On the scaling factors α(z) and δ(z)

We introduce a new method to determine the scaling factors α(z) and δ(z) for the period-doubling route to chaos in dissipative systems, exemplified by the one-dimensional mapping x_n+1=1−λ∣x_n∣^z. With the help of the Feigenbaum universal functions g(x) and h(x) we derive the inequality α^z−α<δ(z)<α^z, implying in particular that δ(z) remains finite (≲30) in the limit z → ∞.     

Further regularities in period-doubling bifurcations

The period-doubling bifurcations of the map x → x′ = f(λ, x) is known to be characterized by a generalized renormalization ground expressing simultaneous scaling in λ and x along the central sequence of bifurcation points (i.e. the sequence which converges to the maximum in f). For other sequences (e.g. the sequence with the largest x in each cycle), the generalized RG is valid only in a well defined subclass of functions S. For other cases, only separate scalings in λ and in x are valid. However, every f can be related to a representative f in S by a linear conjugacy, so that the generalized RG nevertheless reflects a universal regularity. Moreover, the spacings of the bifurcation points are found to be characterized by novel subasymptotic eigenvalues related to δ and α.     

Critical behavior of the Lyapunov number at the period-doubling onset of chaos

For period-doubling bifurcations of 1 d-maps the Lyapunov number λ is calculated explicitly using a renormalization procedure. We find that its slope diverges like (δ/2)^k. In the chaotic regime the unstable cycles yield a continuous curve as an upper bound for the Lyapunov exponent of chaotic bands. The bound has a critical exponent t = 0.449 80…, which is the same as for the chaotic bands.     

57Fe hyperfine interaction parameters and selected magnetic properties of high purity MFe12O19 (M=Sr,Ba)

Much of the confusion regarding the ⁵⁷Fe Mössbauer spectroscopic hyperfine parameters of SrFe₁₂O₁₉ and BaFe₁₂O₁₉ at 300 K has been removed by means of an interlaboratory investigation of well-characterized samples prepared from high purity starting materials. In contrast to previous investigations, the contributions of each of the five Fe sublattices to the Mössbauer spectrum are discernible at 300 K and five components are necessary for an adequate fit of the data. The relative magnitudes of the hyperfine fields, H_n, and isomer shifts, δ, are as follows: for SrFe₁₂O₁₉ and BaFe₁₂O₁₉: H_n(2b)< H_n(12k)<H_n(4f₁) <H_n(2a)<H_n(4f₂), for SrFe₁₂O₁₉ δ(4f₁≤δ(2b)<δ(12k)δ(2a)<δ(4f₂), and for BaFe₁₂O₁₉ δ4f₂), and for BaFe₁₂O₁₉ δ(4f₁<δ(4f₁)<(12k) < δ(4f₂)<δ(2a). The above assignment is based on considerations of both magnetic and crystal/chemical structures.The high purity starting materials seem to have appreciable influences on both hyperfine interaction parameters and bulk magnetic properties as observed by others for spinel and garnet ferrites.     

Universal power spectra for the reverse bifurcation sequence

Many dynamical systems exhibit forward and reverse period-doubling bifurcation sequences, the latter being intrinsically noisy. Feigenbaum has predicted the amplitude of sharp spectral components in the forward sequence from universality arguments. In the same spirit we derive the approximate form of the broad band features in the reverse sequence. Our results give a power-law behavior of the integrated noise spectrum similar to that recently reported by Huberman and Zisook.     

Bound s-state energies for the exponential cosine screened Coulomb potential in Ecker-Weizel approximations

Bound s-state energies of an electron in the exponential cosine screened Coulomb potential, V(r) = ?e²e^?δrcos(δr)/r, are obtained analytically using the Ecker-Weizel approximation. For reasonable values of the screening parameter δ, the predicted results are in excellent agreement with the variational calculation of Lam and Varshni. The effect of screening on the shift of the quantum numbers from their normal values is also studied for various s-states.     

Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps

We review critical situations, linked with period-doubling transition to chaos, which require using at least two-dimensional maps as models representing the universality classes. Each of them corresponds to a saddle solution of the two-dimensional generalization of Feigenbaum-Cvitanovi? equation and is characterized by a set of distinct universal constants analogous to Feigenbaum’s α and δ. One type of criticality designated H was discovered by several authors in 80-th in the context of period doubling in conservative dynamics, but occurs as well in dissipative dynamics, as a phenomenon of codimension 2. Second is bicritical behavior, which takes place in systems allowing decomposition onto two dissipative period-doubling subsystems, each of which is brought by parameter tuning onto a threshold of chaos. Types of criticality designated as FQ and C occur in non-invertible two-dimensional maps. We present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems. In particular, we indicate a possibility for realization of the H type criticality without vanishing dissipation, but with its compensation in a self-oscillatory system. Next, we present a number of examples (coupled Hénon-like maps, coupled driven oscillators, coupled chaotic self-oscillators), which manifest bicritical behavior. For FQ-type we indicate possibility to arrange it in non-symmetric systems of coupled period-doubling subsystems, e.g. in Hénon-like maps and in Chua’s circuits. For C-type we present examples of its appearance in a driven Rössler oscillator at the period-doubling accumulation on the edge of syncronization tongue and in a model map with the Neimark–Sacker bifurcation     

