An Explicit Fractal Interpolation Algorithm for Reconstruction of Seismic Data

Based on our previous investigation and the pioneering work of other researchers, a novel explicit fractal interpolation method based on affine transform is proposed, in which we approximate the vertical scaling factors by the locally explicit expression. Numerical experiments indicate that the explicit fractal interpolation method shows great accuracy of reconstruction of the seismic profile and yields significant improvement over wave-equation based trace interpolation methods （unified approach）.     

A fractal approach to low velocity non-Darcy flow in a low permeability porous medium

In this paper, the mechanism for fluid flow at low velocity in a porous medium is analyzed based on plastic flow of oil in a reservoir and the fractal approach. The analytical expressions for flow rate and velocity of non-Newtonian fluid flow in the low permeability porous medium are derived, and the threshold pressure gradient （TPG） is also obtained. It is notable that the TPG （J） and permeability （K） of the porous medium analytically exhibit the scaling behavior J ~ K-D＇r/（l＋Or）, where DT is the fractal dimension for tortuous capillaries. The fractal characteristics of tortuosity for capillaries should be considered in analysis of non-Darcy flow in a low permeability porous medium. The model predictions of TPG show good agreement with those obtained by the available expression and experimental data. The proposed model may be conducible to a better understanding of the mechanism for nonlinear flow in the low permeability porous medium.     

Fractal Character for Tortuous Streamtubes in Porous Media

An analytical model for fractal dimension of tortuous streamtubes in porous media is derived. The proposed fractal dimension for tortuous streamtubes in porous media is expressed as a function of porosity and scale, and there is no empirical constant in the proposed expression. The model predictions for the fractal dimension of tortuous streamtubes in porous media are in good agreement with those by the box-counting method and with the observations of other researchers.     

Elevation data compression using IFS-based three-dimensional fractal interpolation

This letter presents a novel application of iterated function system (IFS) based three-dimensional (3D) fractal interpolation to compression elevation data. The parameters of contractive transformations axe simplified by a concise fractal iteration form with geometric meaning. A local iteration algorithm is proposed, which can solve the non-separation problem when Collage Theorem is applied to find the appropriate fractal parameters. The elevation data compression is proved experimentally to be effective in reconstruction quality and time-saving.     

Subcooled pool boiling heat transfer in fractal nanofluids:A novel analytical model

A novel analytical model to determine the heat flux of subcooled pool boiling in fractal nanofluids is developed. The model considers the fractal character of nanofluids in terms of the fractal dimension of nanoparticles and the fractal dimen- sion of active cavities on the heated surfaces; it also takes into account the effect of the Brownian motion of nanoparticles, which has no empirical constant but has parameters with physical meanings. The proposed model is expressed as a function of the subcooling of fluids and the wall superheat. The fractal analytical model is verified by a reasonable agreement with the experimental data and the results obtained from existing models.     

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

A New High Performance Realization of Hyperchaotic Modified Canonical Chua Circuit Using JFETs

A new implementation of hyperchaotic modified canonical Chua circuit using junction field-effect transistors （JFETs） is proposed. The design is based on a source coupled JFET circuit to approximate a smooth cubic nonlinearity and a two-terminal negative resistance element containing a p-n-p silicon transistor and an n-channel JFET. The realization is supported by Orcad Pspice simulation and numerical MATLAB results. The hyperchaotic nature is confirmed by two positive Lyapunov exponents associated with the attractor which is a fractal with a Lyapunov dimension between 3 and 4.     

Prediction of heat transfer of nanofluid on critical heat flux based on fractal geometry

Analytical expressions for nucleate pool boiling heat transfer of nanofluid in the critical heat flux (CHF) region are derived taking into account the effect of nanoparticles moving in liquid based on the fractal geometry theory. The proposed fractal model for the CHF of nanofluid is explicitly related to the average diameter of the nanoparticles, the volumetric nanoparticle concentration, the thermal conductivity of nanoparticles, the fractal dimension of nanoparticles, the fractal dimension of active cavities on the heated surfaces, the temperature, and the properties of the fluid. It is found that the CHF of nanofluid decreases with the increase of the average diameter of nanoparticles. Each parameter of the proposed formulas on CHF has a clear physical meaning. The model predictions are compared with the existing experimental data, and a good agreement between the model predictions and experimental data is found. The validity of the present model is thus verified. The proposed fractal model can reveal the mechanism of heat transfer in nanofluid.     

