Fractal basin boundary constructed by superposition of the Weierstrass function

The structure and properties of the basin boundary of a two-dimensional map proposed by Grebogi, Ott and Yorke are investigated. Functional analysis gives the exact shape of the boundary. Using perturbation analysis it is shown that the basin boundary is constructed by superposition of the Weierstrass function which is everywhere continuous but nowhere differentiable. The Lyapunov exponent and the fractal dimension of the boundary are also estimated in terms of the results given by the perturbation method. Perturbation analysis gives a good explanation for the fractal basin boundary.     

On the equality of Hausdorff and box counting dimensions

By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and Hausdorff dimension are expected to be equal.     

受迫Holmes-Duffing系统安全域分形及时滞速度反馈控制

以受迫Holmes-Duffing系统为研究对象, 对系统施加时滞速度反馈控制, 研究周期激励引起的系统安全域的分形侵蚀及时滞速度反馈对分形侵蚀安全盆的控制作用. 利用Melnikov函数法给出时滞受控系统的安全盆的边界分形条件. 再以时滞量为变参数, 运用四阶Runge-Kutta方法和点映射方法数值研究了时滞对受控系统安全盆的影响规律. 结果表明在弱反馈下, 时滞量的增大能够提高安全盆边界分形的阈值, 从而抑制安全盆的分形侵蚀. 说明时滞速度反馈能够有效抑制系统的安全盆侵蚀.     

Fractal basin boundaries

Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.     

弱的参数周期扰动对一非线性系统安全域的影响与分形侵蚀安全域的控制

讨论了弱参数周期扰动对于非线性系统安全域的影响,在一定频率下的参数周期扰动将加速安全域的分形侵蚀,而在另一些频率的扰动下,将抑制安全域的分形侵蚀,并且存在着增进安全域的最优频率.提出了用弱参数周期扰动控制受到分形侵蚀的安全域的方法,并用Melnikov方法进行了分析.最后讨论了这种控制安全域的方法在实际环境中当具有外加噪声时的鲁棒性. 关键词： 分形吸引域边界 参数的周期扰动 控制     

Characterization of Synthetic-Coal Char Particles using fractal dimension analysis

The development and change of surface ruggedness in chars was studied at conditions typical in a pulverized coal furnace. The fractal dimension, a measure of surface ruggedness, of chars was measured using physisorption techniques. By adjusting the temperature encountered (1173 to 1773 K) and residence time (0.1 to 1.5 s) of the synthetic coal (sized to 46–106 μm diameter), chars at different stages of combustion were prepared in a laminar flow (drop-tube) furnace. The particles were quickly cooled and quenched in an inert atmosphere. The samples were examined using a scanning electron microprobe, and their fractal dimensions were determined using gas physisorption. The adsorption data were used to test if the char surface was fractal on a molecular scale, to determine the fractal dimension, and to quantify changes in the fractal dimension during combustion. The fractal dimension of the unburned synthetic coal was approximately 2. The fractal dimension increased as high as 2.85 as the carbon matrix burned away and exposed mineral moieties. However, as combustion continued the carbon burned completely away leaving a mineral fly ash particle with a fractal dimension as low as 2.47.     

Automated Digital Image Based Measurement of Boundary Fractal Dimension for Complex Nanoparticles

There is a growing realization that complex nanoparticles produced by combustion reaction, precipitation, and spray technology using supercritical fluids, are fractally structured. The boundary fractal dimension is linked to the flow, packing and consolidation dynamics of nanopowders. It also contains information on the formation dynamics of the nanoparticles produced by various methods. Extraction of the fractal dimension information embodied in the nanoparticle's fractal structure is hampered by the lack of automated characterization algorithms for processing images of particles. This paper describes an efficient algorithm for analyzing digitized images of fractally structured nanoparticles and presents a computer program that automates the procedure using digital image processing techniques. The program functionality is demonstrated and discussed using digital images of typical pigment, ceramic and pharmaceutical powders.     

Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

近地层大气光学湍流的分形和间歇性特征分析

利用光纤湍流测量系统获得了合肥西郊科学岛上气象观测场内下垫面平坦的水面上方0.48m、草地上方1.8m和23m高处的大气折射率起伏的观测数据,采用R/S分析法计算了近地层大气光学湍流的赫斯特指数和分形维数,统计分析了分形维数的日变化特征及概率分布特征。结果表明:对于一天的不同时段,分形维数在一定范围内动态变化,且中午时段相对稳定;在三种下垫面条件下,全天分形维数的值大多在1.3~1.4之间,其最可几概率位于1.35处,从均值来看,草地上方1.8m的分形维数最大,水面上方0.48m次之,草地上方23m处最小。最后,初步探讨了近地层大气光学湍流分形维数、间歇性指数和湍流发展程度的相关性。     

Exterior dimension of fat fractals

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the “uncertainty exponent” previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.     

