The infinite number of generalized dimensions of fractals and strange attractors

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_q, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_q. We prove that lim _q→0D_q = fractal dimension (D), lim_q→1D_q = information dimension (σ) and D_q=2 = correlation exponent (v). D_q with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_q for any q′ > q. For homogeneous fractals D_q = D_q. A particularly interesting dimension is D_q=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D_∞.     

Fractal dimensions and ƒ(α) spectrum of the Hénon attractor

We measure the generalized fractal dimensions D_q(q⩾0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the D_q yields the ƒ(α) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.     

On the transition to hyperchaos and the structure of hyperchaotic attractors

Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D ₂) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here.     

Characterising chaos-hyperchaos transition using correlation dimension

Transition to hyperchaos is uaually studied by computing the spectrum of Lyapunov Exponents (LE). But such a procedure can be employed mainly when the equations governing the dynamical system are known. However, if the information available on the system is only through time series, the method becomes difficult to implement. We show that the transition to hyperchaos is followed by a sudden change in the topological structure of the underlying attractor. Our numerical results indicate that the transition to hyperchaos can be characterized accurately through the computation of correlation dimension (D ₂) from time series. We use two standard time delayed hyperchaotic systems as examples since, for such systems, D ₂ varies smoothly as a function of the time delay τ which can be used as the control parameter.     

Finite-size crossover in systems with slab geometry

A renormalization group study of the finite-size (dimensional) crossover is carried out with the help pf ε = 4 − d and ε₀ = 3 − d expansion techniques. The finite-size crossover and the invariance relation for the length scale transformation are proven up to the two-loop approximation. The formal equivalence between the finite-size crossover in classical systems and the quantum-to-classical dimensional crossover in certain quantum statistical models is emphasized and exploited. The finite-size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated. It is shown that the shift exponent λ describing the fractional rounding of the critical temperature obeys the relation λ = D − 2, where D is the dimensionality of the system.     

ε-expansion calculations for spin-glasses

The critical properties of the spin-glass transition proposed by Edwards and Anderson are studied using the minimal subtraction method. The universal ratio of the second correction to scaling amplitude to the square of the first for the order parameter susceptibility χ₀ is calculated to first order in ε(ε=6?d). Comparison is made with Fisch and Harris' high temperature series analysis which incorporated Rudnick-Nelson-type corrections to scaling. Within the same formalism the critical exponents are calculated to second order in ε. They agree with the first order ε expansion of Harris, Lubensky and Chen.     

Shot noise in a quantum dot with the coulomb interaction

We study the noise in a quantum dot which is coupled to metallic leads by using the non-equation of motion technique at the Kondo temperature T_K. We compute the out of equilibrium density of states, the current and the shot noise. We find that the shot noise exhibits a nonmonotonic dependence on the voltage when variation of ε_d values of the QD energy in the absence of the external magnetic field occurs. We also find that the amplitude of current exhibits a saturation behavior when driving field is increased.     

Coherent potential approximation for quaternary alloys: Application to phonon spectra in one dimension

The coherent potential approximation is generalized for application to quaternary alloys of the types A_xC_yB_1?x?yD and A_xB_1?xC_yD_1?y. Formalisms are developed for and application is made to the calculation of the phonon spectra of random, mass-disordered quaternary alloys of both types in one dimension.     

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 ...     

Measurement of np and pp asymmetry with an accelerated polarized deuteron beam from 725 to 1000 MeV per nucleon

The accelerated polarized deuteron beam of Saturn II was used to measure the analyzing power for np elastic scattering at five energies. The left-right asymmetries ε = (L + R)/(L + R) for np and for pp elastic scattering were measured simultaneously by CH₂? carbon subtraction using one of the beam-line polarimeters. The analyzing power A_00n0(np) is given by the ratio ε_np^d/ε_pp^d multiplied by the known analyzing power for pp elastic scattering. Experimental evidence is consistent with the underlying assumption that in the kinetmatic region of the experiment the ratio of the np to pp analyzing powers for scattering of quasifree nucleons in deuterons is the same as for scattering of free neutrons and protons, respectively.     

Dilation analyticity in constant electric field

The resolvent of the operatorH ₀(ε, θ)=?Δe ^-20+εx ₁ e ^θ is not analytic in θ for θ in a neighborhood of a real point, if the electric field ε is non-zero. (One manifestation of this singular behavior is that for 0<|Im θ|<π/3,H ₀(ε, θ) has no spectrum in the finite plane.) Nevertheless it is shown that the techniques of dilation analyticity still can be used to discuss the long-lived states (resonances) of a system described by a Hamiltonian of the formH=?Δ+εx ₁+V(x).     

