A renormalization-group analysis is carried out of the long-time behavior of random walks in a one-dimensional lattice consisting of three kinds of bonds generated by a deterministic inflation rule. The random walker is subject to a local drift force, depending on the kind of bond it traverses. The mean-square displacement after time t is found to scale as ${\mathit{t}}^{2\nu }$ where the exponent ν is nonuniversal and depends continuously on the magnitude of the drift force. The model studied is one of a general class of systems that exhibits similar behavior.

This Rapid Communication argues that the strange attractor recently found in the context of the modulated Stokes wave [H. T. Moon, Phys. Fluids A 3, 2709 (1991)] is a universal form and indeed the asymmetric counterpart of the Lorentz attractor. Furthermore, this form appears to reveal another class of universality in that it follows the evolution pattern set by the tent map. This is in marked contrast to the Rössler attractor, which follows the universal sequence of the logistic map.

We prove that the periodic-orbit counting function, a measure of the rate of proliferation of periodic orbits, for a barrier billiard and the π/3-rhombus billiard is of the form ${\mathit{ax}}^2$+bx+c, where x is the length (equivalently, period) up to which periodic orbits are counted and a,b,c are system-specific constants. The generality of our arguments strongly suggests that the law of proliferation given here is a representation of general truth about two-dimensional plane-polygonal billiards.

We present a recursive proportional-feedback (RPF) algorithm for controlling deterministic chaos. The algorithm is an adaptation of the method of Dressler and Nitsche [Phys. Rev. Lett. 68, 1 (1992)] to highly dissipative systems with a dynamics that shows a nearly one-dimensional return map of a single variable X measured at each Poincaré cycle. The result extends the usefulness of simple proportional-feedback control algorithms. The change in control parameter prescribed for the nth Poincaré cycle by the RPF algorithm is given by δ${\mathit{p}}_{\mathit{n}}$=K(${\mathit{X}}_{\mathit{n}}$-${\mathit{X}}_{\mathit{F}}$)+Rδ${\mathit{p}}_{\mathit{n}\mathrm{-}1}$, where ${\mathit{X}}_{\mathit{F}}$ is the fixed point of the target orbit, and K and R are proportionality constants. The recursive term is shown to arise fundamentally because, in general, the Poincaré section of the attractor near ${\mathit{X}}_{\mathit{F}}$ will change position in phase space as small changes are made in the control parameter. We show how to obtain K and R from simple measurements of the return map without any prior knowledge of the system dynamics and report the successful application of the RPF algorithm to model systems from chemistry and biology where the recursive term is necessary to achieve control.

We contrast the effects of uncorrelated power-law noise and linearly correlated Gaussian noise upon the anomalous wandering of directed polymers in random media. In the first instance, we explore the role of the noise on the morphology of the ultrametric tree structure of the ensemble of locally optimal paths, and then, motivated by the work of Zhang [Phys. Rev. Lett. 59, 2125 (1987)] and more recently by Perlsman and Schwartz [Europhys. Lett. 17, 11 (1992)], provide strong evidence for the universality of the model’s river basin patterns. Lastly, we discuss those precise features of the positional and energy fluctuations that could permit a resourceful experimentalist to discern the noise distribution underlying anomalous roughening recently observed in fire fronts, bacterial growth, and fluid flow through porous media.

In an experiment on the parametric excitation of capillary waves (the Faraday instability) we have observed a stable standing-wave pattern with twelvefold orientational order. This ‘‘quasipattern’’ is analogous to a two-dimensional quasicrystal, but it occurs in a macroscopic nonequilibrium fluid-dynamical system. It is observed in containers whose side walls are of various shapes including an irregular shape, and thus it cannot be ascribed to side-wall boundary effects.

We report results from a simulated annealing study of a coarse-grained model for block copolymers. For sufficiently slow cooling rate, the system develops a well-defined lamellar structure by avoiding the formation of defects. We show that the strong segregation limit of the model is computationally accessible when the interface thickness is small enough.

