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At least 325 records · Page 18Linked to original sources

arXiv1997

Noise sustained pattern growth: Bulk versus boundary effects

The effect of thermally generated bulk stochastic forces on the statistical growth dynamics of forwards bifurcating propagating macroscopic patterns is compared with the influence of fluctuations at the boundary of a semiinfinite system, $0<x$. To that end the linear complex Ginzburg-Landau amplitude equation with additive stochastic forcing is solved by a spatial Laplace transformation in the presence of arbitrary boundary conditions for the fluctuations of the pattern amplitude at $x=0$. A situation where the latter are advected with an imposed through-flow from an outside upstream part towards the inlet boundary at $x=0$ is investigated in more detail. The spatiotemporal growth behavior in the convectively unstable regime is compared with recent work by Swift, Babcock, and Hohenberg [Physica A 204, 625 (1994)] where a special boundary condition is imposed.

patt-sol↗

arXiv1996

Convection in rotating annuli: Ginzburg-Landau equations with tunable coefficients

The coefficients of the complex Ginzburg-Landau equations that describe weakly nonlinear convection in a large rotating annulus are calculated for a range of Prandtl numbers $σ$. For fluids with $σ\approx 0.15$, we show that the rotation rate can tune the coefficients of the corresponding amplitude equations from regimes where coherent patterns prevail to regimes of spatio-temporal chaos.

patt-sol↗

arXiv1997

Diffusive Boundary Layers in the Free-Surface Excitable Medium Spiral

Spiral waves are a ubiquitous feature of the nonequilibrium dynamics of a great variety of excitable systems. In the limit of a large separation in timescale between fast excitation and slow recovery, one can reduce the spiral problem to one involving the motion of a free surface separating the excited and quiescent phases. In this work, we study the free surface problem in the limit of small diffusivity for the slow field variable. Specifically, we show that a previously found spiral solution in the diffusionless limit can be extended to finite diffusivity, without significant alteration. This extension involves the creation of a variety of boundary layers which cure all the undesirable singularities of the aforementioned solution. The implications of our results for the study of spiral stability are briefly discussed.

patt-sol↗

arXiv1997

Competition between Spiral-Defect Chaos and Rolls in Rayleigh-Benard Convection

We present experimental results for pattern formation in Rayleigh-Benard convection of a fluid with a Prandtl number, Pr~ 4. We find that the spiral-defect-chaos (SDC) attractor which exists for Pr~1 has become unstable. Gradually increasing the temperature difference from below to well above its critical value no longer leads to SDC. A sudden jump of temperature difference from below to above onset causes convection to grow from thermal fluctuations and does yield SDC. However, the SDC is a transient; it coarsens and forms a single cell-filling spiral which then drifts toward the cell wall and disappears.

patt-sol↗

arXiv1997

Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film

It has been experimentally observed that weakly conducting suspended films of smectic liquid crystals undergo electroconvection when subjected to a large enough potential difference. The resulting counter-rotating vortices form a very simple convection pattern and exhibit a variety of interesting nonlinear effects. The linear stability problem for this system has recently been solved. The convection mechanism, which involves charge separation at the free surfaces of the film, is applicable to any sufficiently two-dimensional fluid. In this paper, we derive an amplitude equation which describes the weakly nonlinear regime, by starting from the basic electrohydrodynamic equations. This regime has been the subject of several recent experimental studies. The lowest order amplitude equation we derive is of the Ginzburg-Landau form, and describes a forward bifurcation as is observed experimentally. The coefficients of the amplitude equation are calculated and compared with the values independently deduced from the linear stability calculation.

patt-sol↗

arXiv1998

Active Walker Model for the Formation of Human and Animal Trail Systems

Active walker models have recently proved their great value for describing the formation of clusters, periodic patterns, and spiral waves as well as the development of rivers, dielectric breakdown patterns, and many other structures. It is shown that they also allow to simulate the formation of trail systems by pedestrians and ants, yielding a better understanding of human and animal behavior. A comparison with empirical material shows a good agreement between model and reality. Our trail formation model includes an equation of motion, an equation for environmental changes, and an orientation relation. It contains some model functions, which are specified according to the characteristics of the considered animals or pedestrians. Not only the kind of environmental changes differs: Whereas pedestrians leave footprints on the ground, ants produce chemical markings for their orientation. Nevertheless, it is more important that pedestrians steer towards a certain destination, while ants usually find their food sources by chance, i.e. they reach their destination in a stochastic way. As a consequence, the typical structure of the evolving trail systems depends on the respective species. Some ant species produce a dendritic trail system, whereas pedestrians generate a minimal detour system. The trail formation model can be used as a tool for the optimization of pedestrian facilities: It allows urban planners to design convenient way systems which actually meet the route choice habits of pedestrians.

cond-mat.stat-mech↗

arXiv1997

Onset of oscillatory instabilities under stochastic modulation

We study the effect of external stochastic modulation on a system with O(2) symmetry that exhibits a Hopf or oscillatory instability in the absence of modulation. The study includes a random component in both the control parameter of the bifurcation and in the modulation amplitude. Stability boundaries are computed by either solving the stationary Fokker-Planck equation in the vicinity of the center manifold of the underlying deterministic system whenever possible, or by direct numerical solution otherwise. If the modulation amplitude has a stochastic component, the primary bifurcation is always to standing waves at a value of the control parameter that depends on the intensity of the fluctuations. More precisely, and to contrast our results with the case of a deterministic periodic forcing, the onset of instability in the standing wave regime is shifted from its deterministic location, and the region of primary bifurcation to traveling waves disappears yielding instead standing waves at negative values of the control parameter.

