Fractal growth kinematics abstracted from snowflakes: topological evolution

Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.     

二维Logistic映射的分岔与分形

理论分析了二维Logistic映射的分岔，并采用相图、分岔图、功率谱、Lyapunov指数和分维数计算的方法，揭示出：二维Logistic映射可按倍周期分岔和Hopf分岔走向混沌；在倍周期分岔过程中，系统在参数空间和相空间中都表现出自相似性和尺度变换下的不变性．对二维Logistic映射的吸引盆及其Mandelbrot-Julia集(简称M-J集)的研究表明：吸引盆中周期和非周期区域之间的边界是分形的，这意味着无法预测相平面上点运动的归宿；M-J集的结构由控制参数决定，且它们的边界是分形的．     

Fractal geometry derived from geometric inversion

In this paper,the conception of fractal geometry derived from geometric inversion isintroduced.A ramified self-inverse fractal with symmetry and a self-inverse fractal dust setare constructed.The authors extend the conception of the fractal osculation and propose anew notion—fractal envelope.Finally,two examples of self-inverse fractal(soap and egg)are given.     

The fractal structure of attractor for the generalized Kuramoto-Sivashinsky equations

I.IntroductionSincethereexistspectralbarriersandspectralgapconditions,theexistenceofaninertialmanifoldformanynonlineardissipativeevolutionequationsisstillamystery.Recently,Edenetal[5]havediscoveredthatfornonlinearsemigroup,definedbynonlineardissipativeevolutionequationsincludingZDNavier-Stokesequations,thereexistsatinliefractaldimensionalinertialsetwhichmayberepresentedbyaunionoffractillsetsandattractor,ifitisLipschitzcontinuousandissqueezingonacompacti,ositiveinvariantset.Ontileotherhand,S…     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

局部高密度钢纤维混凝土疲劳裂纹扩展实验研究

在混凝土弯曲构件底部用高密度钢纤维局部增强称为局部高密度钢纤维混凝土（PHPFRC）,与同样纤维掺量的传统钢纤维混凝土（SFRC）比较,它可以用近似的价格获得高得多的力学性能,本文对PHPFRC缺口试件在循环载下的裂纹扩展规律进行了研究,发现裂纹并非是由一条主裂纹循序渐进地发展,而是随着循环次数的增加,呈多条短裂纹陆续出现,弥散式发展的趋势,说明在高密度纤维的增强下,混凝土的疲劳断裂由脆性向专心性转变,根据实验及分析结果得出了两种纤维掺量下PHPFRC度件的Paris参数,并且用一个分维值D的变化描述了裂纹的演经过程,发现裂纹失稳扩展前的临界分维值D随着纤维掺量的增加而增加。     

AIFS—A Tool for Biomorphic Fractal Modeling

A considerable class of fractal sets can be represented by using the attractors of Iterated Function Systems (Barnsley, 1988), with affine contractive mappings of a metric space . The modeling capabilities of such systems are heavily limited however. For example, it is not easy to predict the location of the attractor nor its global shape. Then, Iterated Systems are not affinely invariant (affine mappings of the elements of the system do not result in affine image of its attractor). In this paper a new setting, the affine invariant Iterated Function System is described in such a way that it removes the mentioned shortcomings and can be used for shape-predictable modeling of fractal based forms. The stress is put on modeling of biological forms and their atributes such as: continuous deformation of the attractor in desired way (like in growing), branching (plants, vascular or alveolar network), gradual changing of fractal dimension from smooth to space-filling fractals. The last is useful for creating images of tissues in different stages of development, symmetry, gradual transformation from one to another form, etc. The fractal images obtained by AIFS are merely to gain resemblance to bio-forms.     

