This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e., not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of ‘‘fully developed’’ chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horse- shoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed.

The spectral densities of the fluctuations of noise-driven underdamped nonlinear oscillators are discussed with particular reference to the large class of systems whose eigenfrequencies vary nonmonotonically with energy. It is shown by analog electronic experiments and theoretically that, astonishingly, the widths of their spectral peaks can sometimes decrease with increasing noise intensity T. The specific system studied, as an example, is the single-well Duffing oscillator in a constant homogeneous field. An explicit expression is derived in terms of T and a field parameter for the shape of the spectral peak. It is shown to be in good agreement with experiment. The possibility of observing such spectral features for localized and resonant vibrations of impurities in solids is discussed.

Amplitude maps that are constructed from a scalar time-series measurement along a trajectory of a dynamical system can be chaotic. This occurs if the underlying flow that generated the time series is chaotic. We explain how amplitude maps of toral flows might give the impression that a transition to chaos has occurred, in that the resulting amplitude map appears complicated and has nontrivial sensitivity to initial conditions. A two-parameter family of aperiodic toral flows is studied and apparently chaotic amplitude maps are found for an open set of parameter values. Such maps are shown to differ from genuinely chaotic maps in that the rate of divergence of orbits is linear, rather than exponential. Three methods of distinguishing them are provided, one analytical and two suitable for use with experimental data.

Recent work in statistical mechanics has developed new analytical and numerical techniques to solve coupled stochastic equations. This paper describes application of the very fast simulated reannealing and path-integral methodologies to the estimation of the Brennan and Schwartz two-factor term-structure (time-dependent) model of bond prices. It is shown that these methodologies can be utilized to estimate more complicated n-factor nonlinear models. Applications to other systems are stressed.

Correlation-dimension calculations have been widely undertaken in attempts to understand various physical and other natural phenomena. Some of these studies have perhaps been incautious in their claims. On the other hand, some authors have been sharply critical of such work. These criticisms hinge on doubts about data-set size. In particular, Smith [Phys. Lett. A 133, 283 (1988)] has suggested that ${42}^{\mathit{D}}$ points are a minimum number of data points, where D is the embedding dimension. He has suggested that, as a result, correlation-dimension calculations are limited to dimensions less than 5 or 6 even on supercomputers. By implication, many published results are placed into doubt, discouraging people from undertaking such calculations. We review the concept of critical embedding dimension and undertake a detailed analysis of data-size requirements, which culminates in tight error estimates for determination of correlation dimension. We conclude that, while data requirements are still substantial, they are not nearly so extreme as has been suggested, making the determination of correlation dimension from data sets feasible. Previously reported ‘‘doubtful’’ calculations of correlation dimension can be reviewed in the light of explicit error estimates.

We study the diffusion-limited reaction A+B→0 analytically by means of fluctuation statistics. Here we consider both the case of equal (${\mathit{A}}_0$=${\mathit{B}}_0$) and that of unequal (${\mathit{A}}_0$<${\mathit{B}}_0$) numbers of reacting particles. We show that the obtained analytical results are in good agreement with computer simulations, in which we focus on reactions in one and two dimensions. Furthermore, as expected from our analytical development, the simulations for different initial concentrations coincide when properly rescaled.

We show that some hard optimization problems studied by Monte Carlo methods, such as simulated annealing, have features that can be estimated by a statistical analysis of the data, well before being actually observed. This applies, for instance, to the estimation of the ground-state energy of the problem. We start by showing that the density of states and the distribution of extremes of energy seen in a given time interval in the Monte Carlo dynamics of combinatorial optimization problems are strongly related to each other through the first-passage-time distribution of the stochastic dynamics of the system. We then introduce a scaling ansatz for this last quantity, which allows an estimate of the ground-state energy. Finally, we demonstrate the method on a ‘‘traveling-salesman’’ problem with known ground-state energy and apply it to the simulated annealing of a graph-bipartitioning problem.

We study the stability of the steady state in an aggregation system with injection for one dimension and the mean-field cases. We prove that for any initial perturbation the system is stable and eventually recovers the power-law steady state. The form of the relaxation follows either a stretched exponential (${\mathit{e}}^{\mathrm{-}\mathit{t}\mathrm{\alpha }}$, α=2,3) or a power law (${\mathit{t}}^{\mathrm{-}\mathrm{\beta }}$, β=1,3/4) depending on the type of injection.