Semi-empirical analysis of magnetic moments of single-particle (or-hole) states around the 208Pb core

Magnetic moments of single-particle (or-hole) states around the ²⁰⁸Pb core were fitted by three state-independent parameters δg_l^(p), δg_l⁽ⁿ⁾ and α, where α is a parameter that is connected closely with δg_S and g_p. The analysis was performed under the assumptions that (i) the state-dependence of radial integral I((nl)²(n'l')²) involved in δg_s and g_p is calculated by the harmonic-oscillator potential, and (ii) δg_s is taken as being equal to ?4g_p, as expected from the δ-force type core polarization. An excellent fit was obtained when δg_l^(p) = 0.10(2), δg_l⁽ⁿ⁾ = ?0.05(1) and α = 1.2(1), which show that the main contributor to δg_s is the M1 core polarization and δg_l⁽ⁿ⁾ is certainly smaller than 0. B(M1)'s for allowed transitions calculated with the use of the same parameters as above are smaller than the observed B(M1)'s whereas α = 1.01(1) explains well the observed B(M1)'s.     

Calculation of Jacobians for inverse radiative transfer: An efficient hybrid method

We present an accurate and numerically efficient procedure of calculating Jacobians by finite difference that consists of two components: (1) a method employing the saving of atmospheric layers that accelerates the solution to the equation of radiative transfer for solvers that use the Discrete Space formulation and (2) a method of perturbing the eigenmatrix spectrum associated with a reduced attenuation matrix. The procedure eliminates the need to call the eigenmatrix package, here, LAPACK a second time and provides insights into the fundamental properties of the attenuation matrix, useful for characterizing the accuracy of the derivatives calculated by finite difference methods. The computational complexity of the perturbation method is 8n³+22n², where n is one half the number of streams in the radiance field as opposed to 16n³ using LAPACK. The method is not limited to the calculation of base state radiances I(ω) and those associated with an ‘infinitesimal’ perturbation I(ω+δω) (from which the numerical derivative of I(ω+δω) with respect to δω may be approximated), but is also useful in the calculation of radiances associated with a ‘finite’ perturbation I(ω+Δω) from which a sensitivity can be calculated.     

On the entropy of melting

The analysis of experimental data shows that the function δS/R = f(δV/V_S), where δS/R is the entropy of melting and δV/V_S the relative change of volume at melting, is the same for argon and sodium. The melting entropy of these substances tends to Rln2, when δV/V_S vanishes.     

Universal Scalings in Homoclinic Doubling Cascades

Cascades of period doubling bifurcations are found in one parameter families of differential equations in ℝ³. When varying a second parameter, the periodic orbits in the period doubling cascade can disappear in homoclinic bifurcations. In one of the possible scenarios one finds cascades of homoclinic doubling bifurcations. Relevant aspects of this scenario can be understood from a study of interval maps close to x↦p+r(1 −x ^β)², β∈ (?,1). We study a renormalization operator for such maps. For values of β close to ?, we prove the existence of a fixed point of the renormalization operator, whose linearization at the fixed point has two unstable eigenvalues. This is in marked contrast to renormalization theory for period doubling cascades, where one unstable eigenvalue appears. From the renormalization theory we derive consequences for universal scalings in the bifurcation diagrams in the parameter plane. Received: 16 June 1999 / Accepted: 24 April 2001     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

The spin of the 2,198 keV level in74Ge and multipole mixing ratios of gamma transitions in the decay of74As

Directional-correlation measurements of several cascades in the⁷⁴As decay have been made with a NaI — Ge(Li) detectors system. In the⁷⁴Ge the following cascades were measured: 608–596 keV, 994–1,204 keV, 994-(608)-596 keV and 887-596 keV. The corresponding results give a unique spin assignment of 2 for the 2,198 keV level and determine theE2/M 1 mixing ratios ofδ(608)=3.4±0.4 andδ(994)=?(2.8 ± 0.2). In the⁷⁴Se it was measured the 635-635 keV cascade. The result obtained assigned the spin of the 1,270 keV level as 2 and fixed the E2/M1 mixing ratio of δ(635)=?(2.6 ± 0.2). Also aγ-intensity study has been made to determine the branching ratios of the transitions that deexcite the 1,270 keV level in⁷⁴Se. From these data aI(635)/I(1,270)=12.5 ± 5.0 is obtained.     

