There is a certain amount of arbitrariness in defining a trajectory scaling function for the circle map for the various quasiperiodic routes to chaos. We show how one may exploit this freedom to select trajectory scaling functions to suit a particular purpose. For comparison with experiment, one might seek a trajectory scaling function that is relatively flat, is smooth, clearly distinguishes between the subcritical and critical regimes of the quasiperiodic route to chaos, and is quite resilient to noise. To allow for experimental difficulties in obtaining (consecutive) exact cycles, we have also developed a trajectory scaling function that is constructed from subcycles within some higher-order cycle. We consider the quasiperiodic routes to chaos with golden-mean and silver-mean (√2 -1) rotation winding numbers.

L=0 doubly excited states, whose mean value of the interelectronic angle is close to π, are studied within the O(4,2) group-theoretical framework. It is conjectured that the energies of these resonances correspond in the limit of large excitation of both electrons to those of the Hamiltonian H=${\mathit{p}}^2$/2+p${\mathcal{’}}^2$/2-Z/r-Z/r’+1/(r+r’) in a space where both particles have fixed value, to be denoted f, for their angular momenta (l=l’=f). The choice for the fixed value f is not important in the limit of infinite principal quantum numbers. In other cases, f must remain smaller than or of the order of the square root of the lowest principal quantum numbers of both electrons. The initial problem with three degrees of freedom thus could be reduced, insofar as energies of resonances are concerned, to a problem with only two degrees of freedom in the limit of large excitation of both electrons. It is of course obvious that the state vectors of the above Hamiltonian with l=l’=f are totally inadequate for describing real two-electron atomic states with interelectronic angles close to π. However, the coefficients of the expansion of these unphysical states with respect to specified Sturmian vectors with l=l’=f approximate the coefficients of the real two-electron states with respect to different specified Sturmian vectors; the latter Sturmian vectors have a large distribution over different values of l (=l’).

A class of one-dimensional Fokker-Planck and Schrödinger equations is defined that are solvable by scalar-continued-fraction calculations for the eigenvalues and for the coefficients in the eigenfunctions. In a selection of preliminary case studies, possibilities and restrictions of the solution method are explored, and a variety of mono-, multi-, and metastable potentials is shown to belong to the class. Different solution-method variants, including the supersymmetry principle, are discussed.

A special, seemingly infinite sequence of saddle-node bifurcations of the driven double-well Duffing oscillator is investigated. It occurs in resonances with even torsion number and shows period-adding behavior. The sequence of saddle-node bifurcations gives rise to a regular window structure of higher and higher period. An empirical law is given of the organization of periods and torsion numbers when proceeding along the sequence to the limit.

The problem of finding the coefficients of a simple series expansion for a quasiconserved quantity K for the Hénon-Heiles Hamiltonian H using a single set of variables is solved. In the past, this type of approach has been problematic because the solution to the equations determining the coefficients in the expansion is not unique. As a result, the existence of a consistent expression for K to all orders had not previously been established. We show how to deal with this arbitrariness in the expansion coefficients for K in a consistent way. Due to this arbitrariness, we find a class of expansions for K, in contrast to the single unique expansion for K generated by the normal-form approach of Gustavson [Astron. J. 71, 670 (1966)]. It may be possible to devise a criterion for deciding which one of our expansions is ‘‘optimally convergent,’’ although we do not deal with this question here. We proceed by introducing a single set of dynamic variables that have simple symmetry properties and that also ‘‘diagonalize’’ the problem of finding the coefficients of K. No canonical transformations are required. A straightforward constructive procedure is given for generating the power series to any order for quantities having the symmetry of the Hamiltonian that are formally conserved. This leads to a very practical method for calculating a quasiconserved quantity in the Hénon-Heiles problem. A comparison is made through several orders of the terms generated by this approach and those generated in the original Gustavson expansion in normal form.

