The geometry of chaos synchronization

Chaos synchronization in coupled systems is often characterized by a map phi between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set--by which we mean graph(phi)--can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. In brief, synchronization sets can in general become nondifferentiable, and in the more severe case of noninvertible dynamics, they might even be multivalued. We suggest two different ways to quantify these features, and we discuss possible failures in detecting chaos synchrony using standard continuity-based methods when these features are present.     

Exponential networked synchronization of master-slave chaotic systems with time-varying communication topologies

The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.     

用滑模控制方法实现具有扇区非线性输入的主从混沌系统同步

分析了一个具有扇区非线性输入且含有参数不确定性以及外部噪声干扰的主从混沌系统的同 步控制问题.设计了一类同步滑模变结构控制器，并从理论上证明了该控制器的有效性.研究 表明该控制器不受受控系统的参数变化和噪声干扰的影响，具有很强的鲁棒性.最后通过对 主从Duffing_Holmes系统的仿真研究，验证了该控制器的有效性. 关键词： 混沌同步 滑模变结构控制 扇区非线性输入 不确定性     

两种调制方式下主从激光器同步性能的理论研究

 以单周期信号为例,数值模拟并讨论了注入频率和注入光场波动强度对主从式半导体激光器同步性能的影响。分别通过直接调制和外部调制两种方式获得单周期信号,并采用改进的参数——相似指数评价系统的同步性能。研究结果表明：当注入强度足够大时,系统实现注入锁定同步,外部调制的相似指数大于内部调制;注入强度恒定时,同步品质几乎不受调制指数的影响,而受调制频率的影响;由于内部、外部调制引起主激光器的输出反相,两种调制方式下同步品质随调制频率的变化趋势也相反。     

两种调制方式下主从激光器同步性能的理论研究

以单周期信号为例,数值模拟并讨论了注入频率和注入光场波动强度对主从式半导体激光器同步性能的影响。分别通过直接调制和外部调制两种方式获得单周期信号,并采用改进的参数——相似指数评价系统的同步性能。研究结果表明：当注入强度足够大时,系统实现注入锁定同步,外部调制的相似指数大于内部调制;注入强度恒定时,同步品质几乎不受调制指数的影响,而受调制频率的影响;由于内部、外部调制引起主激光器的输出反相,两种调制方式下同步品质随调制频率的变化趋势也相反。     

一类多时滞混沌系统的主从容错同步

研究了一类多时滞混沌系统的主从容错同步问题.所设计的同步方法无论混沌系统中是否有故障发生,都可以使混沌主从同步误差系统全局渐近稳定并且满足给定的性能指标.容错同步方法的控制器包含两个部分:状态反馈控制器和故障补偿器.控制器的存在条件是时滞依赖的并可以通过线性矩阵不等式的方法进行求解.最后通过仿真验证了设计方法的有效性.     

Adiabatic small noise fluctuations around anticipated synchronization: A perspective from scalar master-slave dynamics

In this paper we extract and highlight some essential ingredients and properties that characterize the phenomenon of anticipated synchronization when external additive noise sources perturb the master and slave dynamics. Our results rely on a minimal scalar setup able to exhibit the more fundamental aspects of this phenomenon, where the fluctuations around the average dynamics are worked out in a small noise and adiabatic approximations allowing to describe their dynamics through linear delay Langevin equations. In this context, we find necessary conditions that guarantee anticipated synchronization in the mean value. Fluctuations around this condition are studied through the stationary correlation of the delayed difference between the master and slave dynamics. It is shown that external noise properties can be inferred by measuring this object. Conditions for minimizing the dynamical fluctuations around the anticipated synchronization in mean value are found. A detailed analysis of the dependence on the characteristic parameters is presented.     

Impulsive control of time-delay systems using delayed impulse and its application to impulsive master-slave synchronization

This Letter is concerned with impulsive control of a class of nonlinear time-delay systems. Some uniform stability criteria for the closed-loop time-delay system under delayed impulsive control are derived by using piecewise Lyapunov functions. Then the criteria are applied to impulsive master-slave synchronization of some secure communication systems with transmission delays and sample delays under delayed impulsive control. Two numerical examples are given to illustrate the effectiveness of the derived results.     

