We derive a new system of equations that describes the propagation of circularly polarized laser beams in a Kerr medium. Analysis and simulations of this system show that multiple filamentation is suppressed for circularly polarized beams.

The effects of noise on phase synchronization (PS) of coupled chaotic oscillators are explored. In contrast to coupled periodic oscillators, noise is found to enhance phase synchronization significantly below the threshold of PS. This constructive role of noise has been verified experimentally with chaotic electrochemical oscillators of the electrodissolution of Ni in sulfuric acid solution.

We show that turbulent “spirals” and “spots” observed in Taylor-Couette and plane Couette flow correspond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits the phenomenology of coupled noisy Ginzburg-Landau (amplitude) equations with noise. This suggests the existence of a long-wavelength instability of the homogeneous turbulence regime.

We study two coupled spatially extended dynamical systems which exhibit space-time chaos. The transition to the synchronized state is treated as a nonequilibrium phase transition, where the average synchronization error is the order parameter. The transition in one-dimensional systems is found to be generically in the universality class of the Kardar-Parisi-Zhang equation with a growth-limiting term (“bounded KPZ”). For systems with very strong nonlinearities in the local dynamics, however, the transition is found to be in the universality class of directed percolation.

Using precision measurements of tracer particle trajectories in a two-dimensional fluid flow producing chaotic mixing, we directly measure the time-dependent stretching field. This quantity, previously available only numerically, attains local maxima along lines that coincide with the stable and unstable manifolds of hyperbolic fixed points of Poincaré maps. Contours of a passive impurity field are found at each instant to be oriented parallel to the lines that have recently experienced large stretching. The local stretching varies by 12 orders of magnitude.

The crack of a whip is produced by a shock wave created by the supersonic motion of the tip of the whip in the air. A simple dynamical model for the propagation and acceleration of waves in the motion of whips is presented. The respective contributions of tension, tapering, and boundary conditions in the acceleration of an initial impulse are studied theoretically and numerically.

We propose a measure for nonstationarity which is based on the analysis of distributions of temporal distances of neighboring vectors in state space. As an extension of previous techniques our method does not require a partitioning of the time series. Moreover, the deviation of mean recurrence times from frequency distributions that would be expected under stationary conditions allows us to estimate the statistical significance of the method.

We present experimental studies of a new pattern sequence observed in non-Boussinesq convection in a compressible fluid near its gas-liquid critical point (CP). Besides the known hysteretic transitions among conduction state, hexagons, and rolls, another hysteretic transition from rolls to hexagons at higher values of the control parameter is found. This reentrance phenomenon is observed in a rather narrow range of about 60- to $100-\mu \mathrm{m}$ cell heights and is attributed to large compressibility of a fluid near the CP.

We present a highly sensitive nonlinear optical technique to measure optical retardation. The technique is based on second-harmonic generation from thin films using two beams at the fundamental frequency. The sensitive polarization dependence of the process allows measuring optical retardation very precisely. The technique relies on fundamental symmetry principles and does therefore not require complicated experimental arrangement or data analysis. The technique was demonstrated by determining the retardation of a nominal half-wave plate to a precision and repeatability better than $\lambda {/10\mathrm{}}^4$.

Several nonlinear systems such as the Korteweg–de Vries (KdV) and modified KdV equations and $\lambda {\phi }^4$ theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.

Scaling laws governing implosions of thin shells in converging flows are established by analyzing the implosion trajectories in the ( $A,M$) parametric plane, where $A$ is the in-flight aspect ratio, and $M$ is the implosion Mach number. Three asymptotic branches, corresponding to three implosion phases, are identified for each trajectory in the limit of $A,M\gg 1$. It is shown that there exists a critical value ${\gamma }_{\mathrm{cr}}=1+2/\nu$ ( $\nu =1,2\mathrm{}$ for, respectively, cylindrical and spherical flows) of the adiabatic index $\gamma$, which separates two qualitatively different patterns of the density buildup in the last phase of implosion. The scaling of the stagnation density ${\rho }_s$ and pressure $P_s$ with the peak value $M_0$ of the Mach number is obtained.

