A class of differential equations, which captures the logical structure of discrete time logical switching networks composed of many elements, displays deterministic chaos if each element has many inputs. Statistical features of the dynamics are approximated by using a mean field Langevin-type equation with a random telegraph signal as a stochastic forcing function, and also by considering a random walk on an $N$-dimensional hypercube.

We develop a microscopic scattering theory for the electromagnetic response of dielectrics. We derive the Lorentz-Lorenz relation for a hard-sphere fluid and for hard-sphere mixtures by summing rigorously the relevant class of multiple scattering events which incorporates particle correlations. The derivation neither makes use of macroscopic concepts such as local and reaction fields nor does it invoke decoupling schemes for high-order correlation functions.

The interplay of natural and self-induced birefringence leads to a nonlinear polarization symmetry breaking in the process of parametric four-photon scattering. The polarization instability was induced by mixing pump and probe light beams into an optical fiber, and the polarization dependent generation of ultrahigh repetition rate (200 GHz) solitonlike pulse trains was observed in the regime of normal group-velocity dispersion.

We show that the dynamics of spiral waves in excitable media can be reduced to a nonlinear equation of motion for the wave tip that is asymptotically exact in a parameter range where the motion takes place around a large central core region. The onset of meander is shown to occur naturally in this range in the usual free-boundary limit where the medium exhibits an abrupt response to a stimulus. This equation provides a simple understanding of the physical origin of the meandering instability of spiral waves and of the quasiperiodic motion of the wave tip.

A direct transition from two-frequency motion to chaos with a fractal dimension of $d>\mathrm{}3\mathrm{}$ is illustrated. While frequency locking between the two frequencies is a characteristic onset feature when the chaos involved has a lower dimension of $d<\mathrm{}3\mathrm{}$, no such feature is found in the present case.

Acoustic time-reversal experiments usually need large arrays of transducers. It is shown that in a reflecting cavity with negligible absorption one is able to perform a time reversal of elastic waves using a single element. The field is measured at one point over a long period of time and the time-reversed signal is reinjected at the same position. Numerical simulations illustrate the process. Experiments carried out in silicon wafers show that it is possible to obtain an excellent temporal and spatial focusing quality.

Using fluctuating hydrodynamics we describe the slow buildup of long range spatial correlations in a freely evolving fluid of inelastic hard spheres. In the incompressible limit, the behavior of spatial velocity correlations (including $r^{-d}$ behavior) is governed by vorticity fluctuations only and agrees well with two-dimensional simulations up to 50 to 100 collisions per particle. The incompressibility assumption breaks down beyond a distance that diverges in the elastic limit.

We report an unexpected observation of a transition to traveling waves from standing waves. The waves are two dimensional and generated in a rectangular container excited by a horizontal sinusoidal motion along its length. The transition to traveling waves clearly indicates coupling between modes whose natural frequencies do not satisfy obvious resonance conditions. The observed phenomenon is beyond our current understanding of surface waves based on low-dimensional models and demands further study. It implies that care must be taken in developing approximate low-dimensional models for continuous systems in the presence of nonlinearity.

We propose a phenomenological model for pattern formation in a vertically vibrated layer of granular material. This model exhibits a variety of stable cellular patterns, including standing rolls and squares as well as localized objects (oscillons and worms), similar to recent experimental observations [Umbanhowar, Melo, and Swinney, Nature (London) 382, 793 (1996)]. The model is an amplitude equation for the parametrical instability coupled to the mass conservation law. The structure and dynamics of the solutions resemble closely the properties of localized and cellular patterns observed in the experiments.

An analytical cone-jet solution for the electrohydrodynamic atomization of liquids has been found for an asymptotic model assuming an infinitely long and thin emitted jet. Universal expressions for the emitted electric current, jet shape, charge distribution, surface charge, and other essential electrohydrodynamic quantities are obtained as functions of the liquid properties and the emitted liquid flow rate. The agreement with published experiments is good.