We study a two-level system linearly interacting with a set of quantum-mechanical oscillators, referred to as the ‘‘bath.’’ This system is formally equivalent to a magnetic dipole, with spin 1/2, precessing with the Larmor frequency ${\mathrm{\omega }}_0$ around a fixed magnetic field along the z axis and undergoing the influence of a fluctuating field along the x axis. These bath fluctuations are not independent of the state of the spin, and the crucial problem to be studied in this paper is precisely how to take the reaction field, i.e., the influence of the bath on the system, into account. We find a general result based on neglecting a contribution to the reaction field proportional to ${\mathrm{\sigma }}_{\mathit{x}}$(t)-〈${\mathrm{\sigma }}_{\mathit{x}}$(t)〉, where the angle brackets denote averaging on both the spin and the bath space, with a density matrix corresponding to the spin polarized along the x axis. We show that under the special condition that the spin does not significantly depart from its initial state, our general result turns out to coincide with the noninteracting-blip approximation (NBA) of Leggett and co-workers [Rev. Mod. Phys. 59, 1 (1987)]. When we make the assumption that both quantum and thermal fluctuations of the bath can be neglected and that the time scale of the bath is virtually zero (adiabatic assumption), our general result turns out to coincide with the prediction of the discrete nonlinear Schrödinger equation (DNSE) of Davydov and Kislukha [Phys. Status Solidi B 59, 465 (1973)] and Davydov $[—Biology and Quantum Mechanics (Pergamon, Oxford, 1982)] in the two-sites case.

This means that the nonlinear effects proven by Kenkre and co-workers [Phys. Rev. B 34, 4959 (1986); 35, 1473 (1987)] to accompany a significant departure of the spin from the initial state are lost by the NBA. On the other hand, our approach provides a rigorous evaluation of the effects of the oscillator fluctuations on the predictions of the DNSE. It is shown that the quantum fluctuations might have a significant role also in the region of high temperature to which the nonadiabatic corrections of Kenkre and co-workers apply. Finally, it is shown that a still more accurate approximation would be that of neglecting a reaction field proportional to ${\mathrm{\sigma }}_{\mathit{x}}$(t)-〈${\mathrm{\sigma }}_{\mathit{x}}$(t)${\mathrm{〉}}_{\mathit{s}}$, where 〈${\mathrm{\sigma }}_{\mathit{x}}$(t)${\mathrm{〉}}_{\mathit{s}}$ denotes an average carried out only on the spin space. This approximation leads our approach to coincide with the exact result in the special case where both the spin and its bath are replaced by their classical counterparts. Under this approximation, the detrapping would be equivalent to an Arrhenius-like thermal-activation process. When the coupling strength between the system and its bath is decreased, the spin is expected to depart from the initial trapped state. The dynamics of this process is expected to be influenced by the joint action of the bath fluctuations (both thermal and quantum mechanical) and of the nonlinearity stemming from the reaction field, and ignored by the NBA. This intriguing problem should be the subject of further investigations.

A generalized random-phase approximation (RPA) for decoupling the double-time Green’s functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.

Spectral properties, as determined by trace maps, of the one-dimensional chains (layered structures) constructed according to general two-letter substitution rules are investigated. In all trace maps thus obtained an important role is played by the quantity I=${\mathit{x}}^2$+${\mathit{y}}^2$+${\mathit{z}}^2$-xyz-4. However, only a very small fraction of all such trace maps are similar to the Fibonacci golden-mean trace map is that I is their invariant. In addition to the known case of the precious-mean lattices (precious means are ratios of the form 1/2[m+(${\mathit{m}}^2$+4${)}^{1/2}$], m being any positive integer; m=1 gives the golden mean), we have identified two new large classes of substitution rules that give trace maps with invariant I. One of them is a superset of the precious-mean lattices. All other cases represent a vast assortment of different trace maps (and thus the potential for various hitherto unexplored spectral properties) with a unifying feature that the set I=0 plays the role of an attractor in the trace space. In most (but not all) cases, two chains with identical trace maps (and thus identical spectra) are locally isomorphic. Generally, local isomorphism equivalence classes seem to be subsets of identical spectrum equivalence classes.

Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\Lambda }}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\Lambda }}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\Lambda }}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (⊂scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\Lambda }}$. The phase-space representation of the system is realized on scrG/scrH, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of ‘‘quantum chaos’’ via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.

