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，这意味着该系统分为几个部分，这些部分在较慢或较快的时间尺度上或在较短或较长距离上运行。尺度分离在自然界中无处不在，并与各种现象有关；数学生态学和生物学的重要例子包括缺水生态系统中植被模式的形成，脉冲沿神经纤维的传播，以及神经元的周期性爆发节律。在数学上，这些系统经常被用奇异摄动的常微分方程组、偏微分方程组或格点微分方程组的形式的动力系统来建模。这个项目通过发展广泛适用的技术来研究奇异摄动理论，研究这些现象，它们在模型参数变化下的稳健性，以及它们对扰动的稳定性。该项目的一部分包括为本科生提供研究经验的机会。根据不同的应用领域，具体的研究目标被组织为三个部分。贯穿这些应用的共同线索是，它们在概念上由奇异摄动的微分方程描述，其中局部奇异分叉现象和解的整体行为之间存在微妙的相互作用。第一部分是关于半干旱地区坡面上植被条纹格局的形成和弹性。用反应-扩散-平流方程模拟这一过程，重点研究了平流占优势时极限模式的存在性和稳定性分析。第二部分是关于格点微分方程中的周期行波，它模拟了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。这就需要推广常微分方程式的研究方法，当格点微分方程法向双曲性丧失时，无穷维格点微分方程组就失去了正常双曲性。第三部分是关于神经内分泌细胞模型中爆裂液之间的尖峰增加转换。这些转变的构建包括考虑双曲和非双曲动力学，并将解的局部和全局行为联系起来，提供了一个框架，在其中复杂的分支可以在其他系统中被理解。项目目标要求在几何奇异摄动理论、几何去单角化和同宿/异宿分叉理论领域扩展现有的理论方法。该项目的一部分包括为本科生提供研究经验的机会。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.