Localized Structures and Complex Dynamics in Pattern Forming Systems

这个项目的总体目标是帮助理解在许多受驱动的耗散系统中自发产生的时空结构。在数学上，这些系统代表了具有多个自由度的动力系统。PI将聚焦于具有轴向各向异性的二维系统中的波。G.Ahler等人最近的实验给出了直接的动机。关于向列相液晶中的电对流，观察到了三种有趣的模式：1)表现出时空混沌动力学的扩展波，2)局部化成窄而长的斑块(“蠕虫”)的波，以及3)时间上不规则的、空间上局部化的波幅爆发。这些现象是在小波幅的起始点上方立即发现的。因此，它们提供了一个极好的机会来开发和测试数学思想，这些思想将在耗散结构的弱非线性理论中相当普遍地相关。此外，由于各向异性，这个二维系统可以用Ginzburg-Landau型方程系统地描述。不需要求助于斯威夫特-霍亨伯格类型的现象学方程。该项目的主要部分致力于蠕虫。它们不能用直接的渐近分析得到的金兹堡-朗道方程来描述。分析必须扩展到包括某些附加模式，这些模式严格地说不在中心流形上，但已经变得非常相关，已经非常接近阈值。要解决的主要问题是：蠕虫的局部化机制和适当的一维约化(驻波和行波脉冲)，长蠕虫的稳定性，以及从小噪声中蠕虫的成核。该项目将演示如何在像这里描述的情况下，通过包括单一的附加模式来显著提高弱非线性理论的定性和定量相关性。这一方面也适用于许多其他系统。在更具体的层面上，慢模的平流在其他波动系统中也将是重要的。同时，对于电对流系统，将通过与实验者的密切合作，与正在进行的实验进行详细的定性和定量对比，在广泛的系统中自发地出现空间结构。这些模式可以是与时间无关的，也可以是行波或驻波的形式。仅举几个例子，如动物皮毛印记、振动液体表面的波、云街、化学反应和神经传导中的波。如果一个系统的属性在其整个范围内是相同的，人们通常会认为这些模式也覆盖整个系统。然而，在相当多的情况下，已经观察到斑块甚至单个波峰中的图案的自发局部化。在同一实验的重复运行中，补丁出现在随机变化的位置。这表明它们的本地化不是由于该位置的系统的特殊属性。这种局域波已经在振动颗粒材料、混合物中的流体对流和向列相液晶中观察到。所资助的研究将有助于理解导致聚焦于在液晶中观察到的“蠕虫”的局部化的原因。它将详细阐明这种结构的机制，并研究这种结构的混沌破裂，以及它们是如何由流体中的热运动引起的弱噪声产生(成核)的。利用先进的数学方法，这项研究将得出在其他系统中更普遍相关的中心特征。

9804673RieckeThe general goal of this project is to contribute to the understanding of spatio-temporal structures that arise spontaneously in many driven dissipative systems. Mathematically, these systems represent dynamical systems with many degrees of freedom. The PI will focus on waves in two-dimensional systems with axial anisotropy. The immediate motivation is given by recent experiments by G. Ahlers et al. on electroconvection in nematic liquid crystals, where three interesting regimes have been observed: 1) extended waves exhibiting spatio-temporally chaotic dynamics, 2) waves localized into narrow, long patches (`worms'), and 3) temporally irregular, spatially localized bursting of the wave amplitude. These phenomena are found immediately above onset at small amplitudes of the waves. They therefore afford an excellent opportunity to develop and test mathematical ideas that will be relevant quite generally in weakly nonlinear theories of dissipative structures. Furthermore, due to the anisotropy this two-dimensional system can be described systematically by Ginzburg-Landau-type equations. No recourse to phenomenological equations of the Swift-Hohenberg-type is needed. The main part of the project is devoted to the worms. They cannot be described by the Ginzburg-Landau equations derived by straightforward asymptotic analysis. The analysis has to be extended to include certain additional modes that are strictly speaking not on the center manifold but become relevant already very close to threshold. The main questions to be addressed are: the localization mechanism of the worms and suitable one-dimensional reductions (standing- and traveling-wave pulses), the stability of long worms, and the nucleation of worms from small noise. The project will demonstrate how in cases like the one described here the qualitative and quantitative relevance of the weakly nonlinear theory can be drastically increased by including a single additional mode. This aspect carries over to a wide range of other systems. On a more specific level, the advection of a slow mode will be important in other wave systems as well. At the same time, for the electroconvection system detailed qualitative and quantitative comparisons with ongoing experiments will be achieved through close cooperation with the experimentalists.In wide range of systems spatial structures appear spontaneously. These patterns can be time-independent or may have the form of traveling or standing waves. Just a few examples are animal coat markings, waves on the surface of vibrated fluids, cloud streets, waves in chemical reactions and in nerve conduction. If the properties of a system are the same over its whole extent one would usually expect that the patterns also cover the whole system. However, in quite a number of cases a spontaneous localization of the pattern in patches or even single wave crests has been observed. In repeated runs of the same experiment the patches occur in randomly varying locations. This indicates that their localization is not due to special properties of the system at that location. Such localized waves have been observed, e.g., in vibrated granular material, fluid convection in mixtures, and in nematic liquid crystals. The funded research will contribute to the understanding of the causes that lead to localization focussing on the `worms' observed in liquid crystals. It will elucidate the mechanism in detail and investigate the chaotic bursting of such structures and how they are generated (nucleated) from weak noise that is due to the thermal motion in the fluid. Using advanced mathematical methods, the research will bring out the central features that are relevant more generally in other systems as well.