Statistical studies of chaotic wave patterns

Collective motion of macroscopic spheres floating on capillary ripples: dynamic heterogeneity and dynamic criticality

When a densely packed monolayer of macroscopic spheres floats on chaotic capillary Faraday waves, a coexistence of large scale convective motion and caging dynamics typical for glassy systems is observed. We subtract the convective mean flow using a coarse graining (homogenization) method and reveal subdiffusion for the caging time scales followed by a diffusive regime at later times. We apply the methods developed to study dynamic heterogeneity and show that the typical time and length scales of the fluctuations due to rearrangements of observed particle groups significantly increase when the system approaches its largest experimentally accessible packing concentration. To connect the system to the dynamic criticality literature, we fit power laws to our results. The resultant critical exponents are consistent with those found in densely packed suspensions of colloids.

Phase randomization of three-wave interactions in capillary waves

We present new experimental results on the transition from coherent-phase to random-phase three-wave interactions in capillary waves under parametric excitation. Above the excitation threshold, coherent wave harmonics spectrally broaden. An increase in the pumping amplitude increases spectral widths of wave harmonics and eventually causes a strong decrease in the degree of the three-wave phase coupling. The results point to the modulation instability of capillary waves, which leads to breaking of continuous waves into ensembles of short-lived wavelets or envelope solitons, as the reason for the phase randomization of three-wave interactions.

How waves affect the distribution of particles that float on a liquid surface

We study experimentally how waves affect the distribution of particles that float on a liquid surface. We show that clustering of small particles in a standing wave is a nonlinear effect with the clustering time decreasing as the square of the wave amplitude. In a set of random waves, we show that small floaters concentrate on a multifractal set with caustics.

Average patterns and coherent phenomena in wide aperture lasers

Using a realistic model of wide aperture, weakly astigmatic lasers we develop a framework to analyze experimental average intensity patterns. We use the model to explain the appearance of patterns in terms of the modes of the cavity and to show that the breaking of the symmetry of the average intensity patterns is caused by overlaps in the frequency spectra of nonvanishing of modes with different parity. This result can be used even in systems with very fast dynamics to detect experimentally overlaps of frequency spectra of modes.

Microextensive chaos of a spatially extended system

By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky equation for system sizes L in the range 79 < or = L < or = 93, we show that the Lyapunov fractal dimension D scales microextensively, increasing linearly with L even for increments Delta L that are small compared to the average cell size of 9 and to various correlation lengths. This suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity.

Complex Ginzburg-Landau equation in the presence of walls and corners

We investigate the influence of walls and corners (with Dirichlet and Neumann boundary conditions) in the evolution of two-dimensional autooscillating fields described by the complex Ginzburg-Landau equation. Analytical solutions are found, and arguments provided, to show that Dirichlet walls introduce strong selection mechanisms for the wave pattern. Corners between walls provide additional synchronization mechanisms and associated selection criteria. The numerical results fit well with the theoretical predictions in the parameter range studied.

Functional dependence and quasiperiodicity in the spatiotemporal dynamics of yttrium iron garnet films

When thin films of yttrium iron garnet (YIG) are placed in a magnetic field and driven at microwave (rf) frequencies, nonlinear interactions within the material cause the normal microwave spin precession to be modulated at lower frequencies. We measure these lower frequency (kHz) signals at two spatially separated locations on the YIG film and use linear and nonlinear analysis to study the functional dependence of the spin dynamics at one location on the spin dynamics at the other location. We see dynamical states where nonlinear analysis can detect a functional dependence that the linear analysis fails to reveal.

Amplitude measurements of Faraday waves

A light reflection technique is used to measure quantitatively the surface elevation of Faraday waves. The performed measurements cover a wide parameter range of driving frequencies and sample viscosities. In the capillary wave regime the bifurcation diagrams exhibit a frequency independent scaling proportional to the wavelength. We also provide numerical simulations of the full Navier-Stokes equations, which are in quantitative agreement up to supercritical drive amplitudes of epsilon approximately equal 20%. The validity of an existing perturbation analysis is found to be limited to epsilon<2.5%.

