An isotropic passive scalar field $T$ advected by a rapidly varying velocity field is studied. The tail of the probability distribution $P(\theta ,r)$ for the difference $\theta$ in $T$ across an inertial-range distance $r$ is found to be Gaussian. Scaling exponents of moments of $\theta$ increase as $\sqrt{n}$ or faster at large order $n$, if a mean dissipation conditioned on $\theta$ is a nondecreasing function of $|\theta |$. The $P(\theta ,r)$ computed numerically under the so-called linear ansatz is found to be realizable. Some classes of gentle modifications of the linear ansatz are not realizable.

An exact nonperturbative calculation of the fourth-order anomalous correction to the scaling behavior of a random shell model for passive scalars is presented. Importance of ultraviolet (UV) and infrared (IR) boundary conditions on the inertial scaling properties are determined. We find that anomalous behavior is given by the null space of the inertial operator and we prove strong UV and IR independence of the anomalous exponent. A limiting case where diffusive behavior can influence inertial properties is also presented.

Using NMR microscopy we measure the velocity distribution for a wormlike surfactant solution in the gap of a small angle cone-and-plate rheometer. This system, cetylpyridinium chloride/sodium salicylate 100 mM/60 mM, exhibits biphasic shear band structure when the applied shear rate exceeds the critical rate of strain beyond which a plateau is observed in the shear stress. The structure is characterized by two low/high shear interfaces and the region of high shear evolves by increasing width as the average gap shear is increased.

A method to detect the unstable periodic orbits of a chaotic dynamical system is developed. For a given dynamical system our approach allows us to locate the unstable periodic cycles of, in principle, arbitrary length with a high accuracy. Preknowledge of the dynamical system is not required. To demonstrate its reliability as well as efficiency we apply it to several two-dimensional chaotic maps. In the case of short chaotic time series we develop a dynamical algorithm which is based on a mean-field approach via the Voronoi diagram belonging to the time series. This algorithm enables us to detect low period cycles using a very small set of data points. The influence of noise is investigated in some detail.

We calculate the probability distribution of the matrix $Q=-iħS^{-1\mathrm{}}\partial S/\partial E$ for a chaotic system with scattering matrix $S$ at energy $E$. The eigenvalues ${\tau }_j$ of $Q$ are the so-called proper delay times, introduced by Wigner and Smith to describe the time dependence of a scattering process. The distribution of the inverse delay times turns out to be given by the Laguerre ensemble from random-matrix theory.

We demonstrate a laser system with a frequency instability of 0.7 Hz for integration times of 20 s, corresponding to $2.3\times {10}^{-15\mathrm{}}$ fractional instability. This is the lowest short-term instability of any current oscillator. The system uses diode-pumped miniature Nd:YAG lasers stabilized to sapphire Fabry-Pérot reference cavities at liquid helium temperature. Cryogenic optical resonators are promising for precision experiments, such as high-resolution spectroscopy, flywheel oscillators for optical frequency standards, and displacement measurements. As an example, we report a measurement of the ultralow thermal expansion of a sapphire cavity at 1.9 K.

The dynamics of two mutually coupled but nonidentical semiconductor lasers are studied experimentally, numerically, and analytically for weak coupling. The lasers have dissimilar relaxation oscillation frequencies and intensities, and their mutual coupling strength may be asymmetric. We find that the coupled lasers exhibit a form of localized synchronization characterized by low amplitude oscillations in one laser, but large oscillations in the second laser.

We show that the evolution of the average intensity of cw beams in a quasi-phase-matched quadratic (or ${\chi }^{\left(2\right)}$) medium is strongly influenced by induced Kerr effects, such as self- and cross-phase modulation. We prove the existence of rapidly oscillating solitary waves (a spatial analog of the guided-center soliton) supported by the quadratic and induced cubic nonlinearities.

Optical vector wave mixing processes in nematic liquid crystals as a birefringent nonlinear medium have been investigated. Experimental results are explained theoretically taking into account the polarization dependence of the wave mixing effect and the tensor properties of the nonlinearity.

Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems.