Pullback attractor of 2D nonautonomous g-Navier-Stokes equations with linear dampness

The pullback attractors for the 2D nonautonomous g-Navier-Stokes equations with linear dampness are investigated on some unbounded domains. The existence of the pullback attractors is proved by verifying the existence of pullback D-absorbing sets with cocycle and obtaining the pullback D-asymptotic compactness. Furthermore, the estimation of the fractal dimensions for the 2D g-Navier-Stokes equations is given. Key words pullback attractor, g-Navier-Stokes equation, pullback asymptotic compactness, fractal dimension, linear dampness     

Multiple-cell elements and regular multifractals

Based on fractal super fibers and binary fractal fibers, the following objectives are approached in this paper： First, the concept of multiple-cell elements is induced and abstracted. Second, through multiple-cell elements, the constructability of regular multifractals with strict self-similarities is confirmed, and the universality of the con- struction mode for regular multifractals is proved. Third, through the construction mode and multiple-cell elements, regular multifractals are demonstrated to be equivalent to generalized regular single fractals with multilayer fine structures. On the basis of such equivalence, the dimension formula of the regular single fractal is extended to that of the regular multifractal, and the geometry of regular single fractals is extended to that of regular multifractals. Fourth, through regular multifractals, a few golden fractals are constructed.     

二维Logistic映射中的一种新型激变、回滞和分形

研究了二维Logistic映射不动点的性质,给出了在参数空间中二维Logistic映射发生第一次分岔的边界方程。采用相图、分岔图、功率谱、Lyapunov指数计算和分维数计算方法,揭示出具有二次耦合项的二维Logistic映射从规则运动转化到混沌运动所具有的普适特征:①系统是按Pomeau-Manneville途径通向混沌的,且其间歇性与Hopf分岔有关;②系统中存在一种新型循环激变:当参数连续变化时,不稳定周期轨道按固定顺序循环与奇怪吸引子的几个小部分相遇,并导致小部分两两合并,产生出较大的奇怪吸引子;③最大Lyapunov指数的曲线具有“回滞”特征,且回滞现象常伴随循环激变的出现。同时,作者对二维Logistic映射的Mandelbrot-Julia集(简称M-J集)的研究表明:M-J集的结构由控制参数决定,且它们的边界是分形的。     

弹性动力学反问题的非线性及其迭代反演

分析了弹性动力学反问题的非线性及在迭代反演过程中表现出的复杂非线性现象。迭代反演的结果依赖于反演系统参数和迭代初值,而系统参数对应的Mandelbrot集和迭代初值对应的Julia集都是复杂的分形集。随反演系统状态参数的变化,完全确定性的反演系统却可能产生一系列无规则的,不可预测的迭代输出序列。反演迭代过程中出现的分形和混沌现象反映了表面简单的反演迭代后隐藏的复杂性,正是这种复杂性给迭代系统参数的合理选择带来困难,进而使反演迭代不总能给出满意的结果。     

Fractal sets in control systems

In this paper the Hausdorff measure of sets of integral and fractional dimensions is introduced and applied to control systems.A new concept,namely,pseudo-self-similar set is also introduced.The existence and uniqueness of such sets are then proved,and the formula for calculating the dimension of self-similar sets is extended to the psuedo-self-similar case.Using the previous theorem,we show that the reachable set of a control system may have fractional dimensions.We hope that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.     

Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.     

分形力学的数学基础

在分形空间考虑的力学称为分形力学。由于分形理论的特点,分形力学的描述需要引入新的空间概念、新的数学手段和方法。本文从介绍空间、测度和维数入手,引入了分形空间,Ｂｅｓｏｖ空间和分形插值方法,进而介绍了在分形边界和分形体中力学量和力学定律的定义。      

Incorporating truncated fractal distribution into Maxwell model to quantify thermal conductivity and its uncertainty in heterogeneous multiparticle systems

An approach is developed to examine the mean and uncertainty of thermal conductivity of a heterogeneous multiparticle system, where the particle concentration or void fraction is treated as a truncated fractal distribution. The truncated fractal distribution is then integrated into the Maxwell model, which is equivalent to a cell model in which the multiparticle system is conceptualized as a spherical fluid cell that envelopes a solid particle. The developed mean thermal conductivity is compared with four experimental data sets of liquid-saturated media from the literature. The effect of fractal characteristics is quantified and discussed. Incorporating particle concentration or void fraction truncated fractal distribution can better capture scatters in the experimental results. The thermal conductivity and its standard deviation decrease with increasing fractal dimensions. When the void fraction is truncated fractal, the uncertainty increases mostly in the low mean void fraction range and drops more quickly with the increasing mean void fraction compared to the case where the particle concentration is truncated fractal. In a typical case of multiparticle system when the solid particles are more conductive than the fluid, the faster increase rate of standard deviation with the ratio of solid over fluid conductivities occurs when the mean void fraction is smaller.     

