Spectral features and eigenfunctions of the kicked rotor are contrasted with those obtained in the semiclassical limit. For extended eigenfunctions, the difference in the spectra scales as ${ħ}^{3/2}$ in the random matrix theory limit and eigenvalues can be uniquely identified. However, for dynamically localized eigenfunctions, the semiclassical spectrum does not resemble the exact spectrum though numerics indicate that statistical features are reproduced. We suggest that semiclassical periodic orbit theory, even with improved asymptotics, will only provide a statistical description of dynamical localization.

We give a statistical characterization of states with nonzero winding number in the phase turbulence (PT) regime of the one-dimensional complex Ginzburg-Landau equation. We find that states with winding numbers larger than critical ones are unstable in the sense that they decay to states with smaller winding numbers. The transition from phase to defect turbulence is interpreted as an ergodicity breaking transition which occurs when the range of stable winding numbers vanishes. Asymptotically stable states which are not spatiotemporally chaotic are described within the PT regime of a nonzero winding number.

The trajectories of interacting self-guided beams in a homogeneous nonlinear medium are derived directly from the Lorentz force of classical electromagnetic theory, treating the beams as particles with mass. This, to our knowledge, is the first self-consistent particle description of beam interaction, and, in particular, one that does not rely upon interpretation of the wave equation. The method is applicable to any stable beams, in both two and three dimensions.

Relying on contact dynamics simulations, we study the statistical distribution of contact forces inside a confined packing of circular rigid disks with solid friction. We find the following: (1) The number of normal and tangential forces lower than their respective mean value decays as a power law. (2) The number of normal and tangential forces higher than their respective mean value decays exponentially. (3) The ratio of friction to normal force is uniformly distributed and is uncorrelated with normal force. (4) When normalized with respect to their mean values, these distributions are independent of sample size and particle size distribution.

The tribology of a sliding elastic continuum in contact with a disordered substrate is investigated analytically and numerically via a bead-spring model. The deterministic dynamics of this system exhibits a depinning transition at a finite driving force, with complex spatial-temporal dynamics including stick-slip events of all sizes. These behaviors can be understood completely by mapping the system to the well-known problem of a directed path in higher-dimensional random media.

The Chapman-Enskog series for shear stress is summed up in a closed form for a simple model of Grad moment equations. The resulting linear hydrodynamics is demonstrated to be stable for all wavelengths, and the exact asymptotic of the acoustic spectrum in the short-wave domain is obtained.

It was believed that no Taylor vortices would exist in wide spherical shells with an aspect ratio of $\beta >0.24\mathrm{}$. In contrast, we have experimentally generated Taylor vortices in a relatively wide spherical shell with $\beta =0.33\mathrm{}$ using some special initial conditions. It is found that the Taylor vortices remain very stable in a range of the Reynolds number $467<\mathrm{Re}<2100\mathrm{}$, once they are established, in which, normally, the axisymmetric basic state (spherical Couette flow) is preferred. Furthermore, it is interesting that with increasing Reynolds number the Taylor vortices become asymmetric with respect to the equatorial plane.

We consider the sedimentation of high aspect ratio fibers in a Newtonian fluid under Stokes' flow conditions. Measurements of the mean sedimentation velocity, variance of the sedimentation velocity, and the orientation distribution are presented using a modification of the novel experimental technique described by Nicolai et al. [Phys. Fluids 7, 12 (1995)]. We find a steady state where the fibers “clump" together and tend to align in the direction of gravity with occasional “flipping."

When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attractor. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking.

Phase-space representations of the dynamics of the quantal and classical cat map are used to explore quantum-classical correspondence in a $K$ system: As $ħ\to 0$, the classical chaotic behavior is shown to emerge smoothly and exactly. The quantum dynamics near the classical limit displays both exponential separation of adjacent distributions and long time relaxation, two characteristic features of classical chaotic motion.

In pattern-forming systems, such as Rayleigh-Bénard convection or directional solidification, a large number of linearly stable, patterned steady states exist when the basic, simple steady state is unstable. Which of these steady states will be realized in a given experiment appears to depend on unobservable details of the system's initial conditions. We show, however, that weak, Gaussian white noise drives such a system toward a preferred wave number which depends only on the system parameters and is independent of initial conditions. We give a prescription for calculating this wave number, analytically near the onset of instability and numerically otherwise.

We construct a simple semiclassical argument to correlate dynamical localization, observed in several nonintegrable quantum dynamics, with the dominance of diffusive periodic orbits over ballistic ones. We use the kicked rotor (and modifications thereof) dynamics to illustrate the procedure though the argument is valid for any quantum dynamical system fulfilling certain nondegeneracy conditions and which shows deterministic diffusion in the classical limit.

We study the shape of the tail of a heap of granular material. A simple theoretical argument shows that the tail adds a logarithmic correction to the slope given by the angle of repose. This expression is in good agreement with experiments. We present a cellular automaton that contains gravity, dissipation, and surface roughness and its simulation also gives the predicted shape.

A new method is proposed for detecting unstable periodic orbits and their linear stability properties from chaotic experimental time series. Illustrative examples are presented for both numerically and experimentally generated time series. The statistical significance of the results is assessed using surrogate data.

By using a supersymmetric method we derive an explicit expression for the parametric correlation function of densities of eigenphases ${\theta }_a$ of the S matrix in a chaotic quantum system with broken time-reversal symmetry coupled to the continuum via M equivalent open channels; $a=1,\dots ,M$. We use it to find the distribution of derivatives of these eigenphases over the energy (“phase-shift times”) as well as over an arbitrary external parameter. We also find the parametric correlations of Wigner-Smith delay times.

A very general energy conservation law derived from a Lagrangian theory of dielectric crystals is presented. It includes energy propagation from electromagnetic, spin, and acoustic waves. Both linear and nonlinear waves are included as well as various polaritonic combinations. Waves involving nonlocal (wave-vector–dispersive) interactions are also included. An example of the latter for which the Poynting vector is invalid, but which is correctly handled by this theory, is presented.

When a container of fluid is oscillated vertically, capillary waves develop on the surface if the amplitude exceeds a critical value. Experimentally one finds that the motion of small particles on the surface of the fluid is close to Brownian. Here we study the relative motion of particle pairs. The experiment establishes that particle motion is strongly correlated over macroscopic distances. Our observations are in striking agreement with upper-ocean studies, and with theories that appear applicable to this “weak turbulence” problem, and in disagreement with experimental and theoretical results for two-dimensional large-scale atmospheric turbulence.

The basic concept of locality of turbulent interactions in $\overrightarrow{k}$ space is considered. It is suggested that some nonlocal turbulent spectra can be turned into the local ones by virtue of appropriate “hidden parameters.” Based on this idea, a local description is developed for optical turbulence that escaped investigation by standard methods for a while, and a broad set of the Kolmogorov spectra is found explicitly. The developed method is applied then to describe formation of Kolmogorov spectra and evolution of free optical turbulence.

The instantaneous height of the entire surface of a fluid executing high amplitude motion can be obtained by photographing light that has been forced to diffuse through the liquid. This technique has been applied to resolve solitons and observe the frequency and wave-number spectra of ripple turbulence. Higher order correlations and various models of turbulence can be probed with this method.

Chaotic attractors containing periodic orbits with different numbers of unstable directions display fluctuating Lyapunov exponents. We show that the existence of certain nonattracting chaotic sets inside the attractor guarantees the occurrence of this behavior in a persistent manner. These nonattracting sets can be brought inside the attractor via a new type of crisis and may be created, as a parameter is varied, via a sequence of bifurcations out of unstable periodic orbits.