We study a dissipative standardlike map that contains a parameter z that can be used to tune the map from a piecewise-linear map (z=0) to the dissipative standard map (z=∞). When both the tuning parameter z and dissipation are small, the reappearance of an invariant circle after its breakup is observed. This recurrence phenomenon takes place as the nearby mode-locked resonances separate after they overlapped. However, as the dissipation is increased, the number of recurrences gradually decreases, and ultimately reappearance ceases at some dissipation parameter value dependent on z. Scaling behavior at the disappearance and reappearance points is also discussed.

Linear and nonlinear properties of the response of the discrete Ricker system ${\mathit{x}}_{\mathit{n}+1\mathrm{}}$=${\mathit{x}}_{\mathit{n}}$exp[r(1-${\mathit{x}}_{\mathit{n}}$)] against modulating the control parameter r with periodic, aperiodic, and stochastic perturbations ${\xi }_{\mathit{n}}$ via r(1+Δ${\xi }_{\mathit{n}}$) are investigated analytically and numerically in the range where the unmodulated map undergoes its first pitchfork bifurcation from the fixed point ${\mathit{x}}^{\mathrm{*}}$=1 into a period-2 limit cycle. Parametric modulation does not destroy the fixed point. On the contrary, it is stabilized in a universal manner for small modulation amplitudes Δ, irrespective of the kind of perturbations destroyed. However, modulation dramatically changes the period-2 limit cycle of the unmodulated map. A formula for the stability boundary ${\mathit{r}}_{\mathit{c}}$(Δ) of the fixed point is derived and evaluated for various periodic and stochastic modulation types as a function of modulation amplitude Δ. For periodic modulation linear conditions are elucidated for the bifurcation of limit cycles at ${\mathit{r}}_{\mathit{c}}$(Δ) that are harmonic or subharmonic with respect to the modulation. Structure and dynamics of the time-dependent nonlinear solutions that bifurcate out of the fixed point and that depend sensitively on details of the modulation dynamics are investigated analytically in comparison with numerical simulations. The difference in the response against periodic modulation with even and odd periods N is elucidated in particular for N=2 and 3, where the difference is most striking. The peculiarities of the temporal behavior of the map under modulation with large periods are explained. The influence of stochastic perturbations on the period-2 limit cycle of the unmodulated system is determined analytically and numerically by evaluating various averaged quantities that seem appropriate for a statistical description of the stochastic response. Conditions are evaluated under which the flip behavior of successive sign changes, sgn(1-${\mathit{x}}_{\mathit{n}+1\mathrm{}}$)=-sgn(1-${\mathit{x}}_{\mathit{n}}$), of the unmodulated map is preserved under modulation.

We investigate statistics of matrix elements of observables, calculated in an eigenbasis of a quantum-mechanical evolution operator corresponding to a classically chaotic system. We propose a criterion that allows one to predict whether or not such statistics are faithful to random-matrix theory and define a random operator with respect to a given system.

We propose a generalized hierarchical model to study the multifractal properties of the random resistor and random superconductor networks. Analytical expressions for the moments of the voltage and current distributions are derived, and the critical exponents are calculated. The theoretical results are found to be in better agreement with the simulation data than previous studies.

The statistics of random cellular patterns in premixed flames are analyzed. Agreement is found with a variety of topological relations previously found for other networks, namely, Lewis’s law and Aboav’s law. Despite the diverse underlying physics, flame cells are shown to share a broad class of geometric properties with other random networks—metal grains, soap foams, bioconvection, and Langmuir monolayers.

For a two-electron atom, many D=3 states have the same energies as D=5 states of lower angular momentum. Thus the energies of ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}^{\mathit{e}}$, ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}^{\mathit{e}}$, ${}_{}{}^3{}_{}{}^{}$${\mathit{D}}^{\mathit{o}}$, and ${}_{}{}^1{}_{}{}^{}$${\mathit{D}}^{\mathit{o}}$ states for D=3 are respectively identical to those for ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}^{\mathit{e}}$, ${}_{}{}^3{}_{}{}^{}$${\mathit{S}}^{\mathit{e}}$, ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, and ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$ states at D=5. We exploit these interdimensional degeneracies to obtain accurate energies for doubly excited 2pnp ${\mathit{P}}^{\mathit{e}}$ states of helium at D=3, with n=2–6, by calculating energy eigenvalues for the singly excited 1s(n-1)s ${\mathit{S}}^{\mathit{e}}$ states at D=5. We also illustrate how some qualitative aspects of double-excitation spectra can be elucidated in terms of interdimensional degeneracies.