Small angle neutron scattering (SANS) in low density ${}^{86}$Kr gas has been performed in order to measure the small- $k$ behavior of the static structure factor $S\left(k\right)$. Three number densities between 1.52 and $2.42{\mathrm{nm}}^{-3\mathrm{}}$ along the $T=297\mathrm{}\mathrm{K}$ isotherm have been studied. The small- $k$ dependence of the Fourier transform $c\left(k\right)$ of the direct correlation function $c\left(r\right)$ has been derived. The experimental determination of the $k^3$ term in the behavior of $c\left(k\right)$ has led to a direct measurement of the London dispersion interaction in the pair potential of krypton. Also the contribution of the three-body potential in the asymptotic behavior of $c\left(r\right)$ has been observed and related to the magnitude of the three-body interaction potential.

The chaotically driven circle map is considered as the simplest model of phase synchronization of a chaotic continuous-time oscillator by external periodic force. The phase dynamics is analyzed via phase-locking regions of the periodic cycles embedded in the strange attractor. It is shown that full synchronization, where all the periodic cycles are phase locked, disappears via the attractor-repeller collision. Beyond the transition an intermittent regime with exponentially rare phase slips, resulting from the trajectory's hits on an eyelet, is observed.

A general approach for synchronizing pairs of unidirectionally coupled partial differential equations (PDEs) with spatiotemporally chaotic dynamics is introduced. We show that for a large class of PDEs, a pair of PDEs can be synchronized by driving the response system only at a finite number of space points. We also discuss the relevance of our results for control of spatiotemporal chaos.

A semiclassical quantization of the stochastic pump model of Arnold diffusion is presented. The semiclassical model is found to be equivalent to a one-dimensional disordered wire, yielding localized states limiting the extent of diffusion.

Dynamical conditions for the loss of validity of numerical chaotic solutions of physical systems are already understood. However, the fundamental questions of “how good” and “for how long” the solutions are valid remained unanswered. This work answers these questions by establishing scaling laws for the shadowing distance and for the shadowing time in terms of physically meaningful quantities that are easily computable in practice. The scaling theory is verified against a physical model.

We demonstrate the successful control of a periodic orbit associated with two unstable manifolds in a system comprised of two coupled diode resonators. It is shown that due to symmetries generic to spatially extended systems a one-parameter control is not possible. A novel method of determining the local Liapunov exponents utilizing orthogonal control as well as geometric information is presented.

We report a direct observation of space charge waves in photorefractive crystals with point group 23 (sillenites) based on their penetration into an area with uniform light illumination. It is shown experimentally that the quality factor of the waves increases substantially with respect to what current theory predicts [B. Sturman et al., Appl. Phys. A 55, 235 (1992)]. This results in the appearance of strong spontaneous beams caused by space charge wave self-excitation.

It is shown, using a generalized Swift-Hohenberg equation, that a small rotation inhibits the spiral chaos which develops in Rayleigh-Bénard convection at moderate Prandtl number. This is due to the gliding and gradual annihilation of dislocations. For rigid top and bottom boundary conditions, a slow rotation first breaks the chiral symmetry and if sufficient leads to an unfolding of the spirals. This effect is maximum near the critical rotation for the onset of the Küppers-Lortz instability, and, when the horizontal geometry is periodic, straight rolls may even reform. With free-slip boundaries, these structures are subject to a small-angle instability, which leads to the formation of large coherent targets embedded in a turbulence background.

The wavelet transform modulus maxima (WTMM) method is generalized to multifractal image analysis, providing a statistical characterization of the fluctuating roughness of fractal surfaces. This isotropic 2D version of WTMM methodology is calibrated on deterministic self-similar interfaces and random self-affine surfaces (fractional Brownian surfaces and multifractal counterparts). Applications to high-resolution satellite data and simulated radiance fields for stratocumulus clouds are presented.

The finite amplitude purely elastic overstability, for axisymmetric Taylor-vortex flow of highly elastic fluids in the narrow-gap limit, is accurately predicted using the Galerkin projection method. A judicious mode selection is carried out to include the dominant normal stress terms. The resulting twenty-mode dynamical system is capable of capturing the nonlinear behavior observed in the experiment of Muller et al. under conditions of negligible inertia. The model predicts, as experiment suggests, the onset of overstability, the growth of oscillation amplitude of flow, and the emergence of higher harmonics in the power spectrum as fluid elasticity increases beyond a critical level.