Hyper Catastrophe on 4-Dimensional Canards

When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the “symmetry” is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on “hyper finite time line”, or done by using “non-standard analysis”, it is called “Hyper Catastrophe”. In the slow-fast system which includes a very small parameter , it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small , we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number N, which is nonstandard. As and the intervals are linked to use 1/N, the simulation should be done exactly.

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

为离开斜的无限的维的 Hamiltonian 操作符，它有至多可计算的特征值，在 Cauchy 主要价值的意义完全的特徵函数系统的一个必要、足够的条件被使用介绍光谱对称和操作符的新直角的关系。而且，上述结果被扩大到一个更一般的盒子。最后，为从各向同性的飞机 magnetoelectroelastic 固体产生的操作员的特徵函数系统的完全性被描述说明标准的有效性。整个结果为一些力学方程在 Hamiltonian 系统为变量的分离提供理论保证。

High temperature magnetic semiconductors: narrow band gaps and two-dimensional systems

Magnetic semiconductors have been demonstrated to work at low temperatures, but not yet at room temperature for spin electronic applications. In contrast to the p-type diluted magnetic semiconductors, n-type diluted magnetic semiconductors are few. Using a combined method of the density function theory and quantum Monte Carlo simulation, we briefly discuss the recent progress to obtain diluted magnetic semiconductors with both p- and n-type carriers by choosing host semiconductors with a narrow band gap. In addition, the recent progress on two-dimensional intrinsic magnetic semiconductors with possible room temperature ferromangetism and quantum anomalous Hall effect are also discussed.

Three-dimensional Motion of Multichain-Buoy Mooring System in shallow Water and Survival Conditions

An engineering numerical model for three dimensional motion of multichain-buoy mooring system in shallow water and survival condition is given in this paper. Shooting-aim method is employed for solving the dynamic equations of chain system in order to match the computation of buoy motion. The responses of buoy and chain have been computed for different wind-wave-current directions and different rigidity of chain. The results show that the present numerical model is reasonable.

Bifurcations of Periodic Orbits for a Four-Dimensional System

Consider a four-dimensional system having a two-dimensional invariant surface. By analyzing the solutions of bifurcation equations, this paper studied the bifurcation phenomena of a k multiple closed orbit in the invariant surface. Sufficient conditions for the existence of periodic orbits generated by the k multiple closed orbit were given.

The Structure of Reflective Function of Higher Dimensional Differential System

In this article, we discuss the structure of reflective function of the higher dimensional differential systems and apply the results to study the existence of periodic solutions of these systems.

An image encryption scheme based on three-dimensional Brownian motion and chaotic system

At present, many chaos-based image encryption algorithms have proved to be unsafe, few encryption schemes permute the plain images as three-dimensional（3D） bit matrices, and thus bits cannot move to any position, the movement range of bits are limited, and based on them, in this paper we present a novel image encryption algorithm based on 3D Brownian motion and chaotic systems. The architecture of confusion and diffusion is adopted. Firstly, the plain image is converted into a 3D bit matrix and split into sub blocks. Secondly, block confusion based on 3D Brownian motion（BCB3DBM）is proposed to permute the position of the bits within the sub blocks, and the direction of particle movement is generated by logistic-tent system（LTS）. Furthermore, block confusion based on position sequence group（BCBPSG） is introduced, a four-order memristive chaotic system is utilized to give random chaotic sequences, and the chaotic sequences are sorted and a position sequence group is chosen based on the plain image, then the sub blocks are confused. The proposed confusion strategy can change the positions of the bits and modify their weights, and effectively improve the statistical performance of the algorithm. Finally, a pixel level confusion is employed to enhance the encryption effect. The initial values and parameters of chaotic systems are produced by the SHA 256 hash function of the plain image. Simulation results and security analyses illustrate that our algorithm has excellent encryption performance in terms of security and speed.

Generation of transient chaos from two-dimensional systems via a switching approach

A new approach of generating transient chaos from two-dimensional（2D） continuous autonomous systems within finite time is presented.Based on an absolute-value switching law,the phenomenon of transient chaos takes place by switching between three 2D systems.Basic dynamic behavior of the systems is investigated.Numerical examples illustrate the validity of the results.

