Switching among three different kinds of synchronization for delay chaotic systems

We found that the complete synchronization, anticipating synchronization and lag synchronization can be reached by the same kind of one way coupling for a large class of chaotic delay system. By changing the transformation time of the coupling signal we can switch from anticipating synchronization to complete synchronization, and then to lag synchronization. Numerical simulation for three chaotic delay systems were presented, one of them was novel which had two degree of freedoms, and the other two were the well known Ikeda and Mackey–Glass system which are one degree of freedom chaotic delay system. The theoretical analysis and the numerical simulation agreed perfect good.     

Synchronization in multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection

We present the first report of the synchronization regimes in both unidirectionally and bidirectionally coupled multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection. We derive the existence conditions and numerically study the stability for lag, complete, and anticipating synchronization regimes and cascaded synchronization. We also study in detail the effect of parameter mismatches and noise on the synchronization quality. It is emphasized that sensitivity of the synchronization quality to parameter mismatches can lead to a high level of security due to the difficulty in replicating the receiver laser. We show that the injection current and feedback delay times are highly important parameters from this point of view.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

Lag synchronizing chaotic system based on a single controller

Lag synchronization of chaotic system is investigated. Three kinds of schemes are proposed to lag synchronize Chen chaotic system. All the three schemes need only a single controller to realize lag synchronization. Especially in the last two schemes, only one state variable is contained in controller, which is of important significance on using chaos lag synchronization for applications. Finally numerical simulations are provided to show the effectiveness of the developed methods.     

Linear observer based projective synchronization in delay Rössler system

A new type of linear observer based projective, projective anticipating and projective lag synchronization of time-delayed Rössler system is studied. Along with this, the approach arbitrarily scales a drive system attractor and hence a similar chaotic attractor of any desired scale can be realized with the help of a synchronizing scaling factor. A scalar synchronizing output is considered where the output equation includes both the delay and non-delay terms of the nonlinear function. The condition for synchronization is derived analytically and the values of the coupling parameters are obtained. Analytical results are verified through numerical investigation and the effect of modulated time delay in the method is discussed. An important aspect of this method is that it does not require the computation of conditional Lyapunov exponents for the verification of synchronization.     

异结构离散型混沌系统的延迟同步

以异结构离散型混沌系统为研究对象,设计了一种延迟同步控制器实现了离散型Henon混沌系统和Ikeda混沌系统之间的同步控制．根据稳定性定理,确定了延迟同步控制器的结构以及系统状态变量之间的误差方程．设计的延迟同步控制器对于不同的离散型混沌系统具有统一的形式,可以实现任意异结构离散型混沌系统之间的延迟同步．数值仿真模拟进一步验证了该控制器的有效性．     

Integrable lattices

We propose a method for constructing integrable lattices starting from dynamic systems with two different parameterizations of the canonical variables and hence two independent Bäcklund flows. We construct integrable lattices corresponding to generalizations of the nonlinear Schrödinger equation. We discuss the Toda, Volterra, and Heisenberg models in detail. For these systems, as well as for the Landau-Lifshitz model, we obtain totally discrete Lagrangians. We also discuss the relation of these systems to the Hirota equations.     

Unified Orbital Description of the Envelope Dynamics in Two‐Dimensional Simple Periodic Lattices

The propagation of wave envelopes in two‐dimensional (2‐D) simple periodic lattices is studied. A discrete approximation, known as the tight‐binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2‐D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)‐type equations or coupled NLS‐type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.     

