Relative Ding Projective Modules over Formal Triangular Matrix Rings

Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.

Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system

Projective synchronization and generalized projective synchronization have recently been observed in the coupled chaotic systems. In this paper, a new synchronization, called ＂generalized projective synchronization＂, is reported in the chaotic Lorenz system and the chaotic Chen one.

Function projective synchronization in partially linear drive-response chaotic systems

This paper gives the definition of function projective synchronization with less conservative demand for a scaling function, and investigates the function projective synchronization in partially linear drive-response chaotic systems. Based on the Lyapunov stability theory, it has been shown that the function projective synchronization with desired scaling function can be realized by simple control law. Moreover it does not need scaling function to be differentiable, bounded and non-vanished. The numerical simulations are provided to verify the theoretical result.

A general method for modified function projective synchronization

This paper investigates the modified function projective synchronization,which means that the drive system and the response system are synchronized up to a desired scale matrix of function. By the active control scheme,a general method for modified function projective synchronization is proposed. Numerical simulations on chaotic Rssler system and hyper-chaotic Chen system are presented to verify the effectiveness of the proposed scheme.

Projective Changes between Generalized (<i>α</i>, <i>β</i>)-Metric and Randers Metric

Projective change between two Finsler metrics arises from Information Geom-etry. Such metrics have special geometric properties and will play an important role in Finsler geometry. The purpose of the present paper is to find a relation to characterize the projective change between generalized (α, β) - metric ( μ1, μ2 and μ3 ≠ 0 are constants) and Randers metric , where α and are two Riemannian metrics, β and are 1-forms. Further, we study such projective change when generalized (α, β) -metric F has some curvature property.

Generalized projective synchronization of chaotic systems via adaptive learning control

In this paper, a learning control approach is applied to the generalized projective synchronisation （GPS） of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov--Krasovskii functional stability theory, a differential-difference mixed parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronised. The scheme is successfully applied to the generalized projective synchronisation between the Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.

Generalized projective synchronization of a class of chaotic （hyperchaotic） systems with uncertain parameters

In this paper is investigated the generalized projective synchronization of a class of chaotic （or hyperchaotic） systems, in which certain parameters can be separated from uncertain parameters. Based on the adaptive technique, the globally generalized projective synchronization of two identical chaotic （hyperchaotic） systems is achieved by designing a novel nonlinear controller. Furthermore, the parameter identification is realized simultaneously. A sufficient condition for the globally projective synchronization is obtained. Finally, by taking the hyperchaotic L system as example, some numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed technique.

Projective synchronization in coupled fractional order chaotic Rossler system and its control

This paper proposes a method to achieve projective synchronization of the fractional order chaotic Rossler system. First, construct the fractional order Rossler system＇s corresponding approximate integer order system, then a control method based on a partially linear decomposition and negative feedback of state errors is utilized on the new integer order system. Mathematic analyses prove the feasibility and the numerical simulations show the effectiveness of the proposed method.

Controlling projective synchronization coupled chaotic systems

In this paper, a new method for controlling projective synchronization in coupled chaotic systems is presented. The control method is based on a partially linear decomposition and negative feedback of state errors. Firstly, the synchronizability of the proposed projective synchronization control method is proved mathematically. Then, three different representative examples are discussed to verify the correctness and effectiveness of the proposed control method.

Projective synchronization of a complex network with different fractional order chaos nodes

Based on the stability theory of the linear fractional order system, projective synchronization of a complex network is studied in the paper, and the coupling functions of the connected nodes are identified. With this method, the projective synchronization of the network with different fractional order chaos nodes can be achieved, besides, the number of the nodes does not affect the stability of the whole network. In the numerical simulations, the chaotic fractional order Lu system, Liu system and Coullet system are chosen as examples to show the effectiveness of the scheme.

Robust modified projective synchronization of fractional-order chaotic systems with parameters perturbation and external disturbance

Based on fractional-order Lyapunov stability theory, this paper provides a novel method to achieve robust modified projective synchronization of two uncertain fractional-order chaotic systems with external disturbance. Simulation of the fractional-order Lorenz chaotic system and fractional-order Chen＇s chaotic system with both parameters uncertainty and external disturbance show the applicability and the efficiency of the proposed scheme.

Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable

In this paper, the modified projective synchronization （MPS） of a fractional-order hyperchaotic system is inves- tigated. We design the response system corresponding to the drive system on the basis of projective synchronization theory, and determine the sufficient condition for the synchronization of the drive system and the response system based on fractional-order stability theory. The MPS of a fractional-order hyperchaotic system is achieved by transmitting a single variable. This scheme reduces the information transmission in order to achieve the synchronization, and extends the applicable scope of MPS. Numerical simulations further demonstrate the feasibility and the effectiveness of the proposed scheme.

Adaptive projective synchronization of unified chaotic systems and its application to secure communication

In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor between two coupled unified chaotic systems to a desired value, based on the invarianee principle of differential equations. Under this control strategy, one can arbitrarily select the scaling factor. Numerical simulations are given to support the effectiveness of the proposed method and show the robustness against noise. Furthermore, a secure communication scheme based on the adaptive projective synchronization of unified chaotic systems is presented and numerical simulation shows its feasibility.

Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper. Based on Lyapunov stability theory and Barbalat＇s lemma, generalized matrix projective lag synchronization criteria are derived by using the adaptive control method. Furthermore, 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. In addition, numerical simulation results are presented to illustrate the effectiveness of this method, showing that the synchronization speed is sensitively influenced by the adaptive law strength, the network size, and the network topological structure.

Function projective lag synchronization of fractional-order chaotic systems

Function projective lag synchronization of different structural fractional-order chaotic systems is investigated. It is shown that the slave system can be synchronized with the past states of the driver up to a scaling function matrix. According to the stability theorem of linear fractional-order systems, a nonlinear fractional-order controller is designed for the synchronization of systems with the same and different dimensions. Especially, for two different dimensional systems, the synchronization is achieved in both reduced and increased dimensions. Three kinds of numerical examples are presented to illustrate the effectiveness of the scheme.

PROJECTIVELY FLAT MATSUMOTO METRIC AND ITS APPROXIMATION

Abstract In this article, the author studies the projectively flat Matsumoto metric F=α^2/（α -β）, where α=√αijy^iy^j is a Riemannian metric and β =biy^i is 1-form. Theyconclude that α is locally projectively fiat and β is paralled with respect to α. And get the same result for the higher order approximate Matsumoto metric.

Generalized projective synchronization of the fractional-order chaotic system using adaptive fuzzy sliding mode control

An adaptive fuzzy sliding mode strategy is developed for the generalized projective synchronization of a fractional- order chaotic system, where the slave system is not necessarily known in advance. Based on the designed adaptive update laws and the linear feedback method, the adaptive fuzzy sliding controllers are proposed via the fuzzy design, and the strength of the designed controllers can he adaptively adjusted according to the external disturbances. Based on the Lya- punov stability theorem, the stability and the robustness of the controlled system are proved theoretically. Numerical simu- lations further support the theoretical results of the paper and demonstrate the efficiency of the proposed method. Moreover, it is revealed that the proposed method allows us to manipulate arbitrarily the response dynamics of the slave system by adjusting the desired scaling factor λi and the desired translating factor ηi, which may be used in a channel-independent chaotic secure communication.

Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.

PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

This article characterizes projectively fiat Finsler metrics with almost isotropic S-curvature.

Generalized Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions

The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated.According to the stability theory of linear fractional order systems,a sufficient condition to realize synchronization is obtained.The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions.The corresponding numerical results coincide with theoretical analysis.