Collaborative Research: Absolute and essential instabilities in spatially extended systems

NSF奖摘要-DMS-0203301数学科学：合作研究：空间扩展系统中的绝对和本质不稳定性这个项目探索了在远离平衡的空间扩展系统中出现的相干态的几种不稳定性机制，如前锋、脉冲和螺旋波。这些机制的共同主题是，它们涉及由扩散和非线性引起的输运现象。这种转变的例子有锋面的回火不稳定性，螺旋波的倍周期分叉，以及非均匀对螺旋波动力学的影响。虽然交通是最方便的模型和描述使用一个理想的无界区域，边界可以很好地增强或抑制不稳定性通过部分反射或产生波。这个项目的目的是开发技术，帮助同时研究有界域和无界域上的不稳定机制。预期的结果之一将是描述相对于大型反应堆中的典型边界条件是稳健的不稳定机制。从技术上讲，我们的方法是基于空间动力学的方法，允许我们推导出精确的逐点估计，明确地捕捉边界的影响。由于化学反应和扩散的相互作用而产生的复杂的时空模式已经在一些特定的反应中被实验观察到(其中包括Belousov-Zhabotinsky反应和亚氯酸盐-碘-丙二酸反应)。在心脏组织的纤颤过程中也观察到了类似的现象，螺旋波作为复杂时空动力学的组织中心。出现不规则图案的其他例子有振荡介质中脉冲的相互作用，催化反应中激发脉冲的回射，以及非线性光子光栅中的光学双稳态。本项目的重点是用解析的方法分析其中的一些失稳机理。这不仅将加深我们对化学和生物系统中模式形成的理解，而且最终将允许对模式进行系统控制，例如，在一氧化碳的催化氧化和细胞内组织中钙波的传播中。

NSF Award Abstract - DMS-0203301Mathematical Sciences: Collaborative Research: Absolute and essential instabilities in spatially extended systemsAbstract0203301 ScheelThis project explores several instability mechanisms of coherent states, such as fronts, pulses and spiral waves, that occur in spatially extended systems far from equilibrium. The common theme of these mechanisms is that they involve transport phenomena caused by diffusion and nonlinearities. Examples of such transitions are backfiring instabilities of fronts, period-doubling bifurcations of spiral waves, and the effect of inhomogeneities on the dynamics of spiral waves. While transport is most conveniently modeled and described using an idealized unbounded domain, boundaries may well enhance or inhibit the instability through partial reflection or generation of waves. It is the aim of this project to develop techniques that can help to investigate instability mechanisms simultaneously on bounded and unbounded domains. One of the expected outcomes will be a description of instability mechanisms that is robust with respect to typical boundary conditions in large reactors. Technically, our approach is based on methods from spatial dynamics that allow us to derive sharp pointwise estimates which capture explicitly the effects of boundaries.Complicated spatio-temporal patterns that arise due to the interplay of chemical reactions and diffusion have been observed experimentally in a number of specific reactions (among them the Belousov-Zhabotinsky reaction and the chlorite-iodide-malonic acid reaction). Similar phenomena have been observed during fibrillation of cardiac tissue where spiral waves act as organizing centers for the complex spatio-temporal dynamics. Other examples where irregular patterns occur are the interaction of pulses in oscillatory media, backfiring of excitation pulses in catalytic reactions, and optical bistability in nonlinear photonic gratings. The focus of this project is to analyze some of these instability mechanisms by analytical means. This will not only further our understanding of pattern formation in chemical and biological systems, but will eventually allow for a systematic control of patterns, for instance, in the catalytic oxidation of carbon-monoxide and in the propagation of calcium waves in intracellular tissue.