patt-sol↗

arXiv1996

Emergent spatial correlations in stochastically evolving populations

We study the spatial pattern formation and emerging long range correlations in a model of three species coevolving in space and time according to stochastic contact rules. Analytical results for the pair correlation functions, based on a truncation approximation and supported by computer simulations, reveal emergent strategies of survival for minority agents based on selection of patterns. Minority agents exhibit defensive clustering and cooperative behavior close to phase transitions.

adap-org↗

arXiv1996

Sources and sinks separating domains of left- and right-traveling waves: Experiment versus amplitude equations

In many pattern forming systems that exhibit traveling waves, sources and sinks occur which separate patches of oppositely traveling waves. We show that simple qualitative features of their dynamics can be compared to predictions from coupled amplitude equations. In heated wire convection experiments, we find a discrepancy between the observed multiplicity of sources and theoretical predictions. The expression for the observed motion of sinks is incompatible with any amplitude equation description.

patt-sol↗

arXiv1997

Worm Structure in Modified Swift-Hohenberg Equation for Electroconvection

A theoretical model for studying pattern formation in electroconvection is proposed in the form of a modified Swift-Hohenberg equation. A localized state is found in two dimension, in agreement with the experimentally observed ``worm" state. The corresponding one dimensional model is also studied, and a novel stationary localized state due to nonadiabatic effect is found. The existence of the 1D localized state is shown to be responsible for the formation of the two dimensional ``worm" state in our model.

patt-sol↗

arXiv1997

Derivation of Amplitude Equations by Renormalization Group Method

A proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations. The formulation makes it possible not only avoiding a serious difficulty in the previous reduction to amplitude equations by eliminating all of the secular terms but also consistent derivation of higher-order correction to amplitude equations.

patt-sol↗

arXiv1997

Propagation Failure in Excitable Media

We study a mechanism of pulse propagation failure in excitable media where stable traveling pulse solutions appear via a subcritical pitchfork bifurcation. The bifurcation plays a key role in that mechanism. Small perturbations, externally applied or from internal instabilities, may cause pulse propagation failure (wave breakup) provided the system is close enough to the bifurcation point. We derive relations showing how the pitchfork bifurcation is unfolded by weak curvature or advective field perturbations and use them to demonstrate wave breakup. We suggest that the recent observations of wave breakup in the Belousov-Zhabotinsky reaction induced either by an electric field or a transverse instability are manifestations of this mechanism.

patt-sol↗

arXiv1997

Director configuration of planar solitons in nematic liquid crystals

The director configuration of disclination lines in nematic liquid crystals in the presence of an external magnetic field is evaluated. Our method is a combination of a polynomial expansion for the director and of further analytical approximations which are tested against a numerical shooting method. The results are particularly simple when the elastic constants are equal, but we discuss the general case of elastic anisotropy. The director field is continuous everywhere apart from a straight line segment whose length depends on the value of the magnetic field. This indicates the possibility of an elongated defect core for disclination lines in nematics due to an external magnetic field.

cond-mat.soft↗

arXiv1998

Influence of the Soret effect on convection of binary fluids

Convection in horizontal layers of binary fluids heated from below and in particular the influence of the Soret effect on the bifurcation properties of extended stationary and traveling patterns that occur for negative Soret coupling is investigated theoretically. The fixed points corresponding to these two convection structures are determined for realistic boundary conditions with a many mode Galerkin scheme for temperature and concentration and an accurate one mode truncation of the velocity field. This solution procedure yields the stable and unstable solutions for all stationary and traveling patterns so that complete phase diagrams for the different convection types in typical binary liquid mixtures can easily be computed. Also the transition from weakly to strongly nonlinear states can be analyzed in detail. An investigation of the concentration current and of the relevance of its constituents shows the way for a simplification of the mode representation of temperature and concentration field as well as for an analytically manageable few mode description.

patt-sol↗

arXiv1997

Stability of a Nonequilibrium Interface in a Driven Phase Segregating System

We investigate the dynamics of a nonequilibrium interface between coexisting phases in a system described by a Cahn-Hilliard equation with an additional driving term. By means of a matched asymptotic expansion we derive equations for the interface motion. A linear stability analysis of these equations results in a condition for the stability of a flat interface. We find that the stability properties of a flat interface depend on the structure of the driving term in the original equation.

patt-sol↗

arXiv1997

Dynamical Reduction of Discrete Systems Based on the Renormalization Group Method

The renormalization group (RG) method is extended for global asymptotic analysis of discrete systems. We show that the RG equation in the discretized form leads to difference equations corresponding to the Stuart-Landau or Ginzburg-Landau equations. We propose a discretization scheme which leads to a faithful discretization of the reduced dynamics of the original differential equations.

patt-sol↗

arXiv1997

Adiabatic reduction near a bifurcation in stochastically modulated systems

We re-examine the procedure of adiabatic elimination of fast relaxing variables near a bifurcation point when some of the parameters of the system are stochastically modulated. Approximate stationary solutions of the Fokker-Planck equation are obtained near threshold for the pitchfork and transcritical bifurcations. Stochastic resonance between fast variables and random modulation may shift the effective bifurcation point by an amount proportional to the intensity of the fluctuations. We also find that fluctuations of the fast variables above threshold are not always Gaussian and centered around the (deterministic) center manifold as was previously believed. Numerical solutions obtained for a few illustrative examples support these conclusions.

patt-sol↗

arXiv1997

The dynamics of vortex lines in the three-dimensional complex Ginzburg-Landau equation: instability, stretching, entanglement, and helices

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginzburg-Landau equation (CGLE). It is shown that a straight vortex line is unstable with respect to spontaneous stretching and bending in a substantial range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The boundary of the three-dimensional instability in parameter space is determined. Near the stability boundary, the supercritical saturation of the instability is found, resulting in the formation of stable helicoidal vortices.

patt-sol↗