Toward a fractal spectrum approach for neutron and gamma pulse shape discrimination

Accurately selecting neutron signals and discriminating 7 signals from a mixed radiation field is a key research issue in neutron detection.This paper proposes a fractal spectrum discrimination approach by means of different spectral characteristics of neutrons and 7 rays.Figure of merit and average discriminant error ratio are used together to evaluate the discrimination effccts.Different neutron and γ signals with various noise and pulse pile-up are simulated according to real data in the literature.The proposed approach is compared with the digital charge integration and pulse gradient methods.It is found that the fractal approach exhibits the best discrimination performance,followed by the digital charge integration method and the pulse gradient method,respectively.The fractal spectrum approach is not sensitive to high frequency noise;and pulse pile-up.This means that the proposed approach has superior performance for effective and efficient anti-noise and high discrimination in neutron detection.     

Transmission through array of subwavelength metallic slits curved with a single step or mutli-step

The transmission of normally incident plane wave through an array of subwavelength metallic slits curved with a sin- gle step or mutli-step has been explored theoretically. The transmission spectrum is simulated by using the finite-difference time-domain method. The influences of surface plasmon polaritons make the end of finite long sub-wavelength metallic slit behaves as magnetic-reflecting barrier. The electromagnetic fields in the subwavelength metallic slits are the superpo- sition of standing wave and traveling wave. The standing electromagnetic oscillation behaves like LC oscillating circuit to decide the resonance wavelength. Therefore, the parameters of adding step may change the LC circuit and influence the transmission wavelength. A new explanation model is proposed in which the resonant wavelength is decided by four factors： the changed length for electric field, the changed length for magnetic field, the effective coefficient of capacitance, and the effective coefficient of inductance. The effect of adding step is presented to analyze the interaction of two steps in slit with mutli-step. This explanation model has been proved by the transmission through arrayed subwavelength metallic slits curved with two steps and fractal steps. All calculated results are well explained by our proposed model.     

Detection of determinism and randomness in time series: A method based on phase space overlap of attractors

The space overlap of an attractor reconstructed from a time series with a similarly reconstructed attractor from a random series is shown to be a sensitive measure of determinism. Results for the time series for Henon, Lorenz and Rössler systems as well as a linear stochastic signal and an experimental ECG signal are reported. The overlap increases with increasing levels of added noise, as shown in the case of Henon attractor. Further, the overlap is shown to decrease as noise is reduced in the case of the ECG signal when subjected to singular value decomposition. The scaling behaviour of the overlap with bin size affords a reliable estimate of the fractal dimension of the attractor even with limited data.     

一个分段连续系统中的胖奇异集激变

报道一种有特色的激变.这种激变是在一类分段连续力场作用下的受击转子模型中观察到的.描述系统的二维映象定义域中的函数不连续边界随离散时间发展振荡，从而使这个边界的向前象集构成一个承载混沌运动的胖分形.在控制参数的一个阈值下，一个椭圆周期轨道突然出现在此胖混沌奇异集中，使得迭代向它逃逸，胖混沌奇异集因此突然变为一个胖瞬态集.在这种情况下，有可能根据椭圆周期轨道逃逸孔洞，以及胖分形奇异集的测度随参数变化的规律，估算迭代在奇异集中的平均生存时间所遵循的标度规律.直接数值计算和由此估算所得标度因子值可以很好地互相印证. 关键词： 激变 胖分形 分段连续系统 标度律     

基于Julia分形的多涡卷忆阻混沌系统

忆阻器作为一种非线性电子元件,能用作混沌系统中的非线性项,从而提高系统的复杂度.分形与混沌是密切相连的,分别对两者的研究都已成熟,却鲜有将分形过程应用到混沌系统中,以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性,本文首先提出了一个新的忆阻混沌系统,并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性;紧接着,把经典的Julia分形过程应用到该忆阻混沌系统中,产生了新的混沌吸引子,并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统,获得了丰富的混沌吸引子;最后,讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出,分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法,还弥补了使用功能函数方法造成混沌系统不光滑的不足.     