Precision determination of the fine-structure splittings of the 5d and 6d states in potassium

The fine-structure splittings of the inverted 5 and 6² D states of potassium were measured by observing level-crossing signals at high magnetic fields. TheD levels were populated using step-wise excitations, employing two CW tunable dye lasers. For the fine-structure intervals δW we find: δW (5d)= ?15,102.0 (5) MHz andδW(6d) = ?7,965.8 (6) MHz. These values are about 100 times more accurate than previous results for these states. An expression is given, describing the fine-structure splittings in the sequence of potassium²D states.     

Bifurcations in a model of magnetoconvection

Bifurcations from oscillatory solutions are studied in a truncated model of two-dimensional Boussinesq magnetoconvection. The fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period-doubling cascade. After the accumulation point there is a semiperiodic cascade leading to chaotic behaviour. Then the semiperiodic cascade is repeated in reverse, followed by a period-halving cascade and a bifurcation back to symmetry. Finally, the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle-foci. The interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions, each with its own cascade. This pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle-foci with eigenvalues that satisfy Shil'nikov's inequality.     

n(G)-nonextendibility of linear positive maps

Linear positive maps of C¹-algebras into the algebra of all bounded operators acting on a Hilbert space are considered. A special class of n(G)-nonextendible maps, which contains the class of nonextendible positive maps introduced by S.L. Woronowicz [9], is defined and studied. A map is n(G)-positive if it is n-positive or covariant with respect to the action of some group G of automorphisms of a C¹-algebra. n(G)-nonextendible maps are those which are nonextendible in this class in the sense of [9].Any n(G)-positive map can be obtained from an n(G)-nonextendible one by restriction.     

Renormalization group approach to Sterman-Weinberg jets

A renormalization group approach is developed for single-jet Sterman-Weinberg cross sections. For small δ², but large Q²δ², the complete δ², Q² and ? dependence of the cross section for e⁺e^? annihilation is shown to be calculable in QCD systematically in powers of α(Q²). ? is not assumed to be small.     

Quantum effects in the critical properties of the transverse Ising model for 2 < d ⩽ 3

The crossover behaviour of a d-dimensional (2<d?3) Ising model in a transverse field Г is investigated near the multicritical point $\text{[Г, T] = [Г}{}_{\mathrm{c}}\text{(0), 0].}$ A renormalization scheme which removes divergences in the zero-temperature limit is presented. The crossover exponent and scaling function for the longitudinal susceptibility are found.     

Multipole mixing ratios of γ-transltions in 182W

γ-γ directional correlation experiments were performed on 14 cascades in ¹⁸²W populated from the β^? decay of ¹⁸²Ta(115 d). Two Ge(Li) detectors were used in a coincidence arrangement, and the ¹⁸²Ta sources were dissolved in HF acid to minimize extranuclear perturbations. For the 1189keV, 2^? → 2⁺ transition, the measured directional correlation coefficients are consistent only with multipole mixing ratios $\text{δ}\text{(}\text{M}\text{2}\text{E}\text{1)}\text{= 0.45 ± 0.03}\text{and}\text{δ}\text{(E3}\text{E1)}\text{= ?0.67 ± 0.07}$. These mixing ratios are discussed and compared with the known conversion coefficients for the 1189keV transition. The E2/M1 multipole mixing ratios determined are (energy in keV): δ(66) = 0.15 ± 0.15, δ(85) = 0.31 ± 0.05, δ(114) = 0.31 ± 0.05, 0.56 ≦ δ(179) ≦ 1.36, δ(1121) = 12⁺²_?1, and δ(1231) = ?(32⁺¹⁴²_?15). The measured M2/E1 mixing ratios are: δ(68) = 0.03 ± 0.02, δ(152) = 0.014 ± 0.013 and δ(156) = ?0.13 ± 0.19.     

Multipole mixing ratios of transitions in Te

Gamma-gamma directional correlation measurements were made on nine transitions in ¹²⁴Te with a NaI(Tl)-Ge(Li) detector arrangement and multichannel analysis. The multipole mixing ratios obtained were δ(646) = 0.000±0.001, δ(714) = 1.5_−0.3^+0.6, δ(723) = −3.3±0.2, δ(1437) = 3.7_−2.0^+2.7, δ(1489) = −3.4_−1.5^+0.9, δ(968) = −0.03_−0.05^+0.06, δ(1368) = −0.045±0.090, δ(1045) = 0.041_−0.041^+0.047, δ(1691) = −0.02±0.01, and δ(2091) = 0.00_−0.03^+0.02. The first δ is M3/E2, the next three are E2/M1, and the last five are M2/E1. The retardation (a factor of approximately 50) of the crossover to cascade transitions from the 2039 keV, third 2⁺ level to the second and first 2⁺ levels is essentially the same for both the M1 and E2 components. In addition, spin and parity assignments of 2⁺ were made for the 2039 and 2092 keV levels.