三种分形和分数阶导数阻尼振动模型的比较研究

标准的整数阶导数方程不能准确描述粘弹性材料的记忆性参考文献[1]和阻尼的分数次幂频率依赖[2],因此分形导数、分数阶导数及正定分数阶导数被用于描述粘弹性介质中的阻尼振动.该文通过分析模型和数值模拟,比较了三种模型描述的振动过程.结果显示,当p小于约O.75或大于约1.9时(p为非整数阶导数的阶数),分形导数模型衰减最快;当P大于约0.75且小于约1.9时,正定分数阶导数模型衰减最快,衰减最慢的分别为分数阶导数模型(p1).且正定分数阶导数模型衰减快于分数阶导数模型,当p接近2时,两种模型较为相近.     

Representation of robotic fractional dynamics in the pseudo phase plane

This paper analyses robotic signals in the perspective of fractional dynamics and the pseudo phase plane (PPP).It is shown that the spectra of several experimental signals can be approximated by trend lines whose slope characterizes their fractional behavior.For the PPP reconstruction of each signal,the time lags are calculated through the fractal dimension.Moreover,to obtain a smooth PPP,the noisy signals are filtered through wavelets.The behavior of the spectra reveals a relationship with the fractal dimension of the PPP and the corresponding time delay.     

Characterizing the creep of viscoelastic materials by fractal derivative models

In this paper, we make the first attempt to apply the fractal derivative to modeling viscoelastic behavior. The methodology of scaling transformation is utilized to obtain the creep modulus and relaxation compliance for the proposed fractal Maxwell and Kelvin models. Comparing with the fractional derivatives reported in the literature, the fractal derivative as a local operator has lower calculation costs and memory storage requirements. Moreover, numerical results show that the proposed fractal models require fewer parameters, have simpler mathematical expression and result in higher accuracy than the classical integer-order derivative models. Results further confirm that the proposed fractal models can characterize the creep behavior of viscoelastic materials.     

On fractional difference logistic maps: Dynamic analysis and synchronous control

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

     

考虑老化的混凝土粘弹性分数导数模型

混凝土是一种具有分形结构的材料。采用分数微积分模型来研究具有分形结构材料的老化规律目前尚未见到。本文的目的是采用含分数阶导数的类标准线性体来模拟考虑老化的混凝土的蠕变和松弛规律。给出了分数导数与Abel核之间的关系。讨论了类标准线性体的蠕变柔量和松弛模量及其在考虑老化的混凝土中的应用。与传统的混凝土流变模型相比较表明，类标准线性体可以更好地同时拟合混凝土在不同龄期的蠕变和松弛曲线。而且其形式简单、统一，在计算过程中需要调整的参数很少。可以预见，类标准线性体在混凝土的结构设计和计算中将有着广泛的应用前景。     

Fractal analysis and control of the fractional Lotka–Volterra model

Nonlinear Dynamics - This paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. A Julia set of a discrete version of this model is introduced and...     

Viscoelastic properties of fractal media

Temporal fractal sets for analysis of viscoelastic properties of nonhomogeneous media are considered. A fractional derivative directly related to fractal dimension is constructed. The relationship between the diffusion of the relaxation spectrum and the fractal dimension is established. Odessa State Polytechnical University, Odessa 270044, Ukraine. Institute of Molecular Physics, Polish Academy of Sciences, Poznan, Poland. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 162–172, January–February, 2000.     

完全发展的非对称槽道湍流的分形维数研究

湍流的分形维数刻画了湍流的统计正则性 ,本文利用正交子波变换对完全发展的非对称槽道湍流脉动速度的实验数据进行了分析研究。文中首先将脉动速度信号分解到各个尺度 ,然后利用正则法和盒子法两种方法计算了不同尺度信号的分形维数 ,着重于考虑分形维数的变化趋势。研究结果表明 :(1 )随着信号分解尺度的增加 ,速度信号的低频部分和高频部分的维数都逐渐减小 ;(2 )逆输运现象对速度信号的分形维数没有本质的影响 ;(3)正则法计算得到的分形维数要略大于盒子法得到的分形维数     

The fractal dynamics of self-esteem and physical self

The aim of this paper was to determine whether fractal processes underlie the dynamics of self-esteem and physical self. Twice a day for 512 consecutive days, four adults completed a brief inventory measuring six subjective dimensions: global self-esteem, physical self-worth, physical condition, sport competence, attractive body, and physical strength. The obtained series were submitted to spectral analysis, which allowed their classification as fractional Brownian motions. Three fractal analysis methods (Rescaled Range analysis, Dispersional analysis, and Scaled Windowed Variance analysis) were then applied on the series. These analyses yielded convergent results and evidenced long-range correlation in the series. The self-esteem and physical self series appeared as anti-persistent fractional Brownian motions, with a mean Hurst exponent of 0.21. These results reinforce the conception of self-perception as the emergent product of a dynamical system composed of multiple interacting elements.     

On Modeling Flow in Fractal Media form Fractional Continuum Mechanics and Fractal Geometry

In this work we present a model for radial flow in highly heterogenous porous media. Heterogeneity is modeled by means of fractal geometry, a heterogeneous medium is regarded as fractal if its Hausdorff dimension is non-integral. Our purpose is to present a derivation of the model consistent with continuum mechanics, capable to describe anomalous diffusion as observed in some naturally fractured reservoirs. Consequently, we introduce fractional mass and a generalized Gauss theorem to obtain a continuity equation in fractal media. A generalized Darcy law for flux completes the model. Then we develop the basic equation for Well test analysis as is applied in petroleum engineering. Finally, the equation is solved by Laplace transform and inverted numerically to illustrate anomalous diffusion. In this case by showing that the flow rate from fractal systems is smaller than that from the Euclidean system.     

On Λ-fractional elastic solid mechanics

After defining the fractional Λ-derivative, having all the requirements for corresponding to a differential, the fractional Λ-strain is established. Contrary to the common strain, that has a local character, fractional strain access a non-local character, quite important for expressing deformations in non-homogeneous media with microcracks and inhomogeneities, that may change during deformation. The purpose of the present work is the establishement of the principles and laws of the non-linear Λ-fractional Elasticity. The Λ-fractional non-linear stress–strain relations are derived. The restriction into the linear fields is presented. Further, fractional deformation of a fractal bar is discussed. The Fractional deformations and fractional elastic problems are set up with the definition of stresses and displacements in the initial space. Further, the Λ-fractional analysis with its conjugate Λ-fractional space is presented, considering fractional derivatives of both sides in the bending of a cantilever beam under uniform continuously distributed loading.

     

Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media

A theoretical analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media is studied. Using the method of symmetry group of scaling transformations and the H-function, the analytical solutions of concentration distribution are given. At the same time we derive the expressions of scattering function spectrum.The result shows that the scattering function spectra still have the properties of scaling function. The scattering functions of point source, line source and area source in regular Euclidean space can be regarded as particular cases of this paper and are included in this paper. At the end of the paper we discuss the asymptotic behaviors of the solution in detail. The results of this paper can be taken to be the fundamental solutions for every kind of boundary value problems of fractional anomalous diffusion in disordered fractal media.     

A non-Fickian,particle-tracking diffusion model based on fractional Brownian motion

The work is motivated by the recent discovery that ocean surface drifter trajectories contain fractal properties. This suggests that the dispersion of pollutants in coastal waters may also be described using fractal statistics. The paper describes the development of a fractional Brownian motion model for simulating pollutant dispersion using particle tracking. Numerical test cases are used to compare this new model with the results obtained from a traditional Gaussian particle-tracking model. The results seems to be significantly different, which may have implications for pollution modelling in the coastal zone. © 1997 John Wiley & Sons, Ltd.     

Statistical distributions and self-organizing phenomena: what conclusions should be drawn?

Some salient properties of the inverse power law distribution, the exponential distribution, catastrophe distributions, and the relationships among them were explored and compared. Self-organizing events may display any of these distributions. Catastrophe functions and their distributions do not display fractional (fractal) dimensions. Thus it is possible to have self-organization without the fractal. An empirical example from leadership emergence research illustrated a situation where a power law distribution provided a poor characterization of the data, but a swallowtail catastrophe model did so quite well. The results call into question some simplistic assumptions about the relationships among fractals, inverse power laws, self-organization and so-called pink noise.     

Fractal sets in control systems

In this paper the Hausdorff measure of sets of integral and fractional dimensions is introduced and applied to control systems.A new concept,namely,pseudo-self-similar set is also introduced.The existence and uniqueness of such sets are then proved,and the formula for calculating the dimension of self-similar sets is extended to the psuedo-self-similar case.Using the previous theorem,we show that the reachable set of a control system may have fractional dimensions.We hope that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.     

Diffusion in pore fractals: A review of linear response models

A major aspect of describing transport in heterogeneous media has been that of relating effective diffusivities to the topological properties of the medium. While such effective transport coefficients may be useful for mass fractals or under steady state conditions, they are not adequate under transient conditions for self-similar pore fractal media. In porous formations without scale, diffusion is anomalous with the mean-squared displacement of a particle proportional to time raised to a fractional exponent less than unity. The objective of this review is to investigate the nature of the laws of diffusion in fractal media using the framework of linear response theory of nonequilibrium statistical mechanics. A Langevin/Fokker-Planck approach reveals that the particle diffusivity depends on its age defined as the time spent by the particle since its entry into the medium. An analysis via generalized hydrodynamics describes fractal diffusion with a frequency and wave number dependent diffusivity.     

基于分形导数对非牛顿流体层流的数值研究

非牛顿流体具有复杂的流变特性,揭示该流变特性可以更加合理地指导非牛顿流体在工农业生产中的应用.经典的非牛顿流体本构模型往往形式复杂,仅能应用于某些特定的情况.分数阶导数模型具有参数少和形式简单的特点,己成功地应用于描述非牛顿流体的运动.Hausdorff分形导数作为一个备选的建模方法,相比分数阶导数具有更简单的形式以及更高的计算效率.本文基于Hausdorff分形导数改进现有牛顿黏性模型,提出分形黏壶模型.通过研究分形黏壶在常应变率下表观黏度的变化情况,以及在加、卸载条件下的蠕变及恢复特性,发现分形黏壶模型适合于描述具有黏弹性的非牛顿流体(本文称之为分形流体).结合连续性方程及运动微分方程,推导出分形流体在平行板间层流的基本方程.按是否拖动上板和是否存在水平的压力梯度分为3种工况,分别用数值方法计算这3种工况下流速在板间的分布及其随时间变化的情况.通过分析不同工况下的流速分布,发现水平的压力梯度会改变流速随时间变化的形状,且会推迟流速到达稳定的时间.在水平压力梯度不存在的情况下,不同阶数的分形流体具有相同的流速分布或是演变过程.另外,在水平压力梯度存在的情况下,上板速度不影响不同阶数分形流体间稳定速度的差值.     

Fractional derivative reconstruction of forced oscillators

Fractional derivatives are applied in the reconstruction from a single observable of the dynamics of a Duffing oscillator and a two-well experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual information between the observable and its fractional derivative. The ability of this reconstruction method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and fractal dimensions. The reconstruction is compared to the true phase space and the delay reconstruction in order to assess the reconstruction parameters and the quality of results.     

Waves in Fractal Media

The term fractal was coined by Benoît Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with pre-fractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton??s principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.