一类分形曲面的精细计盒维数

研究一类分形曲面的精细计盒维数,得到了星积分形曲面与其生成元的精细计盒维数的关系.     

盒维数的一个等价定义及其应用

给出了盒维数的一个等价定义．该定义与盒维数的现有定义相比,从理论上更容易验证,在应用中更适合于数值计算．据此给出了计算盒维数的一个数值算法．     

一类分形曲面的精细计盒维数公式

本文研究由一个二变元四阶差分方程边值问题生成的分形曲面的精细计盒维数问题,给出了一个自然的维数公式,若该边值问题的边界上的连续函数的图象的精细计盒维数为γ,则该解曲面的精细计盒维数为（1＋γ）。     

Katugampola分数阶积分的阶与Weierstrass函数的分形维数之间的关系（英文）

计算Weierstrass函数的Katugampola分数阶积分的分形维数,如盒维数、K-维数和P-维数.证明了Weierstrass函数的Katugampola分数阶积分的阶与Weierstrass函数的分形维数之间存在线性关系.     

Logistic混沌系统分形特性研究

对Logistic序列进行研究,利用Matlab数值模拟,通过计算不同初值、不同参数对应的混沌序列的计盒维数,得出结论:只要在数据充分的情况下,Logistic系统的分形维数基本由参数λ决定,与系统初值无关;同时计盒维数并非像熵一样随Logistic系统的参数λ增大而增大.     

SG(2,2)上布朗运动的维数性质

本文研究分形集合ＳＧ（２,２）上布朗运动的维数性质,得到了ＳＧ（２,２）上布朗运动的样本图以及象集的Ｈａｕｓｄｏｒｆｆ维数与盒维数。     

Marion集的强正则性

设E=（E1,……,Em)为Marron集（不要求满足分离条件）,本文证明E具有强正则性,即对任意1≤j≤m,dimH Ej=dimB Ej,其中dimH Ej与dimB Ej分别表示Ej的Hausdorff维数与盒维数。     

自仿函数分数阶导数的分形维数

研究一类自仿函数的分数阶导数,获得了自仿函数的Weyl-Marchaud分数阶导数的图像盒维数,证明了分数阶导数的阶与分形维数之间的线性关系.     

基于分形维数的液体爆炸分散中界面破碎形态研究

通过对爆炸抛撒图象的处理,得到液体界面的曲线.采用盒维数的计算方式,计算界面曲线的分形维数.通过对各时刻液体界面分形维数的变化研究,分析爆炸抛撒近场阶段的变化过程,同时观察到蘑菇状尖顶的出现与破碎,以及空化区域的形成和消失现象。     

关于可数点集的盒维数的一些探讨

本文研究了一类可数点集的盒维数的计算问题.通过构造双Lipschitz映射,把原可数点集的盒维数的求解问题转化为求解一类相对简单的可数点集的盒维数.获得了两个单调的可数点集在具有同阶间隔时具有相同的上盒维数和下盒维数的结论.该结论为计算一类可数点集的盒维数提供了方便.     

Category and dimensions for cut-out sets

In this paper, we study the modified box dimensions of cut-out sets that belong to a positive, nonincreasing and summable sequence. Noting that the family of such sets is a compact metric space under the Hausdorff metric, we prove that the lower modified box dimension equals zero and the upper modified box dimension equals the upper box dimension for almost all cut-out set in the sense of Baire category.     

Revisiting the box counting algorithm for the correlation dimension analysis of hyperchaotic time series

We undertake the correlation dimension analysis of hyperchaotic time series using the box counting algorithm. We show that the conventional box counting scheme is inadequate for the accurate computation of correlation dimension (D₂) of a hyperchaotic attractor and propose a modified scheme which is automated and gives better convergence of D₂ with respect to the number of data points. The scheme is first tested using the time series from standard chaotic systems, pure noise and data added with noise. It is then applied on the time series from three standard hyperchaotic systems for computing D₂. Our analysis clearly reveals that a second scaling region appears at lower values of box size as the system makes a transition into the hyperchaotic phase. This, in turn, suggests that correlation dimension analysis can also give information regarding chaos-hyperchaos transition.     

Box dimension of Neimark–Sacker bifurcation

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case.     

填充维数的一个等价定义

本文研究了填充维数与上盒维数的关系.利用Cantor-Bendixson定理的方法,得到了由上盒维数给出的填充维数的等价定义.并证明了齐次Moran集对上盒维数和填充维数的连续性.     

Box dimension and fractional integral of linear fractal interpolation functions

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

The box dimension of completely invariant subsets for expanding piecewise monotonic transformations

We consider completely invariant subsetsA of expanding piecewise monotonic transformationsT on [0, 1]. An estimate of the box dimension of such setsA in terms of a certain pressure function is given, which implies equality of box dimension and Hausdorff dimension ofA.     

Fractal properties of number systems

In this paper we study properties of the fundamental domain F of number systems in the n-dimensional real vector space. In particular we investigate the fractal structure of its boundary F. In a first step we give upper and lower bounds for its box counting dimension. Under certain circumstances these bounds are identical and we get an exact value for the box counting dimension. Under additional assumptions we prove that the Hausdorf dimension of F is equal to its box counting dimension. Moreover, we show that the Hausdorf measure is positive and fnite. This is done by applying the theory of graphdirected self similar sets due to Falconer and Bandt. Finally, we discuss the connection to canonical number systems in number felds, and give some numerical examples.