A Reply to the Comment by Armand Sarkissian and Herbert Levine.

The dynamical behavior of two coupled chaotic oscillators was experimentally studied under superposition of both noise and periodic external forces. Increasing the noise intensity the chaotic behavior was suppressed and the signal-to-noise ratio (SNR) became maximum at a certain noise intensity; that is, the stochastic resonance phenomenon was observed. In order to define the states more quantitatively and compare to the change of SNR the largest Lyapunov exponent was calculated which showed a drastic change from a positive value in the chaotic state to zero in the resonance regimes; noise-induced order was observed.

An algorithm to characterize collective motion as the orbital instability at a macroscopic level is presented, including the introduction of “collective Lyapunov exponent.” By applying the algorithm to a globally coupled map, existence of low-dimensional collective chaos is confirmed, where the scale of (high-dimensional) microscopic chaos is separated from the macroscopic motion, and the scale approaches zero in the thermodynamic limit.

We utilize the dynamical fluctuations in spatiotemporal chaotic systems to elucidate the dynamical role of topological defects. We find that each defect in a representative defect-mediated system is associated with one to two degrees of dynamical freedom. Furthermore, we find that a significant fraction of the total degrees of freedom is not associated with the defects, and, in fact, appear to be related to the background phase turbulence. This new understanding of defect-mediated chaos provides hope that these complicated systems may be amenable to reduced statistical descriptions.

We study the behavior of arrays of unidirectionally coupled Lorenz systems in the chaotic regime. In the case of rings an instability in the uniform synchronized state leads to the appearance of a periodic rotating wave through a symmetric Hopf bifurcation. From theoretical grounds it is argued that this behavior must also manifest in the behavior of linear arrays. Numerically it is shown that this happens in a transient way, leading to unexpected consequences in the velocity of synchronization, that becomes larger.

We report the formation of spatial structures induced by a spatial offset in a two-dimensional nonlinear optical interferometer in which diffractive effects are absent. The observed patterns arise from the destabilization of wide, continuous bands in the Fourier spectrum. The selection of these bands is shown to be determined by the interplay between the amount of spatial offset and the strength of the diffusion in the nonlinear optical material.

For the first time a method for realizing macroscopic quantum optical solitons is presented. Simultaneous photon-number and momentum squeezing is predicted using soliton propagation in an interferometer. Extraction of soliton pulses closer to true quantum solitons than their coherent counterparts from mode-locked lasers is possible. Moreover, it is a general method of reducing photon-number fluctuations below the shot-noise level for nonsoliton pulses as well. It is anticipated that similar reductions in particle fluctuations could occur for other forms of interfering bosonic fields whenever self-interaction nonlinearities exist, for example, interacting ultracold atoms.

Nonlinear frequency conversion in 2D ${\chi }^{\left(2\right)}$ photonic crystals is theoretically studied. Such a crystal has a 2D periodic nonlinear susceptibility, and a linear susceptibility which is a function of the frequency, but constant in space. It is an in-plane generalization of 1D quasi-phase-matching structures and can be realized in periodic poled lithium niobate or in GaAs. An interesting property of these structures is that new phase-matching processes appear in the 2D plane as compared to the 1D case. It is shown that these in-plane phase-matching resonances are given by a nonlinear Bragg law, and a related nonlinear Ewald construction. Applications as multiple-beam second-harmonic generation (SHG), ring cavity SHG, or multiple wavelength frequency conversion are envisaged.

Three-dimensional steady states in plane Couette flow and traveling-wave solutions in plane Poiseuille flow are calculated for stress boundary conditions. The procedure is tied to a self-sustaining mechanism associated with the coherent structures that have been observed in turbulent shear flows. The exact states in both types of flow are remarkably similar to each other and to the coherent structures. They survive down to Reynolds numbers below the critical value for turbulence onset. These results suggest that the underlying process is generic and fundamental to both transitional and developed turbulence.

