Patterns in Continuous Systems

NSF奖摘要-DMS-0072444数学科学：连续系统中的模式摘要这个项目研究对称系统中的分叉和模式形成。这样的系统在各种应用中自然会出现。该项目集中于弱强迫对称破缺扰动的动力学后果，并因此试图确定等变动力学的那些方面是(I)对这种扰动不敏感，以及(Ii)由于它们。在许多情况下，这样的扰动是造成全局分叉的原因，而这些分叉反过来又可能将混沌动力学引入系统。该项目将应用于一些连续介质系统，重点研究流体动力系统中爆发式行为的产生，如对流和法拉第不稳定性，以及声学驱动的液滴和气泡形状振荡的动力学。本项目研究各种物理系统中建立的模式如何随着指定系统的参数的变化而变化，重点是具有不同空间和时间对称性的系统。该方法可以识别新类型的复杂行为以及导致这种行为的机制。例如，应用包括不同形状区域中的水波行为。该项目调查了区域形状的轻微变化的影响，并试图确定行为的哪些方面对这种扰动不敏感，哪些是由于它们造成的。在许多情况下，这样的扰动是将混沌动力学引入系统的原因，否则系统将以简单的方式运行。还将研究出一些其他的应用，包括流体动力系统中的爆发式行为和液滴和气泡的振荡。

NSF Award Abstract - DMS-0072444Mathematical Sciences: Patterns in Continuous SystemsAbstract0072444 KnoblochThis project studies bifurcations and pattern formation in systems with symmetry. Such systems arise naturally in a variety of applications. The project focuses on the dynamical consequences of weak forced symmetry-breaking perturbations, and seeks thereby to identify those aspects of equivariant dynamics that are (i) insensitive to such perturbations, and (ii) due to them. In many cases such perturbations are responsible for the creation of global bifurcations, and these in turn may introduce chaotic dynamics into the system. Applications to a number of continuum systems will be worked out, focusing on the generation of burst-like behavior in hydrodynamical systems such as convection and the Faraday instability, and the dynamics of acoustically driven shape oscillations of drops and bubbles.This project investigates how the patterns set up in various physical systems change as the parameters specifying the system vary, with emphasis on systems having various spatial and temporal symmetries. The approach can identify new types of complex behavior and the mechanisms responsible for it. Applications include, for example, the behavior of water waves in regions of different shapes. The project investigates the effects of slight changes in the shape of the region, and seeks to identify which aspects of the behavior are insensitive to such perturbations, and which are due to them. In many cases such perturbations are responsible for the introduction of chaotic dynamics into systems that would otherwise behave in a simple manner. A number of additional applications will be worked out, including burst-like behavior in hydrodynamical systems and oscillations of drops and bubbles.