Patterns and Bifurcations in Multiple Timescale Dynamical Systems

该项目旨在进一步从理论上理解具有多个尺度的自然系统的行为，这意味着系统分为在较慢或较快的时间尺度或较短或较长的距离上运行的组件。 尺度分离在自然界中普遍存在，并与各种现象联系在一起;数学生态学和生物学中的重要例子包括水有限生态系统中植被模式的形成，脉冲沿着神经纤维的传播，以及神经元中的周期性爆发节律。 在数学上，这些系统经常被建模的动力系统的形式奇摄动普通，部分，或晶格微分方程。 这个项目有助于奇异摄动理论的发展，通过广泛适用的技术来研究这些现象，在模型参数变化下的鲁棒性，以及它们对扰动的稳定性。 该项目的一部分包括本科生的研究经验的机会。具体的研究目标被组织成三个部分的不同应用领域的启发。 通过这些应用程序的共同点是，它们在概念上描述的奇异摄动微分方程，其中一个微妙的相互作用之间发生的局部奇异分岔现象和全局行为的解决方案。 第一部分研究半干旱区坡地植被条带格局的形成与恢复。 这个过程是由反应-扩散-平流方程模拟，重点是在平流占主导地位的极限模式的存在性和稳定性分析。 第二部分是关于晶格微分方程的周期行波模型脉冲在有髓神经纤维中的传播，其中空间离散的FitzHugh-Nagumo方程是一个典型的例子。 这就需要扩展常微分方程的研究中所使用的技术，其中正常双曲性的损失发生在无限维设置的晶格微分方程。 第三部分是关于神经内分泌细胞模型中爆发解之间的加尖峰跃迁。 这些过渡的建设涉及到双曲和非双曲动力学，并连接本地和全球的解决方案的行为，提供了一个框架，在其中复杂的分叉可以理解在其他系统。 该项目的目标需要扩展现有的理论方法在几何奇异摄动理论，几何desingularization，同宿/异宿分岔理论的领域。 该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The project aims to further the theoretical understanding of the behavior of natural systems that have multiple scales, meaning that the system separates into components which operate on slower or faster timescales or over shorter or longer distances. Scale separation occurs ubiquitously in nature and is tied to a variety of phenomena; important examples from mathematical ecology and biology include the formation of vegetation patterns in water-limited ecosystems, the propagation of impulses along nerve fibers, and periodic bursting rhythms in neurons. Mathematically, these systems are frequently modeled by dynamical systems in the form of singularly perturbed ordinary, partial, or lattice differential equations. This project contributes to the theory of singular perturbations through the development of broadly applicable techniques to study these phenomena, their robustness under variation in model parameters, and their stability to perturbation. Part of the project includes research experience opportunities for undergraduate students.The specific research goals are organized into three parts inspired by the different application areas. The common thread through these applications is that they are conceptually described by singularly perturbed differential equations where a delicate interplay occurs between local singular bifurcation phenomena and the global behavior of solutions. The first part concerns the formation and resilience of vegetation stripe patterns on sloped terrain in semiarid regions. This process is modeled by reaction-diffusion-advection equations, and the focus is on existence and stability analysis of patterns in the limit when the advection dominates. The second part is concerned with periodic traveling waves in lattice differential equations that model impulse propagation in myelinated nerve fibers, of which the spatially discrete FitzHugh-Nagumo equation is a prototypical example. This necessitates the extension of techniques used in the study of ordinary differential equations where loss of normal hyperbolicity occurs to the infinite dimensional setting of lattice differential equations. The third part is concerned with spike-adding transitions between bursting solutions in models of neuroendocrine cells. The construction of these transitions involves accounting for both hyperbolic and nonhyperbolic dynamics and linking local and global behavior of solutions, providing a framework in which complex bifurcations can be understood in other systems. The project goals require extensions to existing theoretical methods in the areas of geometric singular perturbation theory, geometric desingularization, and homoclinic/heteroclinic bifurcation theory. Part of the project includes research experience opportunities for undergraduate students.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.