沌比例投影同步方法,提出一种应用于保密通信的混沌掩盖方案.适当分离出混沌系统的线性项与非线性项,构造一个非线性驱动向量函数,混沌状态变量包含用于投影同步的比例因子,把所需传递的有用信息掩盖入其中一个分量上,得到混沌载波信号,提高加密信息的复杂度和解码的困难度.以Lorenz吸引子和超混沌Rössler吸引子为例进行数值仿真,详细分析传输的正弦信息加密解密全过程,给出简单、最优的混沌掩盖方案,数值分析证明比例投影同步方法应用于保密通信领域的有效性.     

Ergodicity of randomly perturbed Lorenz model

We show, by using the Liapunov method, that the Lorenz model perturbed by Gaussian white noise is ergodic for any Rayleigh number. Our theory confirms two properties which have been found by numerical calculation. We also discuss the ergodicity of some other randomly perturbed dissipative systems, a one-dimensional laser, and a homopolar disk dynamo model of the geomagnetism.     

Investigation of Q-S synchronization in coupled chaotic incommensurate fractional order systems

Chaos and synchronization in fractional order systems have received increasing attention in recent years. In this paper, the problem of Q-S synchronization for different dimensional incommensurate fractional order chaotic systems is investigated. Based on Laplace transform and stability theory of linear integer order differential systems, some synchronization schemes are designed to achieve Q-S synchronization between n-D and m-D incommensurate fractional order chaotic systems. Test problems and numerical simulations are used to show the effectiveness of the proposed approach.     

Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems

In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.     

A novel robust proportional-integral （PI） adaptive observer design for chaos synchronization

In this paper, chaos synchronization in the presence of parameter uncertainty, observer gain perturbation and exogenous input disturbance is considered. A nonlinear non-fragile proportional-integral (PI) adaptive observer is designed for the synchronization of chaotic systems; its stability conditions based on the Lyapunov technique are derived. The observer proportional and integral gains, by converting the conditions into linear matrix inequality (LMI), are optimally selected from solutions that satisfy the observer stability conditions such that the effect of disturbance on the synchronization error becomes minimized. To show the effectiveness of the proposed method, simulation results for the synchronization of a Lorenz chaotic system with unknown parameters in the presence of an exogenous input disturbance and abrupt gain perturbation are reported.     

Modified linear-nonlinear decomposition method for chaotic synchronization

This Letter proposes a general linear-nonlinear decomposition method for chaotic synchronization. The method expands the concept of chaotic synchronization based on the stability criterion of linear systems, proposed earlier. Retaining the basic idea of the standard linear-nonlinear decomposition method—the nonlinear part of a given chaotic system is used as a synchronization signal, hence the error system is always linear and allows precise stability analysis—the proposed method allows to design not only one, but a large number of coupling variants, thus offering the researcher the possibility to choose the best possible coupling variant for a given chaotic synchronization problem.     

Projective synchronization of spatiotemporal chaos in a weighted complex network

<正>Projective synchronization of a weighted complex network is studied in which nodes are spatiotemporal chaos systems and all nodes are coupled not with the nonlinear terms of the system but through a weighted connection.The range of the linear coefficient matrix of separated configuration,when the synchronization is implemented,is determined according to Lyapunov stability theory.It is found that projective synchronization can be realized for unidirectional star-connection even if the coupling strength between the nodes is a given arbitrary weight value.The Gray-Scott models having spatiotemporal chaos behaviours are taken as nodes in the weighted complex network,and simulation results of spatiotemporal synchronization show the effectiveness of the method.     

非线性耦合超混沌R(o)ssler系统和网络的同步

研究两个通过非线性函数对称耦合的超混沌Roessler系统的同步问题．通过对超混沌系统的线性项与非线性项的适当分离,构造一个特殊的非线性函数,作为耦合函数,发现在耦合强度α=0．5附近的一小段区域里存在稳定的超混沌同步现象．利用线性系统的稳定性分析准则和条件Lyapunov指数来检验同步状态的稳定性,并进一步研究了由多个超混沌Roessler系统单元通过非线性函数按照完全连接形式组成的网络的混沌同步问题。显示许多耦合单元组成的网络,满足同步稳定性的耦合强度的取值范围可以仅从2个单元组成的网络的参数取值范围估计到。此外发现耦合强度的值与耦合单元数量成反比,数值模拟结果证实所提出方法对超混沌系统和网络的混沌同步是有效的。     

Impulsive control of projective synchronization in chaotic systems

Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed.