The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

On the Hausdorff dimension of fractal attractors

We consider such mappingsx _n+1=F(x_n) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...     

The fractal dimension of the Lorenz attractor

Taken's box-counting algorithm for computing the fractal dimension of a strange attractor is applied to the Lorenz equation. A convergence problem is discussed, and an approximate dimension is computed.     

The fractal dimension of the inflationary universe

In cosmological scenarios of the new inflationary type, inflation never ends completely. The total volume of inflating regions grows exponentially with time, and they form a self-similar fractal of dimension slightly less than 3.     

Plasma resonance — enhanced raman scattering by absorbates on gold colloids: The effects of aggregation

The excitation profile of the 1014 cm^?1 Raman band of pyridine adsorbed on colloidal gold particles, and the extinction and elastic scattering spectra of the colloids, are measured as the colloids slowly aggregate in the presence of pyridine. Transmission electron microscopy shows that the aggregates formed are predominantly strings of particles rather than compact clusters, and the dipolar plasma modes of the aggregates are therefore split into longitudinal and transverse components. It is shown that only for excitation under the longitudinal resonance extinction band is there a large Raman intensity enhancement. The Raman excitation profile maximum corresponding to excitation under this resonance moves progressively to longer wavelengths, increasing substantially in height, as the aggregation proceeds. Thus aggregation is most advantageous for the realization of large Raman signals from these colloids, the Raman intensity at a given excitation wavelength increasing approximately as the square of the absorbance at that wavelength as the aggregation proceeds. These observations are discussed in relation to the electromagnetic field enhancement contribution to the surface Raman effect, with which they are in general agreement, and the large increase in ¦?¦² for gold and silver with increase in wavelength is shown to be a significant factor in accounting for some of these effects of aggregation.     

On the fractal dimension of the Henon attractor

We re-measured the fractal dimension of the Hénon attractor by direct box-counting. We paid special attention to (a) optimal speed and use of storage, and (b) systematic corrections due to the finiteness of the number of iterations. Covering with grids of up to 9600 × 9600 boxes, we observe that the number N(?, n) of boxes visited after n iterations obeys a scaling law N(?, ∞) - N(?, n) ≈ const × ?^-αn^-β (for n → ∞) with α = 2.42 ± 0.15, β = 0.89 ± 0.03. Using this extrapolate to n → ∞, we obtain D = 1.28 ± 0.01 in disagreement with previous box-counting estimates, but in agreement with a recent indirect evaluation.     

分形烟尘簇团粒子的消光特性研究

根据物质的电结构,将烟尘簇团粒子离散为一系列的偶极子,利用离散偶极子近似方法,获得了不同波长电磁波入射情况下分形烟尘簇团粒子不对称因子、吸收和散射效率因子的值.结合偶极子在电磁场中的力学性质,分析了烟尘簇团粒子在外加电磁场中所受的力和力矩,给出三种分形结构烟尘簇团粒子的电磁辐射压力及电磁辐射力矩效率因子随入射波波长变化的数值结果,分析了烟尘粒子对电磁能量吸收的宏观表现和微观机理.结果显示:入射波长、烟尘簇团粒子的分形维数及所含原始微粒数目都将影响烟尘粒子的消光特性.     

The fractal dimension of the sem-picture of slip lines

Computer experiments modelling the plastic slip deformation of an idealized single crystal have shown that under certain conditions the fractal exponentD of the slip line pattern on the crystal surface may differ from the value ofD established on the picture of this surface.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,(), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.