Dynamics and Stability of Spatially Extended Patterns

该研究项目旨在通过分析螺旋波的特性来进一步了解心律失常，螺旋波是心脏组织中可能导致纤颤的异常电化学波。一个特别的重点将是研究如何强大的螺旋波的行为下空间扰动在特定的模型系统。此外，主要研究者将研究不同机制的作用，这些机制可能导致心脏组织模型中复杂的时空动力学。该项目将包括培训研究生和本科生的研究活动。该项目侧重于反应扩散系统中螺旋波的光谱和线性稳定性。在一个空间维度的早期工作的基础上，我们的目标是阐明空间扰动的螺旋波的核心和远场的时空响应。主要的方法将涉及使用空间动力学和时空绿色函数的螺旋波的建设，推导估计的预解核。该项目的第二部分将侧重于使用稳态近似和几何奇异摄动理论的FitzHugh-Nagumo模型中的回火和脉冲分裂的分析方法。

This research project aims to further our understanding of cardiac arrhythmias by analysing the properties of spiral waves, which are anomalous electrochemical waves in cardiac tissue that may lead to fibrillation. A particular focus will be on studying how robust spiral waves behave under spatial perturbations in specific model systems. In addition, the principal investigator will study the role of different mechanisms that can lead to complex spatio-temporal dynamics in models of cardiac tissue. The project will include research activities that will train both graduate and undergraduate students.The project focuses on the spectral and linear stability of spiral waves in reaction-diffusion systems. Building on earlier work in one spatial dimension, the goal is to elucidate spatio-temporal responses to spatial perturbations of spiral waves in the core and far field. The main approach will involve deriving estimates of the resolvent kernel using spatial dynamics and the construction of the spatio-temporal Green's function about a spiral wave. The second part of the project will focus on an analytical approach to backfiring and pulse-splitting in the FitzHugh-Nagumo model using steady-state approximations and geometric singular perturbation theory.