多重分形奇异谱的几何特性I_经典Renyi定义法_周炜星

多重分形奇异谱的几何特性I_经典Renyi定义法_周炜星

Vol.26No.42000-08华东理工大学学报

Journ al of East Chin a University of Science an d T echnology 基金项目:国家重点基础研究发展规划项目(G1999022103)E -mail:****************.cn收稿日期:1999-09-08

作者简介:周炜星(1974-),男,浙江诸暨人,博士生,现从事湍流中非

线性现象的理论和实验研究。

文章编号:1006-3080(2000)04-0385-05

多重分形奇异谱的几何特性

I .经典Renyi 定义法

周炜星*

, 王延杰, 于遵宏

(华东理工大学洁净煤技术研究所,上海200237)

摘要:研究了经典多重分形理论中广义维数和奇异谱的几何特性,通过严格的数学推导证明了

广义维数D ~q 、质量指数S ～(q )、奇异性指数A ～(q )和奇异谱f ～(A ～(q ))的单调性和极限,并提出了判定合理奇异谱f ～(A ～(q ))的准则。

关键词:多重分形;奇异谱;广义维数;经典Renyi 定义法中图分类号:O4;O184

文献标识码:A

Geometrical Characteristics of Singularity Spectra of Multifractals

I .Classical Renyi Definition

ZH OU W ei -x ing *

, W A N G Yan -j ie , YU Zun -hong

(I nstitute of Clean Coal T echnology ECUS T ,Shanghai 200237,China )

Abstract :It is w ell know n that multifractal theo ry is an

effective and w idely applied method to charac-terize a lot of nonlinear physical pheno mena in nature .In this paper ,the geometrical characteristics of sin-gularity spectra of multifractals defined via classical Reny i information are studied.The relev ant pr operties of the generalized dim ensio ns,scaling ex ponents,sing ularity str ength and singularity spectrum are derived rigo rously.It seems that the curve o f generalized dim ensions is sim ilar to that o f singularities w hen para-meter q tends to infinity .Especially ,we sho uld point o ut that singularity spectra cur ve lies in the first quadrant,w hose endpo ints are not necessary to be nought.An analy tical but simple pr ocedure to calculate the asym ptotic value at infinite is presented.It is found that different alg orithm s of first order derivative,and the com putation spacing as w ell ,lead to different multifractal spectrum .Therefore ,a criterion is sug-gested to determ ine the proper sing ular ity spectr um .T his is based on the fact that ,the curve of the m ulti-fractal spectrum is tangent to the diag onal of the first quadr ant,w hich implies that f ≤A for all q .Further-more,there is only one point o f intersection between tw o m ultifractal spectra arising from tw o different sy stems ,w hich is suppor ted by ex perimental and num er ical results .

Key words :multifractal;sing ularity spectrum;generalized dimension;classical Reny i definition

自从M andelbro t 在70年代提出分形概念以来,分形理论在物理、天文、地理、数学、生物、化

学[1～7]、计算机[8]

等科学领域得到了广泛的应用,并取得了大量富有新意的成果。但是,随着理论研究和应用的深入,研究者们越来越清楚地意识到,对于大多数客观存在的分形物体而言,仅用一个分形维数并不能完全刻画其结构。80年代初,Grassber ger 等系统地提出了多重分形理论,用广义维数和