The dynamics of high-dimensional Hamiltonian flows is extensively investigated by means of numerical simulations in the case of the Fermi-Pasta-Ulam (FPU) β model and classical lattice ${\mathit{cphi}}^4$ model; both are considered at N=512 degrees of freedom. This work aims at investigating the major consequences on the dynamical phenomenology of the existence of a strong stochasticity threshold. This threshold corresponds to a transition from two different diffusion regimes in phase space: slow diffusion (along resonances) at low-energy density and fast diffusion (across resonances) at high-energy density. Wave packets are initially excited. The relaxation time ${\mathrm{\tau }}_{\mathit{R}}$ toward equipartition of energy is measured following the time behavior of spectral entropy. A systematic study of ${\mathrm{\tau }}_{\mathit{R}}$=${\mathrm{\tau }}_{\mathit{R}}$(ɛ,n${\mathrm{̃}}_{\mathrm{exc}}$) is reported, where ɛ is the energy per degree of freedom and n${\mathrm{̃}}_{\mathrm{exc}}$ is the average wave number of the initially excited packet.

In the FPU case, it is found that below the strong stochasticity threshold ${\mathrm{\varepsilon }}_{\mathit{c}}$, the equipartition time is an increasing function of n${\mathrm{̃}}_{\mathrm{exc}}$, i.e., high-frequency modes tend to freeze compared to low-frequency modes. This is in qualitative agreement with the predictions of a so-called narrow-packet approximation in which the FPU model is approximated by a nonlinear Schrödinger equation. However, above ${\mathrm{\varepsilon }}_{\mathit{c}}$, the situation is reversed, and initial excitation of high-frequency modes yields quicker mixing. Also, this is in qualitative agreement with some analytical predictions. In the ${\mathit{cphi}}^4$ case, at ɛ>${\mathrm{\varepsilon }}_{\mathit{c}}$ the excitation of high-frequency modes results in exponentially increasing ${\mathrm{\tau }}_{\mathit{R}}$ as a function of n${\mathrm{̃}}_{\mathrm{exc}}$. At ɛ<${\mathrm{\varepsilon }}_{\mathit{c}}$, above some critical n${\mathrm{̃}}_{\mathrm{exc}}$, ${\mathrm{\tau }}_{\mathit{R}}$ is apparently divergent. It is also shown that the crossover in the scaling behavior ${\lambda }_1$(ɛ) of the largest Lyapunov exponent occurs always at ${\mathrm{\varepsilon }}_{\mathit{c}}$ independent of the initial conditions, thus providing a good intrinsic probe of the strong stochasticity threshold.

The second virial coefficient for a D-dimensional polymerized manifold with only mutual repulsive interaction for the same internal D coordinate has been calculated using the perturbative renormalization-group method. This can be carried out to all orders of the perturbative expansion for this model. The exact nontrivial fixed point is located for the D manifolds embedded in d dimensions.

The paper makes two contributions. In the first place, Milne’s integral equation is generalized by assigning, to the surface X=0, a reflection coefficient κ (0≤κ≤1), and the variational method, which has proved exceedingly successful in dealing with a black boundary (κ=0), is put to use. It is found that even the crudest trial function, needing no variation at all, yields results for the extrapolated endpoint (${\mathit{X}}_0$) that are on a par with those found previously by more painstaking procedures. A two-parameter trial function gives ${\mathit{X}}_0$ correct to 0.16 ppm; the largest error in the particle density, slightly under 0.3%, occurs at a black boundary, and decreases rapidly as κ or the distance from the boundary increases. Second, an efficacious strategy is proposed for overcoming the moot problem in the spherical-harmonics method, viz., the choice of the boundary conditions.

