共查询到20条相似文献,搜索用时 0 毫秒
1.
Some results on the behavior and estimation of the fractal dimensions of distributions on attractors
C. D. Cutler 《Journal of statistical physics》1991,62(3-4):651-708
The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimension(x) and the generalized Renyi dimensionsD(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension and information dimension. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure for which corrects for an important bias term (the local measure density) and provides confidence intervals for. The general applicability of this method is illustrated with various numerical examples. 相似文献
2.
研究了二维logistic映射的动力学行为和奇怪吸引子的分形特征.利用分岔图、相图和Lyapunov指数谱分析系统的分岔过程,研究系统通向混沌的道路并确定系统处于混沌运动的参数区间;采用G-P算法计算奇怪吸引子的关联维数和Kolmogorov熵,对奇怪吸引子的分形特征定量刻画;采用逃逸时间算法构造奇怪吸引子的彩色广义M-J集,对奇怪吸引子的分形特征定性表征.结果表明,这些分析方法的配合使用可以更全面、形象地描述奇怪吸引子的分形特征. 相似文献
3.
A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd
f
BB
=1.61±0.01, the chemical dimensionality isd
t=1.40±0.01, and the fractal dimension of the minimum pathd
min=1.15 ± 0.02 for the two-dimensional triangular lattice. 相似文献
4.
P. Grassberger 《Journal of statistical physics》1981,26(1):173-179
We consider such mappingsx
n+1=F(xn) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ... 相似文献
5.
Generalized dimensions of strange attractors 总被引:1,自引:0,他引:1
Peter Grassberger 《Physics letters. A》1983,97(6):227-230
It is pointed out that there exists an infinity of generalized dimensions for strange attractors, related to the order-q Renyi entropies. They are monotonically decreasing with q. For q = 0, 1 and 2, they are the capacity, the information dimension, and the correlation exponent, respectively. For all q, they are measurable from recurrence times in a time series, without need for a box-counting algorithm. For the Feigenbaum map and for the generalized Baker transformation, all generalized dimensions are finite and calculable, and depend non-trivially on q. 相似文献
6.
基于Lorenz,Rssler和H啨non三种典型的奇怪吸引子,全面分析了GrassbergerProcaccia(缩写GP)算法,详细讨论了采样数据量、延迟时间、重构相空间维数和线性区长度等参数对计算关联维数和Kolmogorov熵的影响,结果表明这些关键参数是相互关联的.通过分析关联积分谱的变化趋势,发现延迟时间与重构相空间维数对连续动力系统和离散动力系统的作用效果是不同的,且选择最佳延迟时间对计算关联维数的意义不大.指出了实际中应用GP算法应注意的问题
关键词:
奇怪吸引子
GrassbergerProcaccia算法
关联维数
Kolmogorov熵 相似文献
7.
In this paper we examine in detail the formation and evolution of fractal structure in the chaotic attractors of nonlinear dynamical systems. We explicitly obtain the fractal structure of the underlying chaotic attractors of low-dimensional systems and study their evolution as a system parameter is varied. Using periodic enumeration, dimensional, andf() spectral techniques, we obtain a detailed characterization of the multifractal structure. 相似文献
8.
S. Albeverio G. Gallavotti R. Høegh-Krohn 《Communications in Mathematical Physics》1979,70(2):187-192
We show that for the regularized exponential interaction :e
: ind space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field ifd>2 for all or ifd=2 for all such that ||>0.Partially sponsored by the I.H.E.S. through the Stiftung Volkswagenwerk 相似文献
9.
We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities Dq. We prove that lim q→0Dq = fractal dimension (D), limq→1Dq = information dimension (σ) and Dq=2 = correlation exponent (v). Dq with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > Dq for any q′ > q. For homogeneous fractals Dq = Dq. A particularly interesting dimension is Dq=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D∞. 相似文献
10.
《Physics letters. A》1986,118(4):200-202
A stochastic reorganization model is simulated on a two-dimensional square lattice, where we investigate finite size effects, and on clusters generated by diffusion limited aggregation. We propose a simple mean-field model, which accounts well for the data. 相似文献
11.
We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition. 相似文献
12.
We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-1/2 representation, pointing in various directions, form a sphere. We show that for subsystems with a large number m of local degrees of freedom, the entanglement entropy diverges as d/2 logm, where d is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing d as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder. 相似文献
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15.
A. S. Ul’yanov 《Optics and Spectroscopy》2009,107(6):866-872
The features of the formation of speckle structures under irradiation of a model fractal (Sierpinski carpet) have been investigated.
The relationship between the fractal properties of the diffraction pattern and the scattering structure parameters (model
fractal geometrical sizes, fractal depth) has been analyzed for the irradiation by a focused light beam, whose size is comparable
with that of the irradiated object. The results of the computer simulation of the Gaussian beam scattering in bacterial colonies
are compared with the experimental data. 相似文献
16.
Wiltzius P 《Physical review letters》1987,58(7):710-713
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We calculate the maximal Lyapunov exponent, the generalized entropies, the asymptotic distance between nearby trajectories
and the fractal dimensions for a finite two-dimensional system at different initial excitation energies. We show that these
quantities have a maximum at about the same excitation energy. The presence of this maximum indicates the transition from
a chaotic regime to a more regular one. In the chaotic regime the system is composed mainly of a liquid drop while the regular
one corresponds to almost freely flowing particles and small clusters. At the transitional excitation energy the fractal dimensions
are similar to those estimated from the Fisher model for a liquid-gas phase transition at the critical point.
Received: 16 March 2001 / Accepted: 12 July 2001 相似文献
20.
By the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential, moments, and Fourier transform of exactly self-similar fractal and multifractal charge distributions. Three specific examples are given: the binomial distribution on the middle-third Cantor set, which is a multifractal distribution, the uniform distribution on the Menger sponge, which illustrates the added complication of higher dimensionality, and the uniform distribution on the von Koch snowflake, which illustrates the effect of rotations in the defining transformations. 相似文献