Generalized Exact Boundary Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions. A necessary and sufficient condition for the generalized exact boundary synchronization is obtained, and some results for its generalized exactly synchronizable states are given.     

Induced generalized exact boundary synchronizations for a coupled system of wave equations

Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.     

Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations

For a coupled system of wave equations with Dirichlet boundary controls, this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.     

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.     

一类波动方程耦合系统具Dirichlet边界控制的部分精确边界同步性

作者对具有Dirichlet边界控制的一类波动方程耦合系统提出了部分精确边界同步性的概念,并对其进行了深入的讨论,得到了实现部分精确边界同步性的充要条件.同时对部分同步态进行了讨论,得到了部分同步态不依赖于边界控制的一个充分条件.     

The approximately synchronizable state for a kind of coupled system of wave equations

This paper deals with the approximately synchronizable state by groups for a kind of coupled system of wave equations with Dirichlet boundary controls. So far, the approximate boundary synchronization for a kind of coupled system of wave equations has already been deeply studied; however, the study on the approximately synchronizable state still needs to be done in details. In this paper, we will give some results on the determination of approximately synchronizable state by groups and the attainable set of them.     

一维线性波动方程耦合组的精确边界同步能观性

作者介绍了多种精确同步能观性,并对一维波动方程耦合组在多种边界条件下分别实现了精确边界同步能观性,分组精确边界同步能观性以及分组精确边界零能观性与同步能观性.     

Exact Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.     

A robust method for new fractional hybrid chaos synchronization

In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer‐order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used. Copyright © 2016 John Wiley & Sons, Ltd.     

连续动力系统的广义同步

讨论连续的混沌动力系统之间的广义同步．利用Liapunov稳定性理论,通过构造适当的耦合项,得到了一个关于驱动响应系统广义同步的充分条件．并通过对两个例子的数字模拟,说明了充分条件的有效性．     

Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.     

含有不确定项的混沌系统自适应广义同步

针对一类含有不确定项的混沌系统,设计了广义同步响应系统,利用系统稳定性理论设计了自适应广义同步控制器及自适应率,实现了驱动系统和所设计的响应系统广义同步,所设计的控制策略对外界干扰有较强的鲁棒性,而且通过引入加速因子,可任意配置同步响应速度,具有较高的应用价值,理论分析及仿真结果验证了该方法的有效性。     

离散系统的广义同步

讨论两个离散系统之间的广义同步．通过构造合适的非线性耦合项,导出了驱动响应系统获得广义同步的充分条件．在一个正不变的有界集上,许多混沌映射满足这些充分条件．通过3个例子,说明了充分条件的有效性．     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.     

Local Exact Boundary Synchronization for a Kind of First Order Quasilinear Hyperbolic Systems

In this paper, the synchronization for a kind of first order quasilinear hyperbolic system is taken into account. In this system, all the equations share the same positive wave speed. To realize the synchronization, a uniform constructive method is adopted, rather than an iteration process usually used in dealing with nonlinear systems. Furthermore, similar results on the exact boundary synchronization by groups can be obtained for a kind of first order quasilinear hyperbolic system of equations with different positive wave speeds by groups.     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2014 John Wiley & Sons, Ltd.     

Synchronization analysis of complex‐Variable chaotic systems with discontinuous unidirectional coupling

This article is concerned with the problem of synchronization between two uncertain complex‐variable chaotic systems with parameters perturbation and discontinuous unidirectional coupling. Based on the stability theory and comparison theorem of differential equations, some sufficient conditions for the complete synchronization and generalized synchronization are obtained. The theoretical results show that the two uncertain complex‐variable chaotic systems with discontinuous unidirectional coupling can achieve synchronization if the time‐average coupling strength is large enough. Finally, numerical examples are examined to illustrate the feasibility and effectiveness of the analytical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 343–355, 2016     

Some criteria for the global finite-time synchronization of two Lorenz–Stenflo systems coupled by a new controller

This paper investigates the global finite-time synchronization of two chaotic Lorenz–Stenflo systems coupled by a new controller called the generalized variable substitution controller. First of all, the generalized variable substitution controller is designed to establish the master–slave finite-time synchronization scheme for the Lorenz–Stenflo systems. And then, based on the finite-time stability theory, a sufficient criterion on the finite-time synchronization of this scheme is rigorously verified in the form of matrix and the corresponding estimation for the synchronization time is analytically given. Applying this criterion, some sufficient finite-time synchronization criteria under various generalized variable substitution controllers are further derived in the algebraic form. Finally, some numerical examples are introduced to compare the results proposed in this paper with those proposed in the existing literature, verifying the effectiveness of the criteria obtained.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.