On analytical justification of phase synchronization in different chaotic systems

In analytical or numerical synchronizations studies of coupled chaotic systems the phase synchronizations have less considered in the leading literatures. This article is an attempt to find a sufficient analytical condition for stability of phase synchronization in some coupled chaotic systems. The method of nonlinear feedback function and the scheme of matrix measure have been used to justify this analytical stability, and tested numerically for the existence of the phase synchronization in some coupled chaotic systems.     

缓变系数动力系统的运动稳定性

本文举例说明“冻结”系统稳定性与原变系数系统稳定性之间的各种关系,用函数的方法给出了变系数动力系统运动稳定性的充分条件,用显式确定系数缓变的界限,非线性附加项的界限,带有时滞的系统的时滞的界限.     

变系数动力系统的运动稳定性

<正> §1 序言 随时间t变化的变系数动力系统的运动稳定性在工程和物理中是经常遇到的叫题,例如依靠尾翼来稳定的火箭弹在火箭推力起作用的过程中的运动状态;飞机的稳定与控制.     

Some simple global synchronization criterions for coupled time-varying chaotic systems

Based on the Lyapunov stabilization theory and matrix measure, this paper proposes some simple generic criterions of global chaos synchronization between two coupled time-varying chaotic systems from a unidirectional linear error feedback coupling approach. These simple criterions are applicable to some typical chaotic systems with different types of nonlinearity, such as the original Chua’s circuit and the Rössler chaotic system. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems’ global chaos synchronization.     

A stochastic version of center manifold theory

Summary Random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied. In this situation Lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system. If at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold. The asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold. A dynamical characterization of the stochastic center manifold is given and approximation results are proved.     

线性系统的极限跟踪性

给出了Rⁿ上的线性同构和线性流具有极限跟踪性的特征：线性同构具有极限跟踪性当且仅当其对应的矩阵为双曲的;线性流具有极限跟踪性当且仅当其对应矩阵的所有特征根均具有非零实部.     

Stability criterion for synchronization of linearly coupled unified chaotic systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems. A new stability criterion for asymptotic synchronization is attained using the Lyapunov stability theory and linear matrix inequality (LMI) approach. A numerical example is given to illuminate the design procedure and advantage of the result derived.     

Exponential estimates for singularly perturbed linear fuctional differential equations

Exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts. The exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations. Differences between differential-difference equations and equations with distributed delays are emphasized.     

Analysis and design of anti-controlled higher-dimensional hyperchaotic systems via Lyapunov-exponent generating algorithms

Based on Lyapunov-exponent generation and the Gram-Schimdt orthogonalization, analysis and design of some anti-controlled higher-dimensional hyperchaotic systems are investigated in this paper. First, some theoretical results for Lyapunov-exponent generating algorithms are proposed. Then, the relationship between the number of Lyapunov exponents and the number of positive real parts of the eigenvalues of the Jacobi matrix is qualitatively described and analyzed. By configuring as many as possible positive real parts of the Jacobian eigenvalues, a simple anti-controller of the form $b\sin (\sigma x)$ for higher-dimensional linear systems is designed, so that the controlled systems can be hyperchaotic with multiple positive Lyapunov exponents. Utilizing the above property, one can resolve the positive Lyapunov exponents allocation problem by purposefully designing the number of positive real parts of the corresponding eigenvalues. Two examples of such anti-controlled higher-dimensional hyperchaotic systems are given for demonstration.     

Multifrequency oscillations of singularly perturbed systems

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts.     

Synchronization conditions for chaotic nonlinear continuous neural networks

This paper deals with the synchronization problem of a class of chaotic nonlinear neural networks. A feedback control gain matrix is derived to achieve the state synchronization of two identical nonlinear neural networks by using the Lyapunov stability theory, and the obtained criterion condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. The new sufficient condition can avoid solving an algebraic Riccati equation. The results are illustrated through one numerical example.     

Stability of T. Chan＇s Preconditioner from Numerical Range

A matrix is said to be stable if the real parts of all the eigenvalues are negative. In this paper, for any matrix An, we discuss the stability properties of T. Chan＇s preconditioner cU（An） from the viewpoint of the numerical range. An application in numerical ODEs is also given.     

用Ляпунов-Popov 方法估计电力系统的暂态稳定域

在电力系统暂态稳定问题的研究中,应用直接法与惯用的逐步计算法相比较,有其优越之处:能够提供稳定度指标,能够快速提供临界的重合闸时间的估计.而不必计算各种假定重合闸时间下的摇摆曲线,比较适合于在线的动态安全监控.因此,十几年来,稳定性理论在电力系统中的应用已引起国内外学者的注意,现已有不少研究工作.七十年代以来,现代非线性控制理论的发展,提供了一类构造函数的方法.这一方法首先由 Pai 及 Willems 应用于电力系统的稳定分析中.     

Tensor logarithmic norm and its applications

Matrix logarithmic norm is an important quantity, which characterize the stability of linear dynamical systems. We propose the logarithmic norms for tensors and tensor pairs, and extend some classical results from the matrix case. Moreover, the explicit forms of several tensor logarithmic norms and semi‐norms are also derived. Employing the tensor logarithmic norms, we bound the real parts of all the eigenvalues of a complex tensor and study the stability of a class of nonlinear dynamical systems. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of business cycles in Asada’s two-regional model

A two-regional five-dimensional model describing the development of income, capital stock and money stock, which was introduced by Asada (2004) [2] is analysed. Sufficient conditions for the existence of one pair of purely imaginary eigenvalues and three eigenvalues with negative real parts in the linear approximation matrix of the model are found. Formulae for the calculation of the bifurcation coefficients of the model are derived. The theorem on the existence of business cycles is presented. A numerical example illustrating the gained results is given.     

Master–slave synchronization of continuously and intermittently coupled sampled-data chaotic oscillators

In this paper, we consider the problem of synchronizing a master–slave chaotic system in the sampled-data setting. We consider both the intermittent coupling and continuous coupling cases. We use an Euler approximation technique to discretize a continuous-time chaotic oscillator containing a continuous nonlinear function. Next, we formulate the problem of global asymptotic synchronization of the sampled-data master–slave chaotic system as equivalent to the states of a corresponding error system asymptotically converging to zero for arbitrary initial conditions. We begin by developing a pulse-based intermittent control strategy for chaos synchronization. Using the discrete-time Lyapunov stability theory and the linear matrix inequality (LMI) framework, we construct a state feedback periodic pulse control law which yields global asymptotic synchronization of the sampled-data master–slave chaotic system for arbitrary initial conditions. We obtain a continuously coupled sampled-data feedback control law as a special case of the pulse-based feedback control. Finally, we provide experimental validation of our results by implementing, on a set of microcontrollers endowed with RF communication capability, a sampled-data master–slave chaotic system based on Chua’s circuit.     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical results.     

Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach

Based on stability theory of impulsive differential equation and new comparison theory of impulsive differential system, we study the chaos impulsive synchronization of two coupled chaotic systems using the unidirectional linear error feedback scheme. Some generic conditions of chaos impulsive synchronization of two coupled chaotic systems are derived, and to apply the conditions to typical chaotic system––the original Chua’s circuit. The example illustrates the effectiveness of the proposed result.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

Robust stability of impulsive synchronization in hyperchaotic systems

In this paper the impulsive synchronization of general continuous chaotic and hyperchaotic systems is investigated. The robust stability of the synchronization method is examined in the presence of uncertainties both on linear and nonlinear parts of the system dynamics and the channel noise. Conditions on the impulse distances are derived for different cases. Numerical simulations are presented to show the effectiveness of the method.