【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 τ～ ( q)、奇异性指数 α～ ( q)和奇异谱 f～ ( α～ ( q) )的单调性和极限 ,并提出了判定合理奇异谱 f～ ( α～ ( q) )的准则。更多还原

【Abstract】 It is well known that multifractal theory is an effective and widely applied method to characterize a lot of nonlinear physical phenomena in nature. In this paper, the geometrical characteristics of singularity spectra of multifractals defined via classical Renyi information are studied. The relevant properties of the generalized dimensions, scaling exponents, singularity strength and singularity spectrum are derived rigorously. It seems that the curve of generalized dimensions is similar to that of singularities when para meter q tends to infinity. Especially, we should point out that singularity spectra curve lies in the first quadrant, whose endpoints are not necessary to be nought. An analytical but simple procedure to calculate the asymptotic value at infinite is presented. It is found that different algorithms of first order derivative, and the computation spacing as well, lead to different multifractal spectrum. Therefore, a criterion is suggested to determine the proper singularity spectrum. This is based on the fact that, the curve of the multifractal spectrum is tangent to the diagonal of the first quadrant, which implies that f≤α for all q . Furthermore, there is only one point of intersection between two multifractal spectra arising from two different systems, which is supported by experimental and numerical results.更多还原