Generation of countless embedded trumpet-shaped chaotic attractors in two opposite directions from a new three-dimensional system with no equilibrium point

A new three-dimensional （3D） continuous autonomous system with one parameter and three quadratic terms is presented firstly in this paper. Countless embedded trumpet-shaped chaotic attractors in two opposite directions are generated from the system as time goes on. The basic dynamical behaviors of the strange chaotic system are investigated. Another more complex 3D system with the same capability of generating countless embedded trumpet-shaped chaotic attractors is also put forward.

Study of （2＋1）-Dimensional Higher-Order Broer-Kaup System

Painleve 性质和无限的对称(2+1 )-dimensionalhigher-order Broer—Kaup (HBK ) 系统在这篇论文被学习。用修改直接方法，我们导出一般对称组的定理到(2+1 ) 维的 HBK 系统。基于我们的定理，答案的一些新形式被获得。我们也发现守恒定律的无限的数字(2+1 ) 维的 HBK 系统。

Oscillatory and Asymptotic Behaviour of Solutions of Two Nonlinear Dimensional Difference Systems

This paper deals with the some oscillation criteria for the two dimensional difference system of the form: . Examples illustrating the results are inserted.

New Exact Solution of （N＋1）-Dimensional Burgers System

In this letter, using a Backlund transformation and the new variable separation approach, we find a new general solution of the (N+1)-dimensional Burgers system. The form of the universal formula obtained from many(2+1)-dimensional system is extended.

Design of Discrete-time Repetitive Control System Based on Two-dimensional Model

This paper presents a novel design method for discrete-time repetitive control systems （RCS） based on two-dimensional （2D） discrete-time model. Firstly, the 2D model of an RCS is established by considering both the control action and the learning action in RCS. Then, through constructing a 2D state feedback controller, the design problem of the RCS is converted to the design problem of a 2D system. Then, using 2D system theory and linear matrix inequality （LMI） method, stability criterion is derived for the system without and with uncertainties, respectively. Parameters of the system can be determined by solving the LMI of the stability criterion. Finally, numerical simulations validate the effectiveness of the proposed method.

Three-dimensional fuzzy logic system for process modeling and control

The traditional fuzzy logic system （FLS） can only model and control the process in two-dimensional nature. Many of real-world systems are of multidimensional features, such as, thermal and fluid processes with spatiotemporal dynamics, biological systems, or decision-making processes that contain stochastic and imprecise uncertainties. These types of systems are difficult for the traditional FLS to model and control because they require a third dimension for spatial or probabilistic information. The type-2 fuzzy set provides the possibility to develop a three-dimensional fuzzy logic system for modeling and controlling these processes in three-dimensional nature.

Higher-Dimensional Integrable Systems Arising from Motions of Curves on S^2(R)and S^3(R)

We show that higher-dimensional integrable systems including the (2+1)-dimensional generalized sine-Gordon equation and the (2+1)-dimensional complex mKdV equation are associated with motions of surfaces inducedby endowing with an extra space variable to the motions of curves on S^2(R) and S^3(R).

Guaranteed Cost Consensus for High-dimensional Multi-agent Systems With Time-varying Delays

Guaranteed cost consensus analysis and design problems for high-dimensional multi-agent systems with time varying delays are investigated. The idea of guaranteed cost con trol is introduced into consensus problems for high-dimensiona multi-agent systems with time-varying delays, where a cos function is defined based on state errors among neighboring agents and control inputs of all the agents. By the state space decomposition approach and the linear matrix inequality(LMI)sufficient conditions for guaranteed cost consensus and consensu alization are given. Moreover, a guaranteed cost upper bound o the cost function is determined. It should be mentioned that these LMI criteria are dependent on the change rate of time delays and the maximum time delay, the guaranteed cost upper bound is only dependent on the maximum time delay but independen of the Laplacian matrix. Finally, numerical simulations are given to demonstrate theoretical results.

Exotic interactions between solitons of the （2＋1）-dimensional asymmetric Nizhnik-Novikov-Veselov system

Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.

Novel four-dimensional autonomous chaotic system generating one-,two-,three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.