基于分形导数对非牛顿流体层流的数值研究

非牛顿流体具有复杂的流变特性,揭示该流变特性可以更加合理地指导非牛顿流体在工农业生产中的应用.经典的非牛顿流体本构模型往往形式复杂,仅能应用于某些特定的情况.分数阶导数模型具有参数少和形式简单的特点,己成功地应用于描述非牛顿流体的运动.Hausdorff分形导数作为一个备选的建模方法,相比分数阶导数具有更简单的形式以及更高的计算效率.本文基于Hausdorff分形导数改进现有牛顿黏性模型,提出分形黏壶模型.通过研究分形黏壶在常应变率下表观黏度的变化情况,以及在加、卸载条件下的蠕变及恢复特性,发现分形黏壶模型适合于描述具有黏弹性的非牛顿流体(本文称之为分形流体).结合连续性方程及运动微分方程,推导出分形流体在平行板间层流的基本方程.按是否拖动上板和是否存在水平的压力梯度分为3种工况,分别用数值方法计算这3种工况下流速在板间的分布及其随时间变化的情况.通过分析不同工况下的流速分布,发现水平的压力梯度会改变流速随时间变化的形状,且会推迟流速到达稳定的时间.在水平压力梯度不存在的情况下,不同阶数的分形流体具有相同的流速分布或是演变过程.另外,在水平压力梯度存在的情况下,上板速度不影响不同阶数分形流体间稳定速度的差值.     

两圆柱体表面接触承载能力的分形模型研究

结合M-B分形模型和Hertz接触模型理论,建立了新型两圆柱体分形接触模型,通过与M-B模型和Hertz模型的对比分析及接触强度实例计算,证明模型所反映两接触体的应力状态.结果表明,所建立的模型反映出载荷与面积呈指数关系,指数不仅与分形维数D有关,而且还与圆柱体的接触半径及接触方式有关.此模型的建立可作为相关机械产品（如齿轮和轴承等）的强度分析和优化设计的理论基础.     

复合Logistic映射中的逆分岔与分形

利用分岔图，揭示出复合Logistic映射可按倍周期分岔走向混沌，且混沌区中存在混沌危机及逆分岔现象．同时，分析了复合Logistic映射临界点的轨道，给出了复合Logistic映射Mandelbrot-Julia集(简称M-J集)的定义，推广了Welstead和Cromer所提出的周期点查找技术，并利用该技术，构造出一系列复合Logistic映射的M-J集．在此基础上，研究了M-J集的对称性；探索了M集周期区域分布的拓扑不变性；通过定性地建立M集上J集的整体刻画，发现M集包含了J集构造的大量信息．     

断面分形研究中的几个问题

本文讨论了分形几何应用于断裂研究的几个基本问题．主要内容包括断面的分形特征及分维测量,分维与宏观力学性能（如断裂韧性等）的关系,分形断裂模型及分形断裂的物理机理．      

锡基巴氏合金磨损表面的分形与磨损率

利用结构函数分析法研究了滑动摩擦学系统中金属磨损表面轮廓线的分形特性。结果表明：磨损表面轮廓线在小于Ｓｍ的尺度上具有分形结构。采用结构函数法可以方便地确定粗糙表面轮廓线的分形参数，即垂直于滑动方向上磨损表面轮廓线的分维Ｄ可作为磨损表面分维的特征值，它与金属磨损率的变化有着密切的关系，最佳分维Ｄｏｐｔ值对应于材料的最低磨损率。