The Boussinesq equation is known to be the fundamental nonlinear model, describing wave propagation in a weakly dispersive nonlinear medium. In this paper, we prove the existence of the collapse dynamics for the two basic forms of the Boussinesq equation in the case of the periodic boundary conditions. The sufficient criterion of the blow-up is found analytically.

Spiral waves, a type of ‘‘reentrant excitation,’’ are believed to be associated with the most dangerous cardiac arrhythmias, including ventricular tachycardia and fibrillation. Recent experimental findings have implicated defective regions as a means of trapping spirals which would otherwise drift and (eventually) disappear. Here, we model the myocardium as a simple excitable medium and study via simulation the interaction between a drifting spiral and one or more such defects. We interpret our results in terms of a criterion for the transition between trapped and untrapped drifting spirals.

The influence of parametric noise on a class of a growing-rate-type model is studied. The formal time-dependent solution is found analytically for any linearly coupled white noise. The Mellin-Barnes technique is used in order to extend this solution on a whole parameter space in the case of Gaussian white noise and white shot noise. The asymptotic behavior in different regions of parameter space is examined. A digital averaging is done which supports the analytical results. It is found that, in contrast to the deterministic case, the relaxation in the presence of the noise needs not be purely monotonic.

Quantum coherence and its destruction (dephasing) by coupling to a dissipative environment plays an important role in time-resolved nonlinear optical response as well as nonadiabatic transitions and tunneling processes in condensed phases. Generating functions of density-matrix elements and multitime coordinate and momentum correlation functions related to these phenomena are calculated using a pathintegral approach by performing functional integration. The dissipative environment is assumed to be an ensemble of harmonic oscillators and is taken into account by using Feynman-Vernon influence functional. Closed-form expressions for generating functions in terms of the bath spectral density are derived. The present theory generalizes earlier calculations of these quantities to arbitrary temperatures, any dependence of the transition coupling on coordinates (non-Condon effects), and arbitrary order in the interstate coupling. Conditions for factorizing the Liouville-space generating functions that allow a reduced description based on the classical Langevin equation are established. Possible applications to four-wave-mixing spectroscopy and nonadiabatic rate processes are discussed.

We propose a dynamic model for granular flows. Starting from the equation of motion for a grain together with the equation of continuity, we derive a Langevin equation describing the time evolution of the granular system. We show how the model displays scale invariance and reduces to several existing models in appropriate limits. In particular, we also point out the possibility of different behaviors in granular systems according to their sizes.

We obtain the conditions that ensure the stability of spatially and temporally periodic orbits of coupled-map lattices. The stability matrices can be put in a circulant and block circulant form. This allows us to reduce the problem to smaller matrices corresponding to the building blocks of spatial periodicity. We find that additional conditions are imposed as we expand the size of the lattice. For the traveling-wave solution the analysis is considerably simplified. We have analyzed both the one-dimensional and higher-dimensional lattices.

The time evolution of the mean deviation of initially close trajectories in a stochastic dynamical system is investigated. It is shown both for additive and linearly coupled multiplicative noise that the mean deviation loses its dependence on initial conditions for long times. For shorter times a power law is found for certain types of additive noise processes, in sharp contrast to the exponential separation of initially nearby trajectories in deterministic chaotic systems. Exponential time evolution is obtained for linearly coupled multiplicative noise after an initial transient during which more complex regimes, including a superexponential stage, can take place.