A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied. The coherent mechanism is characterized by locally transverse vortex configurations that are intrinsically helical in both physical and Fourier space when there is a peak of the maximum vorticity $\parallel \omega {\parallel }_{\infty }\left(t\right)$. At no time are nonhelical, antiparallel vorticity elements observed. This event precedes the appearance of the traditional signatures of an energy cascade such as strong growth of the dissipation, spectra approaching $-5/3\mathrm{}$, and strongly Beltramized vortex tubes. Comparing how different large-eddy simulations reproduce these properties demonstrates the importance of properly modeling nonlinear transport of both energy and helicity.

The role of stagnant zones in hydrodynamic dispersion is studied for creeping flow through a fixed bed of spherical permeable particles, covering several orders of characteristic time and length scales associated with fluid transport. Numerical simulations employ a hierarchical model to cope with the different temporal and spatial scales, showing good agreement with our experimental results on diffusion-limited mass transfer, transient, and asymptotic longitudinal dispersion. These data demonstrate that intraparticle liquid holdup in macroscopically homogeneous porous media clearly dominates over contributions caused by the intrinsic flow field heterogeneity and boundary-layer mass transfer.

We report a new mechanism for the secondary instabilities of traveling wave patterns in dissipative systems. In close analogy to nonlinear optics and plasma physics, it is a process of three-wave interaction that leads to sum-frequency generation. The primary traveling wave gives rise to two additional wave components, an upper and a lower frequency. We outline the mechanism and we show that this process does occur for wave patterns obtained in an optical experiment.

A general way to classify stochastic chaos is presented and applied to population dynamics models. A stochastic dynamical theory is used to develop an algorithmic tool to measure the transport across basin boundaries and predict the most probable regions of transport created by noise. The results of this tool are illustrated on a model of virus spread in a large population, where transport regions reveal how noise completes the necessary manifold intersections for the creation of emerging stochastic chaos.

Vortex ripples in sand are studied experimentally in a one-dimensional setup with periodic boundary conditions. The nonlinear evolution, far from the onset of instability, is analyzed in the framework of a simple model developed for homogeneous patterns. The interaction function describing the mass transport between neighboring ripples is extracted from experimental runs using a recently proposed method for data analysis, and the predictions of the model are compared to the experiment. An analytic explanation of the wavelength selection mechanism in the model is provided, and the width of the stable band of ripples is measured.

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions. Comparison with the usual multifractal analysis of measures is presented, and the difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented.

Fluid-mediated interactions between particles in a vibrating fluid lead to both long range attraction and short range repulsion. The resulting patterns include hexagonally ordered microcrystallites, time-periodic structures, and chaotic fluctuating patterns with complex dynamics. A model based on streaming flow gives a good quantitative account of the attractive part of the interaction.

Heat conduction in three types of 1D channels is studied. The channels consist of two parallel walls, right triangles as scattering obstacles, and noninteracting particles. The triangles are placed along the walls in three different ways: (i) periodic, (ii) disordered in height, and (iii) disordered in position. The Lyapunov exponents in all three models are zero because of the flatness of triangle sides. It is found numerically that the temperature gradient can be formed in all three channels, but the Fourier heat law is observed only in two disordered ones. The results show that there might be no direct connection between chaos (in the sense of positive Lyapunov exponent) and normal thermal conduction.

The instability of a period-1 spiral wave resulting in a period-2 spiral wave with a line defect is investigated for the first time in a laboratory system. At the very onset the transition proceeds by an emergence of a spiraling line defect, “breathing” intermittently while retaining its symmetry of a period-1 spiral wave. With a further change in a control parameter, the line defect undergoes a meandering transition producing a compound tip trajectory, following a dynamic shape transition. The observed transitions have a strong analogy to the phase synchronization transition of two coupled nonlinear oscillators and the meandering transition of a period-1 spiral wave.