Patterns In Nonequilibrium Systems Research Articles

Effect of the Polarity of Solvents on Periodic Precipitation: Formation of Hierarchical Revert Liesegang Patterns.

Liesegang pattern (LP) is one example of self-organized periodic precipitation patterns in nonequilibrium systems. Several studies have demonstrated that the LP morphology can track physicochemical environmental conditions (e.g., temperature); however, the polarity effect has not been explored to date. In this study, a copper chromate system is used to reveal the impact of solvent polarity on the evolving LP structure using water/organic solvent mixtures. In the typical case of using water/dimethyl sulfoxide (DMSO) mixtures, two drastic changes in LP morphology with increasing DMSO contents were found: (i) increasing frequency of the original structure and (ii) formation of a hierarchical pattern with the appearance of another, lower-frequency structure. Furthermore, the simulation model operating with a bimodal size distribution, allowing both homogeneous and heterogeneous precipitations showed good agreement with the experimental results. Therefore, this study demonstrated that LP can be tailored by solvent polarity and can be used for designing hierarchical precipitation patterns in a straightforward manner.

Pattern formation in clouds via Turing instabilities

AbstractPattern formation in clouds is a well-known feature, which can be observed almost every day. However, the guiding processes for structure formation are mostly unknown, and also theoretical investigations of cloud patterns are quite rare. From many scientific disciplines the occurrence of patterns in non-equilibrium systems due to Turing instabilities is known, i.e. unstable modes grow and form spatial structures. In this study we investigate a generic cloud model for the possibility of Turing instabilities. For this purpose, the model is extended by diffusion terms. We can show that for some cloud models, i.e special cases of the generic model, no Turing instabilities are possible. However, we also present a general class of cloud models, where Turing instabilities can occur. A key requisite is the occurrence of (weakly) nonlinear terms for accretion. Using numerical simulations for a special case of the general class of cloud models, we show spatial patterns of clouds in one and two spatial dimensions. From the numerical simulations we can see that the competition between collision terms and sedimentation is an important issue for the existence of pattern formation.

Dynamically generated hierarchies in games of competition.

Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in nonequilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows us to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.

DYNAMICAL BIFURCATION OF THE ONE DIMENSIONAL MODIFIED SWIFT-HOHENBERG EQUATION

Abstract. In this paper, we study the dynamical bifurcation of the mod-ified Swift-Hohenberg equation on a periodic interval as the system con-trol parameter crosses through a critical number. This critical numberdepends on the period. We show that there happens the pitchfork bifur-cation under the spatially even periodic condition. We also prove thatin the general periodic condition the equation bifurcates to an attractorwhich is homeomorphic to a circle and consists of steady states solutions. 1. IntroductionThe formation of patterns in non-equilibrium systems is closely related tothe instability ([9]) and the bifurcation analysis plays an important role inunderstanding the instability. Indeed, the instability arises when stable statesare driven into unstable states during phases transition. As a control parameterrelated to the instability passes through critical values, the trivial state losesits stability and bifurcates to nontrivial states which form patterns. Thenthe dynamics of the system after the threshold of bifurcation is completelydetermined by its behavior on the center manifold. In particular, Ma andWang showed in [13] that the system bifurcates to a nontrivial attractor on thecenter manifold which determines the final patterns of the system.The Swift-Hohenberg equation is a widely accepted model in the study of theformation of patterns [1, 12]. It was derived in [18] as an approximate model forthe Rayleigh-B´enardconvection describing the pattern formation in layer fluidsbetween horizontal plates. It has attracted a lot of interest in various areas ofapplication regardingpattern formations such as Taylor-Couetteflow and lasers[5]. In particular, there has been much efforts on the bifurcation analysis as away of understanding pattern formations. See [4, 8, 10, 11, 14, 15, 19, 20] forrecent development in this direction.

Phase dynamics with drift: boundary effects

The description of patterns in nonequilibrium systems is a fascinating subject. It becomes somewhat untractable if one insists to keep the original equations in their complete form, as the Oberbeck–Boussinesq equations for instance in Rayleigh–Bénard convection. Various reduction scheme have been imagined, the ultimate one being the phase picture. I examine a simple version of the phase equation, relevant for systems of travelling rolls or for distributed self-oscillations. The boundary effects limit the ability of this phase to drift freely and yields a well-defined space structure, which can be analyzed in two different limits.

LOCALIZED PATTERNS AND FRONTS IN NONEQUILIBRIUM SYSTEMS

The aim of this paper is to review some of the mechanisms which leads to stable localized patterns in nonequilibrium systems.

Fronts between hexagons and squares in a generalized Swift-Hohenberg equation.

Pinning effects in domain walls separating different orientations in patterns in nonequilibrium systems are studied. Usually, theoretical studies consider perfect structures, but in experiments, point defects, grain boundaries, etc., always appear. The aim of this paper is to perform an analysis of the stability of fronts between hexagons and squares in a generalized Swift-Hohenberg model equation. We focus the analysis on pinned fronts between domains with different symmetries by using amplitude equations and by considering the small-scale structure in the pattern. The conditions for pinning effects and stable fronts are determined. This study is completed with direct simulations of the generalized Swift-Hohenberg equation. The results agree qualitatively with recent observations in convection and in ferrofluid instabilities. \textcopyright{} 1996 The American Physical Society.

Controlling domain patterns far from equilibrium.

A high degree of control over the structure and dynamics of domain patterns in nonequilibrium systems can be achieved by applying nonuniform external fields near parity breaking front bifurcations. An external field with a linear spatial profile stabilizes a propagating front at a fixed position or induces oscillations with frequency that scales like the square root of the field gradient. Nonmonotonic profiles produce a variety of patterns with controllable wavelengths, domain sizes, and frequencies and phases of oscillations.

Rhythms and patterns transitions

We study transitions leading to rhythms and patterns in non-equilibrium systems. Our approach is inspired in Landau's analysis of phase transitions.

Two-dimensional traveling wave patterns in nonequilibrium systems

Traveling wave patterns formed at the oscillatory onset of convection in binary mixtures with free slip and permeable horizontal boundary conditions are theoretically derived by means of an order parameter equation for the case of large aspect ratio systems with circular and rectangular geometry. We obtain in addition to ‘Zipper states’ the so called ‘confined states’, which so far has been observed only experimentally. Our present study of traveling wave patterns in circular systems, which are the first theoretical investigations in this respect, exhibit a characteristic change of behaviour. Close to onset the waves travel in radial direction towards the sidewalls. For higher Rayleigh-numbers the waves are confined to the circular boundary traveling in azimuthal direction. The occurrence of this transition should be confirmed experimentally.

Commensurate-incommensurate transition in nonequilibrium systems.

We consider the effect of a spatially periodic forcing at the onset of one-dimensional periodic patterns in nonequilibrium systems. It is shown that the competition between periodicities can lead to a commensurate-incommensurate transition mediated by solitons as in equilibrium systems. This result is in agreement with a recent experiment.

The shape of stationary dislocations

It is shown that the structure of the stationary dislocations which occur in a wide variety of patterns in nonequilibrium systems is given by a self-similar solution of the universal Cross-Newell equation.