We show that unstable periodic solutions of spatially extended 2D systems, when weakly perturbed, can be stabilized by local feedback without spatial coupling. We argue through this finding that stable solutions can be tracked into unstable and spatiotemporal chaotic regimes by simply utilizing feedback control algorithms established for controlling temporal chaos in spatially constrained systems. We demonstrate numerically that the range of stable traveling waves in a laser system can be greatly extended through one such algorithm. Issues regarding practical implementation of the approach are discussed.

Synchronization of spatiotemporally chaotic extended systems is considered in the context of coupled one-dimensional complex Ginzburg-Landau equations (CGLE). A regime of coupled spatiotemporal intermittency (STI) is identified and described in terms of the space-time synchronized chaotic motion of localized structures. A quantitative measure of synchronization as a function of coupling parameter is given through distribution functions and information measures. The coupled STI regime is shown to disappear into regular dynamics for situations of strong coupling when localized structures become unstable, hence a description in terms of a single CGLE is not appropriate.

We report the experimental observation of the broadening of the frequency spectra of parametrically generated second sound waves by first sound in superfluid ${\mathrm{He}}^4$. The first experimental evidence of Gaussian statistics of amplitudes of parametrically generated waves is presented. Using the kinetic theory of parametric excitation we suggest that the broadening and exponential tails of spectra result from four-wave resonance interaction in a close analogy with similar evidence in parametrically generated spin waves. Universality of the exponential tails in the frequency spectra is suggested.

We examine the linear stability of the Ginzburg-Landau operator with spatially varying coefficients, which mimics strongly nonparallel open flows such as wakes, jets, and boundary layers. The streamwise non-normality of the global eigenmodes explains the observed large transient growths, classically interpreted in terms of local convective instability. The use of pseudospectra provides an exact measure of spatial amplification and aids in the determination of when entrance noise dominates the open-flow dynamics.

We report experimental measurements of a traveling-wave state in rotating Rayleigh-Bénard convection. The fluid was water with a Prandtl number of 6.3 and a dimensionless rotation rate of 274. The marginal and Eckhaus-Benjamin-Feir stability boundaries were determined and the local amplitude and wave number were obtained from demodulation of shadowgraph images. The phase-diffusion coefficient and group velocity were measured in the stable wave number band. This system was found to be well described by the one-dimensional complex Ginzburg-Landau equation.

Existing theories for the surface-tension-driven Bénard convection (Marangoni convection) have considered only thermocapillary effects through the shear stress balance equation at the interface via the surface-tension-gradient-induced shear stresses. Thermocapillary induced surface energy variation has been ignored altogether in these theories. A formulation taking into account this surface energy variation shows that it can have a non-negligible effect on the onset of Bénard convection cells.

We study the statistics of a passive scalar mixed by a turbulent flow that contains coherent structures (Görtler vortices). These structures entrain the passive scalar in such a way that its one-point probability density function (pdf) has a nonstandard shape that can be explained as a superposition of a background Gaussian mixing on the one hand, and the action of the Görtler vortices on the other. We propose a “mean field” approach to predict this pdf. This study (applicable to a wider class of systems) constitutes the first experimental example for which the conditional expectation of the second temporal derivative of the concentration of a passive scalar given the concentration deviates from a linear behavior.

We examine synchronization between identical chaotic systems. A rigorous criteria is presented which, if satisfied, guarantees that the coupling produces linearly stable synchronous motion. The criteria can also be used to design couplings that lead to stable synchronous motion. Analytical results from a dynamical system are presented.

We study synchronization transitions in a system of two coupled self-sustained chaotic oscillators. We demonstrate that with the increase of coupling strength the system first undergoes the transition to phase synchronization. With a further increase of coupling, a new synchronous regime is observed, where the states of two oscillators are nearly identical, but one system lags in time to the other. We describe this regime as a state with correlated amplitudes and a constant phase shift. These transitions are traced in the Lyapunov spectrum.

A method for resampling time series generated by a deterministic chaotic data generating process (DGP) is proposed. Given an observed time series, this method potentially allows one to obtain an arbitrary number of time series of arbitrary length which can be considered as a product of the same unknown DGP. The notion of shadowing and brittleness of the pseudo-orbit proves to be particularly useful in characterizing the conditions for a correct resampling. A simple practical application of the method is shown.