Synchronization effects in rotors partly filled with fluid

In fluid-filled rotors self-excited vibrations occur induced by a surface wave of the fluid. A characteristic property is the instability over the full range of angular velocity above the Eigenfrequency of the system. A possible explanation is the occurrence of synchronization effects between fluid and rotor. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable due to the complexity of fluid modelling. This paper analyses a simple minimal model and shows synchronization effects between fluid and rotor model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Various synchronization phenomena in bidirectionally coupled double scroll circuits

In this paper, the various cases of synchronization phenomena investigated in a system of two bidirectionally coupled double scroll circuits, were studied. Complete synchronization, inverse lag synchronization, and inverse π-lag synchronization are the observed synchronization phenomena, as the coupling factor is varied. The inverse lag synchronization phenomenon in mutually coupled identical oscillators is presented for the first time. As the coupling factor is increased, the system undergoes a transition from chaotic desynchronization to chaotic complete synchronization, while inverse lag synchronization and inverse π-lag synchronization are observed for greater values of the coupling factor, depending on the initial conditions of the state variables of the system. Inverse π-lag synchronization in coupled nonlinear oscillators is a special case of lag synchronization, which is also presented for the first time.     

Variational Bayes and the Reduced Dependence Approximation for the Autologistic Model on an Irregular Grid With Applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices, which are then suitably combined to estimate the constant for the entire lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and that may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios, enabling us to implement a practical and efficient nonsimulation-based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The online supplementary materials include our C++ source code for computing the approximate normalizing constant and simulation studies.     

Contrôlabilité asymptotique et synchronisation asymptotique dʼun système couplé dʼéquations des ondes avec des contrôles frontières de Dirichlet

In this Note, the asymptotic null controllability and various kinds of asymptotic synchronization, considered as some kinds of weakened controllability and synchronization, are introduced and studied for a coupled system of wave equations with Dirichlet boundary controls. Equivalent properties of weak observability are established.     

Lag synchronization of complex networks via pinning control

This paper mainly investigates the lag synchronization of complex networks via pinning control. Without assuming the symmetry and irreducibility of the coupling matrix, sufficient conditions of lag synchronization are obtained by adding controllers to a part of nodes. Particularly, the following two questions are solved: (1) How many controllers are needed to pin a coupled complex network to a homogeneous solution? (2) How should we distribute these controllers? Finally, a simple example is provided to demonstrate the effectiveness of the theory.     

Reciprocal Figures, Graphical Statics, and Inversive Geometry of the Schwarzian BKP Hierarchy

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles that constitute the simplest pair of reciprocal figures representing both a framework and a self‐stress encapsulate the integrable discrete BKP equation and its Schwarzian version. The inherent Möbius invariant nature of the Schwarzian BKP equation is then exploited to define reciprocity in an inversive geometric setting. Integrable pairs of lattices of nontrivial combinatorics consisting of reciprocal triangles and their natural generalizations are discussed. Particular reductions of these BKP lattices are related to the integrable discrete versions of Darboux's (2+1)‐dimensional sine‐Gordon equation and the classical Tzitzéica equation of affine geometry. Furthermore, it is shown that octahedral figures and their hexahedral reciprocals as considered by Maxwell likewise give rise to discrete integrable systems and associated integrable lattices.     

Lag synchronization of chaotic systems with parameter mismatches

The paper studies the effect of parameter mismatch on lag synchronization of chaotic systems. It shall be shown that lag synchronization of coupled systems may weakly achieve, when parameter mismatch is small. The error bound of lag synchronization arising from the parameter mismatch is also estimated by rigorously theoretical analysis. Numerical simulations on Chua oscillator are presented to verify the theoretical results.     

Finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay via periodically intermittent control

In this paper, the issue of finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay (CRDSTD) is considered. A periodically intermittent controller is designed such that drive system and corresponding response system can achieve finite-time lag synchronization. By using graph theory and Lyapunov method, two sufficient criteria are presented to guarantee the finite-time lag synchronization of CRDSTD. Moreover, the time of achieving lag synchronization of CRDSTD is estimated. Finally, a numerical example is given to show the effectiveness of the proposed results.     

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.     

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.     

Efficient option valuation using trees

An algorithm is proposed for the discrete approximation of continuous market price processes that uses trees instead of lattices. It is shown that it is convergent when used for pricing both European and American options and that it is more efficient, for some models, than the usual recombining schemes.