On the chaos synchronization phenomena

Chaos synchronization is an important problem in the nonlinear science. However, several phenomena can be found in the synchronization systems. Here, we discuss several phenomena involved with the chaos synchronization problem. Between the involved phenomena, one can find: Complete, Practical and Partial Synchronization. A feedback controller is used to illustrate such synchronization phenomena. The feedback was recently reported and involves robustness features. Such control actions can induce one more phenomena: the Almost Synchronization (AS). In addition, it is shown that the AS can be found if the master and slave models are strictly different.     

基于微分几何理论的参数失配系统时空同步研究

本文采用状态反馈精确线性化方法,实现了参数失配系统的时空同步控制. 引入微分几何理论,首先采用李导数判定了参数失配系统的仿射模型是否满足状态精确线性化的充要条件,通过非线性坐标变换得到了其线性解耦系统,并结合线性最优控制理论确定了同步控制律,数值仿真证实了该控制方法的有效性,从非线性控制角度探索了混沌系统时空同步的新方法. 关键词： 时空同步 微分几何 精确线性化 参数失配     

On the geometry of stiff knots

We analyse the geometry of a thin knotted string with bending rigidity. Two types of geometric properties are investigated. First, following the approach of von der Mosel [H. von der Mosel, Asymptotic Anal. 18, 49 (1998)], we derive upper bounds for the multiplicity of crossings and braids. Then, using a general inequality for the length of 3D curves derived by Chakerian [G.D. Chakerian, Proc. of the American Math. Soc. 15, 886 (1964)], we analyze the size and confinement of a knot     

On the geometry of coset branes

Coset models and their symmetry preserving branes are studied from a representation theoretic perspective, relating e.g. the horizontal branching spaces to a truncation of the space of bulk fields, and accounting for field identification. This allows us to describe the fuzzy geometry of the branes at finite level.     

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P$P$, there are two approaches to its geometric quantization: one involves a circle bundle Q$Q$ over P$P$ endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P$P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P$P$ is obtained from the Lie groupoid of Q$Q$ via an S¹$S^1$ reduction that preserves both the Lie groupoid and the geometric structures.     

On the geometry of isomonodromic deformations

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇)$(V,\nabla )$ of vector bundles and connections as being obtained by “twists” supported over points of a fixed vector bundle _V0$V_0$ with a fixed connection _∇0${\nabla }_0$; this gives two deformations, one, isomonodromic, of (V,∇)$(V,\nabla )$, and another induced from the isomonodromic deformation of (_V0,_∇0)$(V_0,{\nabla }_0)$. The difference between the two will be Hamiltonian.     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

On the geometry of Lorentz orbit spaces

The orbit space of the Lorentz group acting on the product ofn real, or complex, Minkowski spaces is stratified into subspaces isomorphic to certain products of Grassmann manifolds and varieties of Gram matrices. The Lorentz orbits (of nonzero dimension) are completely classified by the Stiefel manifolds of standard orthogonal bases for the linear subspaces of the Minkowski space. Several representations of the spaces ofn-point Lorentz invariant distributions and differentiable, or analytic, functions onto appropriate spaces of distributions and functions of Lorentz invariant variables are also discussed.On leave of absence from the Institute of Atomic Physics, Bucharest, Romania.     

On the geometry of relativistic magnetofluid flows

This paper deals with the construction of “magnetic vorticity” vector using Greenberg's theory of spacelike congruences for the trajectories of magnetic fields. A set of propagation equations is derived for the geometrical invariants associated with the congruences of magnetic field lines and fluid flow lines. Some applications of these propagation equations are made. A generalization of Ferraro's law of isorotation is obtained employing the propagation equation forω ² along the magnetic field lines.     

On the local geometry of rotating matter

This paper shows that general relativity and ordinary continuum models of matter imply the presence of Cartan torsion. The key concept is that torsion can be viewed as translational holonomy per unit area, in the limit of very small areas. Translational holonomy is introduced as the nonclosure of the development of a space-time loop into a flat space-time. The translational holonomy around a charged rotating black hole is calculated. If a large collection of small rotating objects is approximated by continuous spinning matter, the resulting torsion and spin have the same relation as in Einstein-Cartan theory, except that the torsion traces remain zero for the simple model of spinning matter used here. Finally, this construction adds torsion to the list of nonpropagating fields which can be viewed as continuum density of holonomy around localized space-time boundaries, or around throats which are connected to further local topological structures.