We present computer-simulation results of self-avoiding walks (SAW’s) on a percolation cluster for a square lattice performed very close to the percolation threshold. We specifically consider the disorder averages of SAW’s on all clusters supporting one or more N-step walks and those on a backbone of an infinite cluster and estimate the critical exponents ν and γ that characterize the disorder averages of the end-to-end distance and the number of SAW’s, respectively. Our results for γ indicate a behavior rather similar to SAW’s on fully occupied lattices for both cases, while for ν one of two cases shows different behavior.

We report the calculation of low-energy state-selective electron-capture cross sections in an α-particle collision with molecular hydrogen. The semiclassical molecular-orbital expansion method is used and electron translation effects are taken into account. The calculated ${\mathrm{He}}^{+}$(2p) formation cross sections are in good overall agreement with the recent measurements of Hoekstra et al. [J. Phys. B 22, L603 (1989)]. However, the experimental cross sections show a pronounced plateau between E=0.7 and 2.0 keV/amu. In the same energy region the calculated results exhibit an oscillatory structure that arises from a combination of Landau-Zener and Rosen-Demkov oscillations.

We have calculated a complete set of Fano-Starace parameters for partial photoionization cross sections of He in the vicinity of the 3s2p ${}_{}{}^1{}_{}{}^{}$P° resonance, using a recently proposed multichannel ${\mathit{L}}^2$ method for continuum wave functions [F. Martriaan, A. Riera, and I. Sánchez, J. Chem. Phys. 94, 4295 (1991)]. Our results are in good agreement with recent photoabsorption measurements [D. W. Lindle, T. A. Ferrett, P. A. Heimann, and D. A. Shirley, Phys. Rev. A 36, 2112 (1987)] and show the importance of correctly including interaction between the different open channels to interpret the experimental data.

The C state of ${\mathrm{CO}}^{+}$ has been studied using uv photoelectron spectroscopy. The spectrum was excited by a monochromatized He i${\mathrm{i}}_{\mathrm{\alpha }}$ electron cyclotron resonance source. The vibrational progression in the C state has earlier been obesrved up to v=6, and an anomalous anharmonicity has been inferred. In the present spectrum the progression has been followed up to v=11, and the progression can be described as originating from two nearly harmonic potentials. A perturbation is thus found for the C state at an internuclear distance of about 1.4 Å. Moreover, the progression extends 0.09 eV above the dissociation limit and therefore a potential barrier must exist with a maximum around 2.3 Å.

We demonstrate laser cooling of ${}_{}{}^{172}{}_{}{}^{}\mathrm{Yb}_{}^{+}{}_{}{}^{}$ ions in a rf quadrupole trap using an alternative optical-excitation scheme. The metastable 4${\mathit{f}}^{14}$5${\mathit{d}}^2$${\mathit{D}}_{3/2}$ state of ${\mathrm{Yb}}^{+}$, to which ions decay if the resonance (cooling) transition is driven, is depleted by driving an optical transition at a wavelength of 935 nm, which is much more attractive than the 2.438 μm radiation used previously. This couples the metastable state to a fourth energy level from which the ions decay rapidly to the ground state. Fluorescence line shapes of the cooling transition indicate that small numbers of ${\mathrm{Yb}}^{+}$ ions were cooled in this way to kinetic temperatures below 1 K.

Partial cancellation of the ionization signal in the four-photon-resonant five-photon ionization via J=2 4f and 5f resonances in Xe has been observed at increased gas pressures. Experiments with one and two counterpropagating laser beams provide strong evidence that cancellation is due to destructive interference at the three-photon step between the coherent three- and one-photon excitation channels, the latter involving one vacuum ultraviolet photon of the third-harmonic generation that occurs above the 6s[3/2${]}_1^{\mathit{o}}$ state. The well-known destructive interference for the ‘‘on-resonance’’ case is now shown to occur also for large detuning from the resonance (up to 2000 ${\mathrm{cm}}^{\mathrm{-}1}$) probed by four-photon-resonant five-photon ionization. This type of cancellation may provide a new insight in interference effects involved in multiphoton-excitation processes.