Bifurcations: functional differential equations and waves in inhomogeneous media

分岔：非均匀介质中的泛函微分方程和波

• 批准号: RGPIN-2016-04318
• 负责人: Leblanc, Victor
• 金额: $ 1.31万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676844
• 关键词: Bifurcations; functional; differential; equations; waves

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.**

微分方程是数学模型，它们描述了我们在自然界和日常生活中经历的许多现象。我的研究计划的一部分涉及研究这些类型的方程式，特别注意随着外部参数的变化，这些方程式如何变化。在数学术语中，这称为分叉理论。目的是利用这些知识，以提供对由微分方程建模的物理现象的有益洞察力。我对描述生物组织中电信号传播的微分方程模型特别感兴趣，例如心肌或组成大脑和神经系统的神经元。在这种情况下，溶液描述了在生物学介质中传播的波。这些现象的非常简单的模型假设传播的介质是均匀的（或均质和各向同性）。 但是，现实要复杂得多。瑕疵（例如患病组织）可能导致病理状况，例如心脏组织中的重分波。这是心动过速和心室纤颤的常见原因，可能是致命的。我在此提案中描述的研究计划的一部分将涉及研究不均匀性和/或各向异性对令人兴奋的媒体（例如心肌或神经系统）中波传播的影响。神经系统是这种系统的奇妙例子。在这种情况下，感觉器官对信号的感知，该信号向大脑传播，其治疗和大脑的处理，然后在身体的其他部位之间涉及时间延迟。这些方程式还用于模拟患者中的药物输送，工具的机器chat不休，疾病暴发，疫苗接种策略等。我在这一领域的努力涉及开发分析工具来研究这些方程，目的是阐明这些方程模型的生物或物理系统的行为。在提案中，我描述了一个程序，该程序将使我过去的研究扩展到结构化延迟系统的领域。这些模型经常用于研究人群，并考虑到人口的发展和/或大小的各个阶段，例如少年与成年人。 **该研究计划将继续为知识的发展做出贡献，以及对几位本科，硕士，博士和博士后学生的科学培训，如提案中所述。** **。