Synchronization of chaotic systems with different orders

A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state variables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.     

Lyndon-like and V-order factorizations of strings

We say a family of strings is an UMFF if every string has a unique maximal factorization over . Then is an UMFF iff and y non-empty imply . Let L-order denote lexicographic order. Danh and Daykin discovered V-order, B-order and T-order. Let R be L, V, B or T. Then we call r an R-word if it is strictly first in R-order among the cyclic permutations of r. The set of R-words form an UMFF. We show a large class of B-like UMFF. The well-known Lyndon factorization of Chen, Fox and Lyndon is the L case, and it motivated our work.     

Using Non‐Destructive Analysis Techniques to Examine the Artworks of Taiwanese Artist Cheng‐Po Chen (1895‐1947)

This study employed the non‐destructive techniques to analyze and examine the artworks of the Taiwanese artist Cheng‐Po Chen (1895‐1947) for disclosing the damage to Chen's artworks and determine his creation habits. Ten pieces of artworks that required restoration were examined using ultraviolet examination techniques. Based on the differences in their fluorescence characteristics, various parts of these works were shown to have been retouched, thus, original and retouched regions were distinguished. Examining these works using Infrared instrument revealed that Chen did not use charcoal to prepare drafts, and he directly applied oil paint to the canvases. Subsequently, these two examination techniques were adopted to examine sketches. The results showed that residues imprinted on them cause of storage methods were organic materials from charcoal and not carbon granules. The X‐ray examination identified numerous creations beneath the surface of several artworks. The X‐ray fluorescence spectrometer was used for elemental analysis of the pigments. The results corresponded with the pigments listed in Chen's purchase form, thus, we summarized the composition of palettes used by Chen. In general, this study acquired a considerable amount of information from the pigment examinations and established a database of the pigments that Chen commonly employed in his artworks.     

Adaptive fuzzy control‐based projective synchronization of uncertain nonaffine chaotic systems

This article aims to introduce a projective synchronization approach based on adaptive fuzzy control for a class of perturbed uncertain multivariable nonaffine chaotic systems. The fuzzy‐logic systems are employed to approximate online the uncertain functions. A Lyapunov approach is used to design the parameter adaptation laws and to demonstrate the boundedness of all signals of the closed‐loop system as well as the convergence of the synchronization errors to bounded residual sets. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of the proposed synchronization system based on fuzzy adaptive controller. © 2014 Wiley Periodicals, Inc. Complexity 21: 180–192, 2015     

基于非线性控制的超混沌Chen系统混沌同步

研究了基于非线性控制的超混沌Chen系统的混沌同步问题.基于Lyapunov稳定性理论，设计了非线性控制器，改进了Jiang和Huang等设计的同步误差系统的Lyapunov函数形式，理论证明了超混沌Chen系统的自同步和超混沌Chen系统与超混沌R?ssler系统的异结构同步.数值模拟进一步验证了所提出方案的有效性. 关键词： 超混沌Chen系统 自同步 异结构同步 非线性控制器     

超混沌Chen系统和超混沌Roessler系统的异结构同步

用两种不同的方法——主动控制同步法和自适应控制同步法实现超混沌Chen系统和超混沌Roessler系统的异结构同步，各自设计了不同的控制器，使得响应系统与驱动系统同步．当参数已知时，采用主动控制法，方法简单有效且不需要构造Lyapunov函数，实现同步的时间短；当系统参数未知或结构不确定时，基于Lyapunov稳定性理论，给出自适应同步控制器的系统设计过程和参数自适应律，使得系统达到同步同时识别未知参数，数值模拟验证了两种方法的有效性。     

Modified Projective Synchronization of a New Hyperchaotic System via Nonlinear Control

In this paper, a nonlinear control scheme of two identical hyperchaotic Chert systems is developed to realize their modified projective synchronization. We achieve modified projective synchronization between the two identical hyperchaotic systems by directing the scaling factor onto the desired value. With symbolic computation system Maple and Lyapunov stability theory, numerical simulations are given to perform the process of the synchronization.     

伪欧氏空间的伪球面子流形

将子流形的位置向量分解成水平分量和垂直分量，运用活动标架法研究伪欧氏空间的伪球面子流形。得到紧致类空子流形是伪球面子流形的两个充要条件：1)子流形的支撑函数是常值函数；2)子流形位置向量的水平分量是调和的。同时，给出具有平行平均曲率向量场的子流形是伪球面子流形的一个充要条件。特别地，对于Chen子流形，若它具有非迷向的平行平均曲率向量场且其支撑函数有固定符号，则它是伪球面子流形。     

N-lines and chromium-pairs in the luminescence spectra of the spinels ZnAl2O4:Cr3+ and MgAl2O4:Cr3+

It is shown that the N-lines in the luminescence spectra of the two spinels ZnAl₂O₄:Cr³⁺ and MgAl₂O₄:Cr³⁺ exhibit quite similar dependencies on chromium concentration, excitation frequency, and thermal treatment of the samples. While most of these lines are structure dependent, the line N₄ at $\text{ν}{}_{\mathrm{R}}\text{?}\text{ν}\text{≈ 400}\text{cm}{}^{\mathrm{?1}}$ and two very weak lines are in both cases due to chromium-pairs. The exchange Hamiltonian H_ex = JS₁ · S₂ + j(S₁ · S₂)² used for the ground-state splitting is fitted by the parameters J = 40.9 cm^?1, j = 1.5 cm^?1 and J = 45.6 cm^?1, j = 2.0 cm^?1 for ZnAl-spinel and MgAl-spinel, respectively. The differences between the spectra of low-doped and high-doped samples are in both cases caused by the existence of a phonon sideband of the N₄-line, which is in many respects similar to the well-known phonon side band of the R-line.