The mechanism of the increase of the generalized dimension of a filtered chaotic time series

The determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes. While this is expected from the Kaplan-Yorke conjecture, we show that for one-dimensional maps a weak filter can induce a self-similarity which is responsible for the increase of the Hausdorff dimension. We are able to calculate the increase of the generalized dimensionD _q for the filtered time series of the logistic mapx _i +1=rx _i (1–x _i ) atr=4 analytically.     

Oscillations in the Determination of Renyi Dimension

To describe the degree of chaos of a strange attractor created by a nonlinear map, GRASSBERGER [1] revived Renyi Dimension D_q which is a common generalization of both metric capacity D₀ and information dimension D₁. But often, the numerical determination of D_q is complicated by an oscillating effect first investigated by BADII and PUOLITI [5] for a dimension similar to D₀. This effect is studied in the present paper. We consider the conditions for the appearance of oscillations and the dependence on q, and we give some numerical examples.     

Time evolution of a fractal distribution: Particle concentrations in free-surface turbulence

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal structure, a rare manifestation of a strange attractor observable in real space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions D_q(t) (q=1 to 6) of the floating particles starting with the uniform distribution D_q(0)=2 for Taylor Microscale Reynolds number Re_λ?160. Focus is on the time evolution of the correlation dimension D₂(t) as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.     

Die Hyperfeinstruktur des übergangs Ag I, λ=19372 å

The hyperfinestructure of the transition AgI, 4d ⁹ 5s ^{2 2} D _3/2-4d ¹⁰ 5p ² P _1/2,λ=19372 å has been investigated with a photoelectric recording Fabry-Perot interferometer and digital data processing. The isotope shiftδ Ν _IS and the magnetic splittings factorsA have been determined to beδ Ν _IS=35,77 (12) mK,A(¹⁰⁹Ag, 5p ² P _1/2)=?7,00 (45) mK, andA(¹⁰⁹Ag, 4d ⁹ 5s ^{2 2} D _3/2)=?12,18 (23) mK. The influence of shielding effects on the value of the volume effect of the isotope shift and the influence of core polarisation on the splitting factors are discussed.     

Measurement of the neutron-proton spin correlation coefficient Ayy at 90° c.m. by elastic scattering of 13.7 MeV polarized neutrons from a polarized proton target

The neutron-proton spin correlation coefficient A_yy was measured at 90° c.m. by elastic scattering of 13.7 MeV polarized neutrons from a polarized proton target. The target polarization of about 60% was produced in a LMN single crystal by dynamic nuclear orientation using 70 GHz microwaves. The target was thin enough to allow the detection of the recoil protons in coincidence with the associated scattered neutrons. Our result for A_yy is the first experimental determination of a neutron-proton spin correlation parameter at such a low energy. To determine the influence of A_yy on the ³S₁-³D₁ mixing parameter ε₁, a single-energy n-p phase shift analysis was performed using the experimentally determined value of A_yy= 0.078 ± 0.014 along with analyzing power and cross section data. As expected, the result for ε₁ changes drastically upon the inclusion of A_yy. An unexpectedly small value of ε₁ = −0.16°±0.54° has been obtained in the present analysis.     

Stark-Effekt-Untersuchungen in der Hyperfeinstruktur der 3s 2 3d 2 D 5/2- und2 D 3/2-Terme des Al I-Spektrums

The resonance fluorescence of the transitions 3d ² D _5/2,3/2 3p ² P _3/2,1/2 in the Al I-spectrum was observed as a function of a magnetic field. Adding an electric field parallel to the magnetic field the shifts of level crossing signals caused by the Stark effect of the electric field were used to separate overlapping signals of the 3d ² D _5/2- and 3d ² D _3/2-states. The following values of the Stark parametersβ of both states and the hyperfine structure constantsA andB of the 3d ² D _3/2-states were deduced: 3d ² D _3/2∶ ¦A¦=99(1) Me/sec · g_J/0,8,B/A=?0,22(12), ¦β¦=0.45 (8) Mc/sec/(kV/cm)² · g_J,/0.8, A/β< 0 3d²D_5/2∶ ¦β¦=0.16 (4) Mc/sec/(kV/cm)² · g_J/1.2, A/β>0. Some qualitative aspects of interconfiguration mixing in the 3d²D-states are discussed.