Inverting Chaos: Extracting System Parameters from Experimental Data

Given a set of experimental or numerical chaotic data and a set of model differential equations with several parameters, is it possible to determine the numerical values for these parameters using a least-squares approach, and thereby to test the model against the data? We explore this question (a) with simulated data from model equations for the Rossler, Lorenz, and pendulum attractors, and (b) with experimental data produced by a physical chaotic pendulum. For the systems considered in this paper, the least-squares approach provides values of model parameters that agree well with values obtained in other ways, even in the presence of modest amounts of added noise. For experimental data, the “fitted” and experimental attractors are found to have the same correlation dimension and the same positive Lyapunov exponent

Temporal Modulation of the Control Parameter in Electroconvection in the Nematic Liquid Crystal I52

I report on the effects of a periodic modulation of the control parameter on electroconvection in the nematic liquid crystal I52. Without modulation, the primary bifurcation from the uniform state is a direct transition to a state of spatiotemporal chaos. This state is the result of the interaction of four, degenerate traveling modes: right and left zig and zag rolls. Periodic modulations of the driving voltage at approximately twice the traveling frequency are used. For a large enough modulation amplitude, standing waves that consist of only zig or zag rolls are stabilized. The standing waves exhibit regular behavior in space and time. Therefore, modulation of the control parameter represents a method of eliminating spatiotemporal chaos. As the modulation frequency is varied away from twice the traveling frequency, standing waves that are a superposition of zig and zag rolls, i.e. standing rectangles, are observed. These results are compared with existing predictions based on coupled complex Ginzburg-Landau equations

Superconducting Diamagnetic Fluctuations in MgB2

The fluctuating diamagnetic magnetization Mfl at constant field H as a function of temperature and the isothermal magnetization Mfl vs H are measured in MgB2, above the superconducting transition temperature. The expressions for Mfl in randomly oriented powders are derived in the Gaussian approximation of local Ginzburg-Landau theory and used for the analysis of the data. The scaled magnetization Mfl/H^{1/2}*T is found to be field dependent. In the limit of evanescent field the behaviour for Gaussian fluctuations is obeyed while for H>~ 100 Oe the field tends to suppress the fluctuating pairs, with a field dependence of Mfl close to the one expected when short wavelength fluctuations and non-local electrodynamic effects are taken into account. Our data, besides providing the isothermal magnetization curves for T>Tc(0) in a BCS-type superconductor such as MgB2, evidence an enhancement of the fluctuating diamagnetism which is related to the occurrence in this new superconductor of an anisotropic spectrum of the superconducting fluctuations.Comment: Tex file, 4 pages, 3 ps figures, submitted to Phys. Rev. Let

Superlattice Patterns in Surface Waves

We report novel superlattice wave patterns at the interface of a fluid layer driven vertically. These patterns are described most naturally in terms of two interacting hexagonal sublattices. Two frequency forcing at very large aspect ratio is utilized in this work. A superlattice pattern ("superlattice-I") consisting of two hexagonal lattices oriented at a relative angle of 22^o is obtained with a 6:7 ratio of forcing frequencies. Several theoretical approaches that may be useful in understanding this pattern have been proposed. In another example, the waves are fully described by two superimposed hexagonal lattices with a wavelength ratio of sqrt(3), oriented at a relative angle of 30^o. The time dependence of this "superlattice-II" wave pattern is unusual. The instantaneous patterns reveal a time-periodic stripe modulation that breaks the 6-fold symmetry at any instant, but the stripes are absent in the time average. The instantaneous patterns are not simply amplitude modulations of the primary standing wave. A transition from the superlattice-II state to a 12-fold quasi-crystalline pattern is observed by changing the relative phase of the two forcing frequencies. Phase diagrams of the observed patterns (including superlattices, quasicrystalline patterns, ordinary hexagons, and squares) are obtained as a function of the amplitudes and relative phases of the driving accelerations.Comment: 15 pages, 14 figures (gif), to appear in Physica

Theory of the Diamagnetism Above the Critical Temperature for Cuprates

Recently experiments on high critical temperature superconductors has shown that the doping levels and the superconducting gap are usually not uniform properties but strongly dependent on their positions inside a given sample. Local superconducting regions develop at the pseudogap temperature ($T^*$) and upon cooling, grow continuously. As one of the consequences a large diamagnetic signal above the critical temperature ($T_c$) has been measured by different groups. Here we apply a critical-state model for the magnetic response to the local superconducting domains between $T^*$ and $T_c$ and show that the resulting diamagnetic signal is in agreement with the experimental results.Comment: published versio

