A twist localizes three-dimensional patterns

A mechanism for the localization of spatially periodic, self-organized patterns in anisotropic media which requires systems extended in all three spatial dimensions is presented: When the anisotropy axis is twisted the pattern becomes localized in planes parallel to the anisotropy axis. An analytic description of the effect is developed and used to interpret recent experiments in the high-frequency regime of electroconvection by Bohatsch and Stannarius [Phys. Rev. E {\bf 60}, 5591 (1999)]. The localization width is found to be of the order of magnitude of the geometrical average of pattern wavelength and the inverse twist.Comment: 7 pages, 2 figures, submitted to PRE; minor changes in resubmissio

On the limits of spectral methods for frequency estimation

An algorithm is presented which generates pairs of oscillatory random time series which have identical periodograms but differ in the number of oscillations. This result indicate the intrinsic limitations of spectral methods when it comes to the task of measuring frequencies. Other examples, one from medicine and one from bifurcation theory, are given, which also exhibit these limitations of spectral methods. For two methods of spectral estimation it is verified that the particular way end points are treated, which is specific to each method, is, for long enough time series, not relevant for the main result.Comment: 18 pages, 6 figures (Referee did not like the previous title. Many other changes

A frequency measure robust to linear filtering

A definition of frequency (cycles per unit-time) based on an approximate reconstruction of the phase-space trajectory of an oscillator from a signal is introduced. It is shown to be invariant under linear filtering, and therefore inaccessible by spectral methods. The effect of filtering on frequency in cases where this definition does not perfectly apply is quantified.Comment: 10 pages, 2 figure

Pattern Formation from Defect Chaos --- A Theory of Chevrons

For over 25 years it is known that the roll structure of electroconvection (EC) in the dielectric regime in planarly aligned nematic liquid crystals has, after a transition to defect chaos, the tendency to form chevron structures. We show, with the help of a coarse-grained model, that this effect can generally be expected for systems with spontaneously broken isotropy, that is lifted by a small external perturbation. The linearized model as well as a nonlinear extension are compared to simulations of a system of coupled amplitude equations which generate chevrons out of defect chaos. The mechanism of chevron formation is similar to the development of Turing patterns in reaction diffusion systems.Comment: 17 pages, Latex, 11 PS-figures, submitted to Physica

Weakly Nonlinear Theory of Pattern-Forming Systems with Spontaneously Broken Isotropy

Quasi two-dimensional pattern forming systems with spontaneously broken isotropy represent a novel symmetry class, that is experimentally accessible in electroconvection of homeotropically aligned liquid crystals. We present a weakly nonlinear analysis leading to amplitude equations which couple the short-wavelength patterning mode with the Goldstone mode resulting from the broken isotropy. The new coefficients in these equations are calculated from the hydrodynamics. Simulations exhibit a new type of spatio-temporal chaos at onset. The results are compared with experiments.Comment: 4 pages, RevTeX, 4 PS-figures, to appear in PR

Food Webs: Experts Consuming Families of Experts

The question what determines the structure of natural food webs has been listed among the nine most important unanswered questions in ecology. It arises naturally from many problems related to ecosystem stability and resilience. The traditional view is that population-dynamical stability is crucial for understanding the observed structures. But phylogeny (evolutionary history) has also been suggested as the dominant mechanism. Here we show that observed topological features of predatory food webs can be reproduced to unprecedented accuracy by a mechanism taking into account only phylogeny, size constraints, and the heredity of the trophically relevant traits of prey and predators. The analysis reveals a tendency to avoid resource competition rather than apparent competition. In food webs with many parasites this pattern is reversed.Comment: 16 pages, 3 figures, 1 table + Appendix of 36 pages, 18 figures. movie available from http://ag.rossberg.net/matching.mp

Modulated structures in electroconvection in nematic liquid crystals

Motivated by experiments in electroconvection in nematic liquid crystals with homeotropic alignment we study the coupled amplitude equations describing the formation of a stationary roll pattern in the presence of a weakly-damped mode that breaks isotropy. The equations can be generalized to describe the planarly aligned case if the orienting effect of the boundaries is small, which can be achieved by a destabilizing magnetic field. The slow mode represents the in-plane director at the center of the cell. The simplest uniform states are normal rolls which may undergo a pitchfork bifurcation to abnormal rolls with a misaligned in-plane director.We present a new class of defect-free solutions with spatial modulations perpendicular to the rolls. In a parameter range where the zig-zag instability is not relevant these solutions are stable attractors, as observed in experiments. We also present two-dimensionally modulated states with and without defects which result from the destabilization of the one-dimensionally modulated structures. Finally, for no (or very small) damping, and away from the rotationally symmetric case, we find static chevrons made up of a periodic arrangement of defect chains (or bands of defects) separating homogeneous regions of oblique rolls with very small amplitude. These states may provide a model for a class of poorly understood stationary structures observed in various highly-conducting materials ("prechevrons" or "broad domains").Comment: 13 pages, 13 figure