Spiral Waves and Target Patterns

许多生物、化学和物理过程表现出螺旋波和目标模式。例如，在自催化化学反应中，在变形虫的细胞信号传导过程中，以及在心脏组织中，螺旋波通常是心律失常的原因，也是心室颤动和心脏骤停的前兆。同样，靶图案在自催化反应中也很突出，而且在具有小空间不均匀性的系统中也经常发现。这个项目的重点是了解螺旋波和目标模式何时出现，在什么条件下可以观察到它们，以及什么机制导致它们改变形状或消失。研究者将开发分析、计算和几何工具来回答这些问题，并将研究结果应用于心脏组织中自催化化学反应和波传播的模型。研究生和本科生将参与这些研究活动。这个项目的重点是通过空间动力学方法的发展来理解螺旋波和目标模式的存在性、多样性、稳定性和分岔，这些方法可以应用于广泛的反应扩散系统。研究者将利用空间动力学建立反应扩散系统中一维螺旋波和目标模式的多重结果，并将这些结果应用于Brusselator模型。第二个项目的中心是螺旋波和靶型的横向不稳定性。横向不稳定性对平面结构的光谱没有贡献，尽管它们确实引起线性不稳定性。目的是研究这些不稳定性如何在大型有界磁盘上表现出来。第三个项目的重点是通过开发依赖于适当的指数权重的预调节器来独立于域的大小降低解析的范数，从而使螺旋谱的数值计算更加鲁棒。在最后一个项目中，将发展奇异摄动方法来理解当模型中的某些扩散系数趋近于零时平面螺旋波和目标图案的轮廓和光谱的极限。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many biological, chemical, and physical processes exhibit spiral waves and target patterns. Spiral waves occur, for instance, in autocatalytic chemical reactions, during cellular signaling of amoebae, and in cardiac tissue where they are often the cause of cardiac arrhythmias and the precursor to ventricular fibrillation and cardiac arrest. Similarly, target patterns appear prominently in autocatalytic reactions and are also often found in systems with small spatial inhomogeneities. This project focuses on understanding when spiral waves and target patterns arise, under which conditions they can be observed, and what mechanisms cause them to change their shape or disappear. The investigator will develop analytical, computational, and geometric tools to answer these questions and apply the findings to models of autocatalytic chemical reactions and wave propagation in cardiac tissue. Graduate and undergraduate students will be engaged in these research activities.This project focuses on understanding existence, multiplicity, stability, and bifurcations of spiral waves and target patterns through the development of spatial-dynamics methodologies that can be applied to a broad range of reaction-diffusion systems. The investigator will establish multiplicity results for one-dimensional spiral waves and target patterns in reaction-diffusion systems using spatial dynamics and apply these results to the Brusselator model. The second project centers on transverse instabilities of spiral waves and target patterns. Transverse instabilities do not contribute to the spectra of planar structures, though they do cause linear instabilities. The aim is to investigate how these instabilities manifest themselves on large bounded disks. The third project focuses on making the numerical computation of spiral spectra more robust by developing preconditioners that rely on appropriate exponential weights to reduce the norm of the resolvent independently of the size of the domain. In the last project, singular perturbation methods will be developed to understand the limits of profiles and spectra of planar spiral waves and target patterns when some of the diffusion coefficients in the model approach zero.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.