一般二维二次映射中的混沌与分形

研究了一般二维二次映射不动点的性质,给出了在参数空间中一般二维二次映射发生第一次分岔的边界方程,通过数值计算方法分析了一般二维二次映射非线性动态行为的普适特征,并利用Ｌｙａｐｕｎｏｖ指数作判据,构造了该映射的奇吸引子,又根据Ｌｙａｐｕｎｏｖ指数求出了奇怪吸引子的分维数。同时,对一般二维二次映射的分形图研究表明,控制参数不同,分形图互不相同,且它们的边界是分形的。     

二维Logistic映射中混沌与分形的研究

研究了二维Ｌｏｇｉｓｔｉｃ映射非线性动态行为的普适特征后发现,控制参数不同,二维Ｌｏｇｉｓｔｉｃ映射可分别按Ｆｅｉｇｅｎｂａｕｍ途径、Ｒｕｂｅｌｌｅ－Ｔａｋｅｎｓ－Ｎｅｗｈｏｕｓｅ方案和Ｐｏｍｅａｕ－Ｍａｎｎｅｖｉｌｌｅ途径走向混沌,且在控制参数空间中的较大区域,其通向混沌的道路与Ｈｏｐｆ分岔有关,在这些途径上可观察到锁相和准周期运动。同时对二维Ｌｏｇｉｓｔｉｃ映射的分形研究表明,控制参数不同,分     

一类不连续映射的分岔分析

研究一类一维分段不连续映射的边界碰撞分岔问题,推导了周期n解的边界碰撞分岔曲线及fold分岔条件,通过数值仿真验证了这些条件的正确性.研究发现系统存在周期增加序列和周期叠加序列.最后,对分段不连续映射进行三参数分岔研究,揭示了系统各参数对其动力学行为的综合影响.     

基于混沌动力学构造二岔树结构的新方法

运用混沌动力学中的抛物线映射方程构造了二岔树数据结构，运用Feigenbaum分岔原理、MSS序列，周期轨道，周期窗口及暗线方程精确计算出低维倍周期分院岔点（二岔树节点）的数值，为二岔树常用的操作，二岔树遍历和二岔搜索树等提供了新方法，为进一步研究二岔树和树的分形结构及混沌特性奠定了基础。     

高次logistic映射的混沌运动

logistic映射具有简单的形式,但却有复杂的动力学性质.构造了一种高次logistic映射,对这类映射的动力学特性做了研究.通过稳定性分析,得到了定点的稳定条件.数值迭代发现,此类映射都具有相似的稳定定点经倍周期分岔进入混沌的道路,反映了典型的Feigenbaum途径.当参数超过某个临界值时存在混沌吸引子的突然消失和突然膨胀,分别称为边界激变和内部激变.特别的当参数取一定值时会出现一种内部激变,使得混沌吸引子突然增大并且关于x轴对称,这是logistic映射所没有的.分析高次logistic映射中的这种激变对动力学的研究具有一定的意义.     

一类分段线性神经元系统的动力学行为分析

针对一类平面上三分段连续线性神经元模型,研究了边界平衡点的持续性分岔(persistence)、非光滑折分岔(non-smooth fold)的存在条件及一类跨边界周期解.最后,通过施加缓慢变化的周期外激励研究其对边界分岔和系统周期放电的影响.     

复映射z←(z)-a+c(a≥2)的广义M集及其对称周期检测法

研究了指数为负实数的非解析复映射z←(z)^-a+c(a≥2)的广义Mandelbrot集.分析和证明了(取不同值时该映射的广义M集所具有的性质,严格地给出了(为正整数时复映射周期1轨道稳定区域边界的参数方程.提出了对称周期检测法,根据各参数点的周期值对M集进行着色,并充分利用M集的对称性来减少绘制过程中计算周期时所需要的迭代运算.实验结果表明,新算法在获得高质量M分形图的同时具有较高的绘制速度.进一步地,新算法可以推广到其他M(Mandelbrot)集和J(Julia)集的绘制.     

二维Logistic映射分岔控制

利用系统的状态反馈和参数调节方法,精确控制了二维Logistic映射发生第一、二次分岔时分岔点位置,从而在不改变系统原来分岔特性的基础上,只需选择合适的调控参数,就可实现二维Logistic映射的混沌控制和反控制.     

广义M-J集的边界构造及分维数计算

讨论逃逸时间算法、Lyapunov指数法、反函数法和同胚分形插值法等方法的适用范围,提出构造广义M-J集分形边界的一种通用算法即边界检测算法.利用设计的检测模板将广义M-J集的边界部分提取出来并计算这种分形边界的盒维数 .实验结果表明,与几种现有的算法相比较,新算法具有普适性和精确性,用于构造任意形式复映射的广义M-J集的分形边界.     

复映射z←(-z)-a+c(a≥2)的广义M集及其对称周期检测法

研究了指数为负实数的非解析复映射z←(-z)-a+c(a≥2)的广义Mandelbrot集.分析和证明了(取不同值时该映射的广义M集所具有的性质,严格地给出了(为正整数时复映射周期1轨道稳定区域边界的参数方程.提出了对称周期检测法,根据各参数点的周期值对M集进行着色,并充分利用M集的对称性来减少绘制过程中计算周期时所需要的迭代运算.实验结果表明,新算法在获得高质量M分形图的同时具有较高的绘制速度.进一步地,新算法可以推广到其他M(Mandelbrot)集和J(Julia)集的绘制.     

