Bifurcations: functional differential equations and waves in inhomogeneous media

微分方程是描述我们在自然界和日常生活中所经历的许多现象的数学模型。我的研究计划的一部分是研究这些类型的方程，特别注意这些方程的解如何随着外部参数的变化而变化。在数学术语中，这称为分叉理论。我们的目标是利用这些知识来提供对微分方程式所模拟的物理现象有帮助的洞察力。我对描述电信号在生物组织中传播的微分方程式模型特别感兴趣，生物组织如心肌或组成大脑和神经系统的神经元。在这种情况下，解决方案描述的是在生物介质中传播的波。对于这些现象，非常简单的模型假定传播介质是均匀的(或均匀的和各向同性的)。然而，现实要复杂得多。缺陷(如病变组织)可导致病理情况，如心脏组织中的折返波。这是心动过速和室颤的常见原因，这些情况可能是致命的。我在这份提案中描述的研究计划的一部分将涉及研究非均质性和/或各向异性对波在可兴奋介质(如心肌或神经系统)中传播的影响。**与上述计划密切相关的是，我建议继续研究一类特殊的微分方程，称为延迟-微分方程，它经常被用作存在时间延迟的生物系统的模型。神经系统就是这种系统的一个很好的例子。在这种情况下，感觉器官对信号的感知、将信号传输到大脑、大脑对其进行处理和处理，然后到达身体的其他部位之间存在时间延迟。这些方程也被用来模拟患者的药物输送、工具的机器振动、疾病爆发、疫苗接种策略等。我在这一领域的努力包括开发分析工具来研究这些方程，目的是揭示由这些方程模拟的生物或物理系统的行为。在提案中，我描述了一个计划，该计划将把我过去的研究扩展到结构化时滞系统领域。这些模型经常用于研究人口，并考虑到人口发展的不同阶段和/或人口的大小，例如青少年与成人。**这项研究计划将继续(像过去一样)促进知识的进步，并按照提案中的描述，为几名本科生、硕士、博士和博士后的科学培训做出贡献。**

Differential equations are mathematical models that describe many of the phenomena we experience in nature and in our every day lives. Part of my research program involves studying these types of equations, with particular attention to how the solutions of these equations change as external parameters are varied. In mathematical terminology, this is called bifurcation theory. The goal is to use this knowledge in order to provide helpful insight into the physical phenomenon that is being modeled by the differential equations. I am particularly interested in differential equations models which describe the propagation of electrical signals in biological tissue, such as the heart muscle or the neurons that compose the brain and nervous system. In this case, solutions describe waves which propagate in the biological medium. Very simple models for these phenomena suppose that the medium of propagation is uniform (or homogeneous and isotropic). However, reality is much more complicated than that. Imperfections (such as diseased tissue) can lead to pathological conditions, such as re-entrant waves in cardiac tissue. This is a common cause of tachycardia and ventricular fibrillations, conditions which can be fatal. Part of my research program described in this proposal will involve studying the effects of inhomogeneities and/or anisotropy on the propagation of waves in excitable media such as the heart muscle or nervous system.**Closely related to the program described above, I propose to continue studying a special class of differential equations, called delay-differential equations, which are frequently used as models for biological systems in which time-delays are present. The nervous system is a marvelous example of such a system. In this case, there is a time delay involved between the perception of a signal by the sensory organs, transmission of this signal to the brain, its treatment and processing by the brain, and then on to other parts of the body. These equations are also used to model drug delivery in patients, machine chattering of tools, disease outbreaks, vaccination strategies, etc. My efforts in this area have involved developing analytical tools to study these equations, with the goal of shedding light on the behavior of the biological or physical system being modeled by these equations. In the proposal, I describe a program which would extend my past research into the area of structured delay systems. These are models that are frequently used to study populations, and take into account the various stages of development and/or sizes of the population, e.g. juveniles vs adults. **This research program will continue to contribute (as it has in the past) to the advancement of knowledge, and to the scientific training of several undergraduate, masters, doctoral and post-doctoral students, as is described in the proposal.**