一种识别关联维数无标度区间的新方法

在计算关联维数过程中, 为了减少人为因素识别无标度区间带来的误差, 提出一种基于模拟退火遗传模糊C均值聚类识别无标度区间的新方法. 该方法根据无标度区间对应曲线的二阶导数在零附近波动的变化特征, 利用分类算法进行识别. 首先对双对数关联积分的离散数据进行二阶差分; 然后利用模拟退火遗传模糊C均值聚类方法对该数据进行分类, 选出在零附近波动的数据; 再剔除粗大误差保留有效数据; 最后进行统计分析识别出线性度最好的作为无标度区间. 应用新方法对两个著名的混沌系统Lorenz 和Henon 进行了仿真, 计算结果与理论值非常符合. 实验表明, 所提出的新方法与主观识别、K-means和2-means方法比较, 可以有效自动识别无标度区间, 减少误差, 计算结果更加精确.     

On the transition to hyperchaos and the structure of hyperchaotic attractors

Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D ₂) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here.     

结合Levenberg-Marquardt算法的垂直视差消减方法

垂直视差的存在是影响立体视频观视舒适度的主要因素。为了在不影响水平视差的条件下实现对垂直视差的消减,本文引入Levenberg-Marquardt(L-M)非线性算法实现变换矩阵的精确求解。首先用抗缩放、旋转及仿射变换的SIFT(Scale-invariant feature transform)特征匹配算法检测出双目图像对的特征匹配点,然后根据匹配点的坐标位置运用L-M算法计算可消减垂直视差的变换矩阵,将变换矩阵作用于目标图像,计算出该视图每个像素点的新坐标位置。实验结果表明:与利用线性算法求解二维射影变换矩阵的垂直视差消减方法相比,本文提出的求解方法在垂直视差消减上比该算法提高了约0.029 1~0.323 2个像素,对水平视差的影响比该算法降低了约0.118 7~1.139 1个像素。因此本文提出的方法对垂直视差的消减起到了优化作用。     

基于分形自仿射的混沌时间序列预测

从混沌与分形的关系出发，基于奇怪吸引子的分形结构和时间序列的自仿射特性，提出了一种混沌时间序列的预测方法.采用迭代函数系统跟踪混沌的局部运动轨迹，由此确定统计意义上仿射性能最优的时间序列段，并根据吸引子定理和拼贴定理建立预测模型.以Mackey-Glass混沌系统、脑电信号和Lorenz混沌系统等三种混沌系统为例进行预测试验，结果表明本方法能对混沌时间序列进行准确预测，且对混沌时间序列先验知识要求少，具有广泛的实用性. 关键词： 自仿射 迭代函数系统 混沌时间序列 预测     

Fractal Poisson processes

The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results — extending them from affine to general nonlinear scaling schemes.     

Statistical-thermodynamic approach to a chaotic dynamical system: Exactly solvable examples

The static and dynamic properties of a chaotic attractor of a two-dimensional map are studied, which belongs to a particular class of piecewise continuous invertible maps. Coverings of a natural size to cover the attractor are introduced, so that the microscopic information of the attractor is written on each box composing the cover. The statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated. Analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor. The statistical thermodynamics of local Lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local Lyapunov exponents is invetigated. For the nonhyperbolic example, the free energy and entropy functions of local Lyapunov exponents are obtained in analytic forms. These results display the existence of phase transitions. A phase transition is seen in the thermodynamics of scaling indices also.