A new image encryption algorithm based on non-adjacent coupled map lattices

We propose a new image encryption algorithm which is based on the spatiotemporal non-adjacent coupled map lattices. The system of non-adjacent coupled map lattices has more outstanding cryptography features in dynamics than the logistic map or coupled map lattices does. In the proposed image encryption, we employ a bit-level pixel permutation strategy which enables bit planes of pixels permute mutually without any extra storage space. Simulations have been carried out and the results demonstrate the superior security and high efficiency of the proposed algorithm.     

Biomorphic Dynamical Networks for Cognition and Control

Advances in understanding the neuronal code employed by cortical networks indicate that networks of parametrically coupled nonlinear iterative maps, each acting as a bifurcation processing element, furnish a potentially powerful tool for the modeling, simulation, and study of cortical networks and the host of higher-level processing and control functions they perform. Such functions are central to understanding and elucidating general principles on which the design of biomorphic learning and intelligent systems can be based. The networks concerned are dynamical in nature, in the sense that they compute not only with static (fixed-point) attractors but also with dynamic (periodic and chaotic) attractors. As such, they compute with diverse attractors, and utilize transitions (bifurcation) between attractors and transient chaos to carry out the functions they perform. An example of a dynamical network, a parametrically coupled net of logistic processing elements, is described and discussed together some of its behavioural attributes that are relevant to elucidating the possible role for coherence, bifurcation, and chaos in higher-level brain functions carried out by cortical networks.     

Generalized Julia set in coupled map lattice and its coupling and synchronization

This paper discusses first the basic theory and fractal characteristics of generalized Julia set in coupled map lattice (CML Julia set for short). Then, two different generalized CML Julia sets are coupled, which makes two different generalized CML Julia sets change to the same one. Using gradient control and optimal control, the synchronization of two different such generalized CML Julia sets is, respectively, also realized, which makes one CML Julia set change to another. The coupling and synchronization of two no-identical generalized CML Julia sets are accomplished via coupling and synchronizing their trajectories. To verify the feasibility of coupling and gradient and optimal control methods, we consider the generalized CML Julia sets, whose lattice length is 2, as examples to achieve their coupling and synchronization. Finally, digital simulations are carried out for α=3 and α=4 and the results verify the effectiveness of the coupling and synchronization.     

Anomalous Statistics for Type-III Intermittency

The statistics for the distribution of laminar phases in type-IIIintermittency is examined for the map . Due to a strongly nonuniform reinjectionprocess, characteristic deviations from the normal statistics are observed.There is an enhancement of relatively long laminar phases followed by anabrupt cut-off of laminar phases beyond a certain length. The paper alsoexamines the bifurcation structure of two symmetrically coupled maps, eachdisplaying a subcritical period-doubling bifurcation. The conditions forsuch a pair of coupled maps to exhibit type-II intermittency are discussed.     

Modeling urban growth in Atlanta using logistic regression

This study applied logistic regression to model urban growth in the Atlanta Metropolitan Area of Georgia in a GIS environment and to discover the relationship between urban growth and the driving forces. Historical land use/cover data of Atlanta were extracted from the 1987 and 1997 Landsat TM images. Multi-resolution calibration of a series of logistic regression models was conducted from 50 m to 300 m at intervals of 25 m. A fractal analysis pointed to 225 m as the optimal resolution of modeling. The following two groups of factors were found to affect urban growth in different degrees as indicated by odd ratios: (1) population density, distances to nearest urban clusters, activity centers and roads, and high/low density urban uses (all with odds ratios < 1); and (2) distance to the CBD, number of urban cells within a 7 × 7 cell window, bare land, crop/grass land, forest, and UTM northing coordinate (all with odds ratios > 1). A map of urban growth probability was calculated and used to predict future urban patterns. Relative operating characteristic (ROC) value of 0.85 indicates that the probability map is valid. It was concluded that despite logistic regression’s lack of temporal dynamics, it was spatially explicit and suitable for multi-scale analysis, and most importantly, allowed much deeper understanding of the forces driving the growth and the formation of the urban spatial pattern.     

A boundary equation of the principal period-2 component in the degree-n bifurcation set

The degree-n bifurcation set is a generalized Mandelbrot set for the complex polynomial P _c (z)=z ⁿ +c. The boundary of the principal period-2 component in the degree-n bifurcation set is first defined and then formulated by a parametrization of its image, which is the unit circle under the multiplier map. We investigate the boundary equation using the geometric symmetry of the degree-n bifurcation set with respect to rays of symmetry in the complex plane. In addition, an algorithm drawing the boundary curve with Mathematica codes is proposed.     

On the determination of boundaries to manipulator workspaces

The accumulation of two independent, broadly applicable formulations for determining the boundary to manipulator workspaces, presented elsewhere, are compared in this paper. Insights gained from one method are used to explain behavior exhibited in the other. Results are also compared and validated. A numerical formulation based on continuation methods is used to map curves that are on the boundary of a manipulator workspace. Analytical criteria based on row rank deficiency criteria of the manipulator's analytical Jacobian are used to map a family of one-dimensional solution curves on the boundary. The other formulation, based on a similar rank-deficiency criteria, yields analytic boundaries parametrized in terms of surface patches on the boundary. Results concerning the applicability of the numerical method to open- and closed-loop systems are compared with those limited to the open-loop for the analytical method. Conclusions regarding the behavior of the manipulator on geometric entities characterized by singular curves, higher-order bifurcation points, and surfaces inside the workspace are drawn. Applicability of both methods and their limitations are also addressed.