时滞位移反馈对Helmholtz振子系统的分形侵蚀安全域的控制

对一个Helmoholtz振子系统引入线性时滞位移反馈,研究时滞位移反馈对系统的安全盆侵蚀的控制作用.将Melnikov函数法推广到时滞系统,从而给出时滞受控系统的安全盆的边界分形条件,在此基础上得出了安全盆分形的激励振幅阈值.再以时滞量为变参数,运用四阶Rung-Kutta方法和点映射方法从定量上研究了时滞对受控系统安全盆的影响规律.结果表明在弱反馈下,时滞量的增大能够提高安全盆边界分形的阈值,进而有效地抑制安全盆的分形侵蚀.说明时滞位移反馈是控制系统的安全盆侵蚀的良好方法. 关键词： 安全盆 分形侵蚀 时滞 反馈     

Exact solution of Kramers' problem for piecewise parabolic potential profiles

This paper presents a general procedure for deducing exact time characteristics of Brownian diffusion in the high damping limit in piecewise parabolic pontential profiles.Our approach to solve this problem is based on the Laplace transform method for obtaining the desired time characteristics. For some concrete examples of piecewise parabolic profiles' exact values of metastable state decay times and bistable system relaxation times are found. These results are valid for any height of the potential barrier and coincide with Kramer's results for a high potential barrier. Conditions for the validity of Kramer's formula are also derived.     

P-adic coverage method in fractal analysis of showers

Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/k parts of the space studied.     

Diffusion-limited aggregation: A continuum mean field model

Mean field theory is used as a basis for a new approach to analyzing fractal pattern formation by diffusion-limited aggregation. A coarse time scale is introduced to take into account the discrete nature of DLA clusters. A system of equations is derived and solved numerically to determine the fractal dimension and density of a cluster as a function of distance from its center. The results obtained are in good agreement with direct numerical simulations.     

Error analysis of the adding/doubling method for radiative scattering in inhomogeneous media

The doubling method is a fast and exact procedure for calculating radiative transfer in a homogeneous, scattering, plane-parallel medium. It can also be used in the adding mode for an inhomogeneous medium that is approximated by a finite number of homogeneous sublayers with different radiative properties. The errors caused by this approximation are analyzed in this paper through comparison with invariant imbedding calculations that are slow but exact for inhomogeneous media. A procedure is developed so errors can be estimated and controlled when using the faster adding/doubling calculations.     

The scattering of electromagnetic waves in fractal media

Abstract

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

基于分形理论的多孔介质导热系数研究

本文根据分形理论,对多孔介质几何结构进行了描述,计算出了不同孔隙率多孔介质简化模型的剖面面积分布分形维数d,并利用分形维数结合土壤导热模型推出了土壤的有效导热系数表达式,计算结果表明,该方法是符合客观实际的．     

The Upper Critical Dimension of the Abelian Sandpile Model

The existing estimation of the upper critical dimension of the Abelian Sandpile Model is based on a qualitative consideration of avalanches as self-avoiding branching processes. We find an exact representation of an avalanche as a sequence of spanning subtrees of two-component spanning trees. Using equivalence between chemical paths on the spanning tree and loop-erased random walks, we reduce the problem to determination of the fractal dimension of spanning subtrees. Then the upper critical dimension d _u=4 follows from Lawler's theorems for intersection probabilities of random walks and loop-erased random walks.     

Multi-dimensioned intertwined basin boundaries and the kicked double rotor

Using two examples, a four-dimensional kicked double rotor and a simple noninvertible one-dimensional map, we show that basin boundary dimensions can be different regions of phase space. For example, they can be fractal or not fractal depending on the region. In addition, we show that these regions of different dimension can be intertwined on arbitrarily fine scale. We conjecture, based on these examples, that a basin boundary typically can have at most a finite number of possible dimension values.     

分形维数海豚哨声信号检测方法

针对海豚哨声信号自动检测的问题,提出一种基于分形维数的自适应阈值海豚哨声信号检测方法。对待检测声信号计算盒分形维数,根据得到的盒分形维数特征值,通过模糊C均值聚类自适应确定检测阈值,实现海豚哨声信号的自动检测。文中对水池录制的海豚声信号进行了数据分析,利用哨声信号盒分形维数对哨声信号段与非信号段进行检测,并与基于谱熵的方法进行对比,获得了较高的检测率以及较低的虚警率,可以适用于海豚哨声信号的自动检测与分割。