Collaborative Research: Absolute and essential instabilities in spatially extended systems

NSF Award Abstract - DMS-0203301 Mathematical Sciences：Collaborative Research：Absolute and Essential Instabilities in Spatially Extended SystemsAbstract 0203301 Scheel该项目探讨了相干态的几种不稳定机制，如波前、脉冲和螺旋波，它们发生在远离平衡的空间扩展系统中。这些机制的共同主题是，它们涉及由扩散和非线性引起的传输现象。这种转变的例子是回火不稳定性的前线，倍周期分岔的螺旋波，和不均匀性的影响，螺旋波的动力学。虽然运输是最方便的建模和描述使用一个理想化的无界域，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。这是本项目的目的是开发技术，可以帮助调查不稳定机制，同时对有界和无界域。其中一个预期的结果将是描述的不稳定机制，是强大的，在大型反应堆的典型边界条件。从技术上讲，我们的方法是基于从空间动力学的方法，使我们能够得到尖锐的逐点估计捕捉明确的影响boundarys. Complex的时空模式，由于化学反应和扩散的相互作用而产生的实验中观察到一些特定的反应（其中包括Belousov-Zhabotinsky反应和亚氯酸盐-碘化物-丙二酸反应）。在心脏组织纤颤期间已经观察到类似的现象，其中螺旋波充当复杂时空动力学的组织中心。不规则图案出现的其他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回火，以及非线性光子光栅中的光学双稳态。本项目的重点是通过分析手段来分析其中的一些失稳机制。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.