A Reply to the Comment by M. M. Sigalas and E. N. Economou.

Dynamical behaviors of a dissipative particle in a periodic potential subject to chaotic noise are reported. We discovered a macroscopic symmetry breaking effect of chaotic noise on a dissipative particle in a multistable system emerging, even when the noise has a uniform invariant density with parity symmetry and white Fourier spectrum. The broken parity symmetry of the multistable potential is not necessary for the dynamics with broken symmetry. We explain the mechanism of the symmetry breaking and estimate the average velocity of a particle under chaotic noise in terms of unstable fixed points.

We study the interaction of counterpropagating traveling waves in 2D nonequilibrium media described by the complex Swift-Hohenberg equation (CSHE). Direct numerical integration of CSHE reveals novel features of domain walls separating wave systems: wave-vector selection and transverse instability. Analytical treatment is based on a study of coupled complex Ginzburg-Landau equations for counterpropagating waves. At the threshold we find the stationary (yet unstable) solution corresponding to the selected waves. It is shown that sources of traveling waves exhibit long wavelength instability, whereas sinks remain stable. An analogy with the Kelvin-Helmholtz instability is established.

Two-dimensional turbulence normally relaxes through vortex merger and filamentation, with energy flowing to large scales and enstrophy dissipated on fine scales. Experiments on magnetized electron columns show that this relaxation can be arrested by spontaneous “cooling” of the chaotic vortex motions, leading to regular lattices of vortices within a uniform background of weaker vorticity.

An invariant measure is introduced to quantify the disorder in extended locally striped patterns. The measure is invariant under Euclidean motions of the pattern, and vanishes for a uniform array of stripes. Irregularities such as point defects and domain walls make nonzero contributions to the measure. Analysis of patterns generated in a reaction-diffusion system suggests two additional properties of the measure: (1) Apart from small fluctuations, it is invariant for distinct patterns generated at fixed control parameters. (2) It exhibits a jump at the onset of pattern dynamics.

Electronic efficiency close to the value calculated for a perfect electron beam in a type of free-electron maser based on the stimulated electron cyclotron radiation has been obtained for the first time. The cyclotron-autoresonance-maser oscillator utilizing a 500 kV, 100 A electron beam with rotational velocity $0.55c$ yielded coherent radiation at 7.9 mm wavelength with a power of 13 MW. The corresponding electronic efficiency was 26%.

Employing third sound in a superfluid helium film as an acoustic wave, transmission spectra are measured experimentally in the Penrose lattice. The spectra are extremely sensitive to the impinging direction of third sound. The transmissivity is suppressed strongly at certain frequency ranges. A transparent direction of the Penrose lattice is found simultaneously. We attribute the origin of the prominent suppressions of transmissivity to the multiple diffraction of third sound in the Penrose lattice.

It has been recently suggested that the various $q$th order velocity structure functions in turbulent flows are related to each other through well defined scaling laws, which extend outside the usual inertial domain. Even if theoretical models provide different scaling laws for fluid and magnetohydrodynamic turbulent flows, no attempt has been made up to now to furnish experimental evidence for these differences. By using measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, we show that the differences can be observed only by looking at the high-order velocity structure functions.

The turbulent velocity fluctuations in a homogeneous shear flow (mean velocity $\overrightarrow{U}=Sy\hat{x}$) are studied numerically, in the range of Reynolds number $2600\le \mathrm{Re}\le 11300$. The skewness of the $z$ component of vorticity is independent of the Reynolds number in this range, suggesting that some small scale anisotropy remains at very high Reynolds numbers. This anisotropy is not seen in the two-point correlation functions. An analogy is drawn with earlier results on turbulent mixing of a passive scalar with a mean gradient. The relevance of this work for turbulent boundary layers is discussed.

Usually, diffusive coupling of nonlinear oscillators in one dynamical variable leads to synchronization of oscillators. We study a model of coupled neural oscillators in which simple diffusive coupling in voltage, counterintuitively, leads to dephasing of oscillators. We examine the general conditions under which dephasing through diffusive interaction will occur. We show that such systems with dephasing limit cycles lead to a new burstinglike behavior: oscillators switch between high and low oscillation amplitude. This occurs because the interaction is such that oscillators tend to synchronize for sufficiently small oscillation amplitude, while they tend to desynchronize once their oscillation amplitude has become large.

A Comment on the Letter by Raymond Y. Chiao and Jack Boyce Phys. Rev. Lett. 73, 3383 (1994).

A general method is presented to determine regular orbits in the presence of irregular trajectories in phase space for a dynamical system of $n$ degrees of freedom. A cost functional is introduced with a control field to guide the system towards regular orbits. Iteration of the optimizing algorithm naturally seeks out regular orbits with the control field turned off. If the system is completely chaotic in some region, the method chooses the best initial condition to achieve a nearly regular orbit. An illustration is presented for $n=3\mathrm{}$ and $n=4\mathrm{}$ dimensional dynamical systems.

We present an effective approach for controlling chaos by using a differential geometric method. It has been shown that the proposed method can control chaotic motion not only to a steady state but also to any desired periodic orbit. The main characteristic of the method is to algebraically transform a nonlinear dynamics into a linear one, so that linear control techniques can be applied. To demonstrate the feasibility of our proposed method, a Lorenz system under different designed requirements is illustrated.

Combining the Reynolds equations with the difference-quotient turbulence model a simple analytical theory of plane turbulent Couette flow has been obtained. In a very natural manner an order parameter of turbulence appears, which is related to the production rate of turbulent kinetic energy in the whole domain. Good agreement with experimental data has been achieved.

A Comment on the Letter by Aaron S. Manaka et al., Phys. Rev. Lett. 73, 1789 (1994). The authors of the Letter offer a Reply.

A Reply to the Comment by J. H. Eberly.

We observe coherently enhanced reflection (Bragg scattering) of a near resonant laser beam from cesium atoms ordered by a periodic array of optical potential wells known as an optical lattice. The reflected light is enhanced by a factor of $\sim {10}^5$ over scattering from a disordered atomic sample. We measure the magnitude of the reflectivity and its detuning dependence as functions of the density and the degree of localization of the atoms. The change in lattice constant due to the frequency-dependent effective index of refraction seen by the lattice light field is observed.

We have conducted a time-resolved, two-dimensional light scattering study of the nonlinear dynamics of field-induced structures in an electrorheological fluid subjected to oscillatory shear. We have developed a theoretical description of the observed dynamics by considering the response of a fragmenting and aggregating particle chain to the prevailing hydrodynamic and electrostatic forces. This structural theory is then used to describe the nonlinear rheology of electrorheological fluids.

We investigate stationary nonequilibrium states of particles moving according to Hamiltonian dynamics with Maxwell demon “reflection rules” at the walls. These rules simulate, in an energy but not phase space volume conserving way, moving boundaries. The resulting dynamics may or may not be time reversible. In either case the average rates of phase space volume contraction and macroscopic entropy production are shown to be equal for stationary hydrodynamic shear flows, i.e., when the velocity distribution of particles incident on the walls is a local Maxwellian. Molecular dynamic simulations of hard disks in a channel produce a steady shear flow with the predicted behavior.

It is proposed that the spoiling of high-Q whispering gallery modes in deformed dielectric spheres can be understood as a transition to chaotic ray dynamics which can no longer be confined by total internal reflection. This is a Kolmogorov-Arnold-Moser/Lazutkin transition for light. The theory is applied to explain the lasing properties of droplets.