A study of secondary instabilities in ac-driven electroconvection of a planarly aligned nematic liquid crystal is presented. At low frequencies one has a transition from normal rolls to a zigzag pattern and at high frequencies a direct transition from normal to abnormal rolls. The crossover defines a codimension-2 point. This point is also the origin of a line of homoclinic bifurcations, which enable the transition from zigzags to abnormal rolls at lower frequencies.

A Comment on the Letter by J. Kasparian et al., Phys. Rev. Lett. 78, 2952 (1997). The authors of the Letter offer a Reply.

A Reply to the Comment by D. Carroll and X. H. Zheng.

According to diffraction theory the size of a focused spot is determined by wavelength, numerical aperture, and aperture shape. But even when these parameters are fixed it is possible to achieve super-resolution and reduce the spot size even further. It is shown, however, that the three-dimensional spot size cannot be arbitrarily reduced. The minimum focal spot able to probe high-resolution volumetric information in an optical system is about half of the diffraction-limited spot size.

We investigate the properties of a model of granular matter consisting of $N$ Brownian particles on a line, subject to inelastic mutual collisions. This model displays a genuine thermodynamic limit for the mean values of the energy, and the energy dissipation. When the typical relaxation time $\tau$ associated with the Brownian process is small compared with the mean collision time ${\tau }_c$ the spatial density is nearly homogeneous and the velocity probability distribution is Gaussian. In the opposite limit $\tau \gg {\tau }_c$ one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the Gaussian one.

The spinodal decomposition of binary mixtures in uniform shear flow is studied in the context of the time-dependent Ginzburg-Landau equation, approximated at one-loop order. We show that the structure factor obeys a generalized dynamical scaling with different growth exponents ${\alpha }_x=5\mathrm{}/4$ and ${\alpha }_y=1\mathrm{}/4$ in the flow and in the shear directions, respectively. The excess viscosity $\Delta \eta$ after reaching a maximum relaxes to zero as ${\gamma }^{-2\mathrm{}}{t}^{-3\mathrm{}/2}$, $\gamma$ being the shear rate. $\Delta \eta$ and other observables exhibit log-time periodic oscillations which can be interpreted as due to a growth mechanism where stretching and breakup of domains cyclically occur.

We study the effect of nonlinear gradient terms on breathing localized solutions in the complex Ginzburg-Landau equation. It is found that even small nonlinear gradient terms—which appear at the same order as the quintic term—can cause dramatic changes in the behavior of the solution, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, thus causing the solution to breathe periodically or chaotically on only one side or the solution to rapidly spread.

Very narrow fingers or needles are observed in viscous fingering experiments using lyotropic lamellar phases as the viscous liquid and air as the less viscous fluid. The highly anisotropic flow-induced texture of the lamellar phase is strongly correlated with the onset of needle formation. The needles are observed to have a width which decreases to zero while the velocity of the tip increases dramatically with time, reminiscent of a finite-time singularity.

This article demonstrates the existence of a nonuniform stationary measure (referred to as the $w$-invariant measure) associated with the statistical space-filling properties of the unstable invariant foliation in 2D differentiable area-preserving systems and derives a sequence of analytical approximations for it. In fluid-dynamic problems, the $w$-measure coincides with the intermaterial contact area density.

A thin solid, burning against an oxidizing wind, develops a fingering instability. The effect is observed in a narrow gap geometry, where free convection is suppressed. Focusing on the developed nonlinear state, we find that two length scales coexist. The spacing between fingers is determined by the Péclet number, and the finger width is determined by heat losses. Dense fingers develop by tip splitting. A phenomenological model accurately predicts the fingers' spacing, and is generally applicable to diffusion limited systems. We suggest that the effect is a new, accurately controllable, version of the thermal-diffusive instability.

Phase synchronized states can emerge in the collective behavior of an ensemble of chaotic coupled map lattices, due to a mean field interaction. This type of interaction is responsible for synchronized chaotic global activity of the lattices, while the local activity of each map remains unsynchronized. The resulting collective dynamics is called “weak synchronization.” The transition to such a state is characterized in an ensemble of one-dimensional lattices of logistic maps, in terms of the distance in phase among the different lattices. Its robustness against a small difference in the map parameters is proved. We show that this phenomenon can be associated with pattern formation.