Fractal basin boundary constructed by superposition of the Weierstrass function

The structure and properties of the basin boundary of a two-dimensional map proposed by Grebogi, Ott and Yorke are investigated. Functional analysis gives the exact shape of the boundary. Using perturbation analysis it is shown that the basin boundary is constructed by superposition of the Weierstrass function which is everywhere continuous but nowhere differentiable. The Lyapunov exponent and the fractal dimension of the boundary are also estimated in terms of the results given by the perturbation method. Perturbation analysis gives a good explanation for the fractal basin boundary.     

The geometry of chaotic dynamics — a complex network perspective

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ε-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ε-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ε-recurrence networks exhibit an important link between dynamical systems and graph theory.     

量纲分析在大学物理热力学部分中的教学应用

本文首先简单介绍了量纲分析中的齐次定理和∏定理,然后具体例举了量纲分析在大学物理热力学部分的教学应用.我们的求解过程是:分析例举可能相关的物理量;写出各个物理量的量纲表;分析量纲表,确定相关的物理量和独立量纲;设定无量纲量等式;列出量纲指数方程组并求解.     

Fractional differentiability of nowhere differentiable functions and dimensions

Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the "critical order" 2-s and not so for orders between 2-s and 1, where s, 1相似文献   

Fat fractals in the lung? A comment on a paper by grebogi et al.

An exponent characterising the scaling behaviour of a branching structure occupying volume (an example of a “fat” fractal), introduced recently by Grebogi et al., is examined using data on the arteries, veins and airways of mammalian lungs. The results cast doubt on the usefulness of the exponent in this context.     

On determining the dimension of chaotic flows

We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques. We show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time-varying quantity is available for analysis. These methods are applied to several dissipative dynamical systems.     

物理量单位的智能导出算法及其应用

推导物理量的单位是物理作业中不可缺少的重要环节。提出一种物理量单位的智能导出算法,即计算机根据物理量的运算式自动推导出待求物理量的单位。实验结果表明,物理量单位导出及换算算法的结果正确,对算术运算式中的物理量单位导出及换算的正确率为100%,且物理量数值的运算正确。     

Pointwise dimensions of the lorenz attractor

We discuss a connection between two complementary views of the Lorenz attractor: the first is the accepted view where the attractor has a smooth measure on a fractal support. This complex system is alternatively manifest as a self-similar curve for the pointwise dimension alpha. We describe why the latter approach is accessible for the analysis of an experimental signal.     

Upper Bounds on Coarsening Rates

 We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as in the first case and in the second. We prove a weak, one-sided version of this assertion – showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument. Received: 20 September 2001 / Accepted: 5 February 2002 Published online: 12 August 2002     

Constructive proof of localization in the Anderson tight binding model

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.Work supported in part by National Science Foundations grant MCS-8108814 (A03).Work supported in part by National Science Foundation grant DMR 81-00417.     

Tracking unstable fixed points in parametrically dynamic systems

The method of Ott, Grebogi and Yorke is extended to control a two-parameter system when one of the parameters is time dependent and the other is used as the control-parameter. As one of the parameters changes, the unstable fixed point follows its branch of the bifurcation tree. We control a chaotic orbit such that it tracks this “moving” unstable fixed point using an adaptive control method.     

Local control of chaotic motion

On the basis of the Ott, Grebogi and Yorke method (OGY) of controlling chaotic motion by stabilizing unstable periodic orbits we propose a control method which allows a nearly continuous adjusting of the control parameter and which therefore is capable also for controlling noisy systems. Any motion which is a solution of the system's equation of motion can be stabilized, unstable periodic orbits as well as chaotic trajectories. We demonstrate the feasibility of the method by stabilizing experimentally arbitrarily chosen chaotic trajectories of a driven damped pendulum affected by noise.     

On the shear coefficient in Timoshenko's beam theory

Some existing formulations for the shear coefficient in Timoshenko's beam theory are discussed, especially through evaluation of the accuracy to which natural frequencies of simply supported, prismatic, thin walled beams can be obtained. The main conclusion drawn is that if a consistent expression for the shear coefficient, such as those given by Cowper [1] or Stephen [2], is used in Timoshenko's beam theory, then very high accuracies can be expected for the natural frequencies, even for wavelengths of the same magnitude as the transverse dimension of the beam. It is noted that no reduction of the moment of inertia due to shear lag effects should be made as these effects are included in the consistent formulas for the shear coefficient. Finally, some apparently paradoxical results indicating that a reduction in shear stiffness occurs in rare cases if more material is added to a section are discussed and explained as resulting from the use of integrated rather than pointwise deflection measures in the derivation of consistent shear coefficient expressions. The results are discussed in the light of the importance of the shear stiffness of the hull girder in ship hull vibration analysis.     

Harmonic Analysis on Quantum Tori

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded L _p Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman’s H₁-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.     

闪电的分形特征研究及其在自动识别中的应用

利用2009年夏季青海地区的快电场变化测量仪的野外观测资料, 对120例地闪和77例云闪辐射场信号的分形特征进行了深入研究, 结果表明地闪辐射场信号的分形维数与云闪辐射场信号的分形维数有明显的差别, 再利用闪电的分形维特征, 构造了5个特征值, 将其作为支持向量机的输入进行地闪和云闪不同放电类型的识别, 有效识别率达到95%以上; 通过构造地闪辐射场时间序列信号的分形维数轨迹图表明分形维数最低点对应于原时间序列的回击位置, 利用分形维数轨迹中的最低点的位置能够快速准确地对地闪辐射场信号的回击点进行检测, 检测率可达到100%. 分形维是闪电的一种具有鉴别性的特征, 可用于闪电的智能分析与自动化处理. 关键词： 闪电信号 分形维数 支持向量机 自动识别     

DESCENING SEQUENCE OF GAUGE ANOMALIES

In this paper, the descending equations of cochains defined on simplexes are generalized to the general chains. We have also explained geometrically in some detail that the topological non-trivialness of physical quantities can descend from higher dimension to lower dimension through the surface term.