Collaborative Research: Absolute and essential instabilities in spatially extended systems

数学科学：协同研究：空间扩展系统的绝对和基本不稳定性[摘要]0203301 scheel本项目探讨了在远离平衡的空间扩展系统中出现的相干态（如前沿、脉冲和螺旋波）的几种不稳定机制。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。这种转变的例子是锋面的逆射不稳定性，螺旋波的周期加倍分岔，以及非均匀性对螺旋波动力学的影响。虽然输运是最方便的建模和描述使用理想的无界域，边界可以很好地增强或抑制不稳定性通过部分反射或波的产生。这个项目的目的是开发技术，可以帮助研究不稳定机制同时在有界和无界的领域。预期的结果之一将是描述相对于大型反应堆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，使我们能够得出明确捕获边界影响的尖锐的点估计。由于化学反应和扩散的相互作用而产生的复杂时空模式已在一些特定反应（其中包括Belousov-Zhabotinsky反应和氯酸碘-丙二酸反应）中通过实验观察到。在心脏组织的纤颤中也观察到类似的现象，其中螺旋波作为复杂时空动力学的组织中心。其他发生不规则图案的例子是振荡介质中脉冲的相互作用，催化反应中激发脉冲的反射，以及非线性光子光栅中的光学双稳性。本项目的重点是用分析方法分析其中的一些不稳定机制。这不仅将进一步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.