We show that for linear polymers a collapse transition exists on a truncated six-simplex lattice but not on a truncated five-simplex lattice. The existence of the collapsed phase is determined not only by the connectivity but also by the geometrical form of the first-order unit of the fractal lattice. We report exact results for the critical exponents at the transition for the six-simplex lattice.

As an example of a nonperiodically forced system, the dynamics of the Fermi accelerator (ball bouncing on piston oscillating periodically in time) subject to viscous friction is studied numerically and analytically. Its evolution is determined from a two-dimensional (2D) map, realized as two coupled difference equations. Dimensional reduction is achieved analytically in a simple high-dissipation approximation yielding a version of the circle map. Numerical experiments using this map compare quantitatively and qualitatively well with the exact 2D map in the high- and medium-dissipation regimes, respectively. Quasiperiodic, periodic, and chaotic behavior are observed in the region around the critical line of the circle map in accordance with the Arnol’d tongues scheme. For higher values of the control parameter, an infinite number of ‘‘chaos-period doubling-crisis’’ sequences characterize the bifurcation diagram of the system. The mechanisms for the sudden death and expansion of chaotic attractors are easily understood via return maps of the circle. The dynamics of the weakly damped Fermi accelerator is qualitatively well described by a simplified map that is a nonstandard dissipative 2D map with a nonconstant Jacobian. The possibility of the damped Fermi accelerator not obeying universal properties of a certain class of dissipative systems is discussed.

A classical phase-space structure of two coupled Morse oscillators is studied. An alternative method to construct the Poincaré surfaces of section is presented. The method includes the location of the Farey-tree families of the hyperbolic periodic orbits. The Farey-tree organization of resonances provides a very useful decomposition of the phase space. The pattern of the hyperbolic orbits of the Farey-tree families exhibits a self-similar (fractal) structure. This feature enables one to locate the resonances and provides a certain organization inherent in the apparent chaotic motion.

The wave-function localization in a classically, strongly chaotic system—quadric oscillator model—is discussed in the adiabatic approach. A comparison is made between the coordinate-based traditional method and the periodic-orbit-based scheme of Eckhardt, Hose, and Pollak [Phys. Rev. A 39, 3776 (1989)] for channel localized wave functions. A second type of localization of the ring type, observed in the model, is accounted for by an adiabatic approach in polar coordinates.

We have measured the critical exponent y characterizing the divergence of the viscosity η∝‖T-${\mathit{T}}_{\mathit{c}}$${\mathrm{\Vert }}^{\mathrm{-}\mathit{y}}$ for carbon dioxide and xenon. The values of y for both fluids fall within the range y=0.041±0.001 and are consistent with the range y=0.042±0.002 spanned by our earlier data for four binary liquid mixtures. This agreement is the strongest evidence that pure fluids and binary liquids are in the same dynamic universality class; however, the results for y are inconsistent with the recent theoretical value of 0.032.

Mode-coupling theory is used to calculate the Green-Kubo time correlation functions for the two-dimensional Frisch-Hasslacher-Pomeau lattice-gas cellular automaton. An intermediate algebraic decay ∼1/t and an asymptotic decay ∼1/[t √ln(t) ] are found. The amplitudes for these decays are calculated and it is found that the contribution from diffusive modes associated with spurious conservation laws is significant. For the computer simulations in cellular automata fluids only the algebraic tail is relevant; the asymptotic tail dominates only on time scales large compared to ${10}^{10}$ mean free times.

Near a bifurcation point a system far from thermal equilibrium can be described by use of generalized Ginzburg-Landau equations. We present a systematic method to derive the nonlinear interaction terms of these equations in real space reflecting the selection rules as well as the stabilization of different patterns intrinsic in the basic equations of the systems under consideration. Our work treats the case of periodic instabilities of a homogeneous state in space as well as in time, where the interacting patterns are represented by traveling-wave trains having arbitrary orientations in a two-dimensional plane. Numerical solutions of two-dimensional pattern formation and wave propagation are presented using a system that allows for a backward Hopf bifurcation as is the case for the convection instability of a binary fluid mixture. The stability of the emerging traveling-wave structures is discussed in terms of phase-diffusion equations.

The spatial profile of the amplitude of linear convective traveling waves in binary fluid mixtures is sensitive to the local solute concentration and can thus be used to measure it. I exploit this effect to look for changes in the lateral concentration profile produced by confined states of nonlinear traveling waves. For several different confined states, no effect can be detected, even for fluid layers that are grossly asymmetric with respect to reflection through the midplane of the cell.