Dynamic scaling and quasi-ordered states in the two dimensional Swift-Hohenberg equation

The process of pattern formation in the two dimensional Swift-Hohenberg equation is examined through numerical and analytic methods. Dynamic scaling relationships are developed for the collective ordering of convective rolls in the limit of infinite aspect ratio. The stationary solutions are shown to be strongly influenced by the strength of noise. Stationary states for small and large noise strengths appear to be quasi-ordered and disordered respectively. The dynamics of ordering from an initially inhomogeneous state is very slow in the former case and fast in the latter. Both numerical and analytic calculations indicate that the slow dynamics can be characterized by a simple scaling relationship, with a characteristic dynamic exponent of $1/4$ in the intermediate time regime

A phase-field model of Hele-Shaw flows in the high viscosity contrast regime

A one-sided phase-field model is proposed to study the dynamics of unstable interfaces of Hele-Shaw flows in the high viscosity contrast regime. The corresponding macroscopic equations are obtained by means of an asymptotic expansion from the phase-field model. Numerical integrations of the phase-field model in a rectangular Hele-Shaw cell reproduce finger competition with the final evolution to a steady state finger the width of which goes to one half of the channel width as the velocity increases

A comparative study of high-field diamagnetic fluctuations in deoxygenated YBa2Cu3O(7-x) and polycrystalline (Bi-Pb)2Sr2Ca3O(10)

We studied three single crystals of YBa2Cu3O{7-x} with Tc= 62.5, 52, and 41 K, and a textured specimen of (Bi-Pb)2Sr2Ca2Cu3O10 with Tc=108 K, for H//c axis. The reversible data were interpreted in terms of 2D lowest-Landau-level fluctuation theory. The data were fit well by the 2D LLL expression for magnetization obtained by Tesanovic etal., producing reasonable values of kappa but larger values of dHc2/dT. Universality was studied by obtaining a simultaneous scaling of Y123 data and Bi2223. An expression for the 2D x-axis LLL scaling factor used to obtain the simultaneous scaling was extracted from theory, and compared with the experimental values. The comparison between the values of the x-axis produced a deviation of 40% which suggests that the hypothesis of universality of the 2D-LLL fluctuations is not supported by the studied samples. We finaly observe that Y123 magnetization data for temperatures above $T_c$ obbey a universal scaling obtained for the diamagnetic fluctuation magnetization from a theory considering non-local field effects. The same scaling was not obbeyed by the corresponding magnetization calculated from the two-dimensional lowest-Landau-level theory.Comment: 7 pages 5 figures, accept in Journ. Low Temp. Phy

Modulation of Localized States in Electroconvection

We report on the effects of temporal modulation of the driving force on a particular class of localized states, known as worms, that have been observed in electroconvection in nematic liquid crystals. The worms consist of the superposition of traveling waves and have been observed to have unique, small widths, but to vary in length. The transition from the pure conduction state to worms occurs via a backward bifurcation. A possible explanation of the formation of the worms has been given in terms of coupled amplitude equations. Because the worms consist of the superposition of traveling waves, temporal modulation of the control parameter is a useful probe of the dynamics of the system. We observe that temporal modulation increases the average length of the worms and stabilizes worms below the transition point in the absence of modulation.Comment: 4 pages, 4 figure

Shear-banding in a lyotropic lamellar phase, Part 1: Time-averaged velocity profiles

Using velocity profile measurements based on dynamic light scattering and coupled to structural and rheological measurements in a Couette cell, we present evidences for a shear-banding scenario in the shear flow of the onion texture of a lyotropic lamellar phase. Time-averaged measurements clearly show the presence of structural shear-banding in the vicinity of a shear-induced transition, associated to the nucleation and growth of a highly sheared band in the flow. Our experiments also reveal the presence of slip at the walls of the Couette cell. Using a simple mechanical approach, we demonstrate that our data confirms the classical assumption of the shear-banding picture, in which the interface between bands lies at a given stress $\sigma^\star$. We also outline the presence of large temporal fluctuations of the flow field, which are the subject of the second part of this paper [Salmon {\it et al.}, submitted to Phys. Rev. E]