Characteristic recurrence phenomena that are strongly dependent on initial conditions but do not depend much on the numbers of degrees of freedom are observed in the study of a system consisting of identical harmonic oscillators coupled with the Nosé-Hoover thermostat. The time evolution of the energy of each oscillator in this system at large Q (Q is a parameter controlling the speed of the response of the thermostat) is composed of a secular periodic motion with frequency of order 1/Q and fast fluctuations around it with frequency of order 1. The latter is the natural frequency of oscillators. The secular part depends very sensitively on initial conditions. The dynamical behaviors of the system are analyzed by the perturbation treatment with respect to 1/Q, and a Hamiltonian H’={${\mathit{tsum}}_{\mathit{i}}$${\mathit{tsum}}_{\mathit{j}}$ ${\mathrm{p}}_{\mathit{i}}$${\mathit{p}}_{\mathit{j}}$cos[2(${\mathit{q}}_{\mathit{i}}$-${\mathit{q}}_{\mathit{j}}$)]}/16, which describes the behavior of the secular part in the original system, is derived of first order in 1/Q. The coordinate ${\mathit{q}}_{\mathit{i}}$ and the conjugated momentum ${\mathit{p}}_{\mathit{i}}$ in the Hamiltonian H’ are the slowly changing part of the phase ${\mathrm{\phi }}_{\mathit{i}}$ of the oscillator and the energy ${\mathit{H}}_{\mathit{i}}$. I found that this Hamiltonian H’ is completely integrable and can be solved analytically. The solutions thus obtained describe very well the dynamical behaviors in the original system.

A restricted variational formulation for extended irreversible thermodynamics is proposed. Then, it is applied to a heat- and electricity-conducting viscous fluid, and the resulting equations are reduced to order two to be compared with those well known of kinetic theory. A better understanding of the phenomenological coefficients than in previous reported works is obtained.

We explore the rich behavior of vibrated powders, focusing on the relaxation of transients between vibrational steady states, on the diffusion of independent particles, and on ‘‘global’’ and ‘‘local’’ angular distributions of contacting particles. When examined as a function of vibrational intensity, these are shown to be explicable in terms of a model of independent-particle and collective excitations. Our data provide a wealth of information about topics as diverse as consolidation under packing, the roughness of granular piles, and the local structure of bridges and arches in shaken packings.

We consider mean first-passage times (MFPT’s) for systems driven by non-Markov gamma and McFadden dichotomous noises. A simplified derivation is given of the underlying integral equations and the theory for ordinary renewal processes is extended to modified and equilibrium renewal processes. The exact results are compared with the MFPT for Markov dichotomous noise and with the results of Monte Carlo simulations.

We study the probability distribution of a passive scalar undergoing turbulent mixing in the presence of a mean scalar gradient. Kerstein’s model, which describes the turbulent mixing process as a collection of instantaneous local rearrangements of the scalar, is argued to be a plausible approximation when the size of the system is much larger than the velocity correlation length, and after the description has been coarse grained over a correlation volume. In the physical range of parameters, we find numerically that the fluctuations of the scalar θ are close to exponentially distributed when a linear mean scalar gradient is imposed. The phenomenological mean-field-like theory of Pumir, Shraiman, and Siggia [Phys. Rev. Lett. 66, 2984 (1991)] is derived heuristically beginning with the Kerstein model. This theory predicts strictly exponential tails for the probability distribution of the scalar fluctuations P(θ), i.e., P(θ)∼exp(‖θ‖). We also consider a simplified version of the Kerstein model which is analytically fully tractable and gives qualitatively similar results numerically. Under conditions of spatial homogeneity and imposed linear mean scalar gradient, this simplified model and the full Kerstein model are described by the same mean-field theory. However, for the simplified model, the large-‖θ‖ asymptotic behavior of P(θ) is Poisson-like, i.e., P(θ)∼exp[-‖θ‖ln‖θ/(const)‖].

In recent numerical experiments on series arrays of overdamped Josephson junctions, Nichols and Wiesenfeld [Phys. Rev. A 45, 8430 (1992)] discovered that the periodic states known as splay states are neutrally stable in all but four directions in phase space. We present a theory that accounts for this enormous degree of neutral stability. The theory also predicts the four non-neutral Floquet multipliers to within 0.1% of their numerically computed values. The analytical approach used here may be appli- cable to other globally coupled systems of oscillators, such as multimode lasers, electronic oscillator circuits, and solid-state laser arrays.