Singularities in the Boussinesq equation and in the generalized Korteweg-de Vries equation

In this paper, two kinds of analytic singular solutions (finite-time and infinite-time singular solutions) of two classical wave equations (the Boussinesq equation and a generalized Korteweg-de Vries equation) are obtained by means of the improved homogeneous balance method and a nonlinear transformation. The solutions show that special singular wave patterns exist in the classical models of shallow water wave problem.

Scarred patterns in surface waves

Surface wave patterns are investigated experimentally in a system geometry that has become a paradigm of quantum chaos: the stadium billiard. Linear waves in bounded geometries for which classical ray trajectories are chaotic are known to give rise to scarred patterns. Here, we utilize parametrically forced surface waves (Faraday waves), which become progressively nonlinear beyond the wave instability threshold, to investigate the subtle interplay between boundaries and nonlinearity. Only a subset (three main types) of the computed linear modes of the stadium are observed in a systematic scan. These correspond to modes in which the wave amplitudes are strongly enhanced along paths corresponding to certain periodic ray orbits. Many other modes are found to be suppressed, in general agreement with a prediction by Agam and Altshuler based on boundary dissipation and the Lyapunov exponent of the associated orbit. Spatially asymmetric or disordered (but time-independent) patterns are also found even near onset. As the driving acceleration is increased, the time-independent scarred patterns persist, but in some cases transitions between modes are noted. The onset of spatiotemporal chaos at higher forcing amplitude often involves a nonperiodic oscillation between spatially ordered and disordered states. We characterize this phenomenon using the concept of pattern entropy. The rate of change of the patterns is found to be reduced as the state passes temporarily near the ordered configurations of lower entropy. We also report complex but highly symmetric (time-independent) patterns far above onset in the regime that is normally chaotic.

Phase turbulence in rayleigh-Benard convection

We present a three-dimensional simulation of Rayleigh-Benard convection in a large aspect ratio Gamma=60 with stress-free boundaries for a fluid Prandtl number sigma=0.5. We find that a spatiotemporal chaotic state (phase turbulence) emerges immediately above onset, which we investigate as a function of the reduced control parameter epsilon. In particular we find that the correlation length for the vertical velocity field, the time averaged convective current, and the mean square vorticity have power law behaviors near onset, with exponents given by -1/2, 1, and 5/2 respectively. We also find that the time averaged vertical velocity and vertical vorticity fields have the same (disordered) spatial characteristics as the corresponding instantaneous patterns for these fields, and that there is no long-term phase correlation in the system. Finally, we present simple theoretical explanations for the time averaged convective current as a function of the control parameter, and for the fact that the time dependence of three global quantities (characterizing the dissipation of kinetic energy, the release of internal energy by buoyancy, and entropy flow) is essentially the same.

Shock structures in time-averaged patterns for the kuramoto-sivashinsky equation

The Kuramoto-Sivashinsky equation with fixed boundary conditions is numerically studied. Shocklike structures appear in the time-averaged patterns for some parameter range of the boundary values. Effective diffusion constant is estimated from the relation of the width and the height of the shock structures.

Forecasting confined spatiotemporal chaos with genetic algorithms

A technique to forecast spatiotemporal time series is presented. It uses a proper orthogonal or Karhunen-Loève decomposition to encode large spatiotemporal data sets in a few time series, and genetic algorithms to efficiently extract dynamical rules from the data. The method works very well for confined systems displaying spatiotemporal chaos, as exemplified here by forecasting the evolution of the one-dimensional complex Ginzburg-Landau equation in a finite domain.

Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system

By computing the Karhunen-Loeve decomposition (KLD) correlation length xi(KLD) of a reaction-diffusion system in the extensive chaos regime, we show that it is a sensitive measure of spatial dynamical inhomogeneities. It reveals substantial spatial nonuniformity of the dynamics at the boundaries and can also detect slow spatial variations in system parameters. The intensive length xi(KLD) can be easily computed from small local subsystems and is found to have a similar parametric dependence as the two-point correlation length computed over the full system size.

Order and disorder in fluid motion

The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characterized statistically, and methods for doing so are evolving. Transport and mixing phenomena can also lead to spatial complexity, but the degree depends on the significance of molecular or thermal diffusion.