Patterns and Bifurcations in Multiple Timescale Dynamical Systems

多时间尺度动力系统中的模式和分岔

• 批准号: 2016216
• 负责人: Paul Carter
• 金额: $ 9.08万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-10-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2016216&HistoricalAwards=false
• 关键词: Patterns; Bifurcations; Multiple; Timescale; Dynamical

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.

该项目的目的是进一步理解具有多个量表的自然系统行为的理论理解，这意味着该系统将其分离成较慢或更快的时间尺度或更短或更长的距离的组件。 比例分离在性质上无处不在，并与多种现象相关。数学生态学和生物学的重要例子包括在水限制的生态系统中形成植被模式，沿神经纤维的冲动的传播以及神经元中的定期爆发节奏。 从数学上讲，这些系统通常以动态系统的形式建模，以奇异的普通，部分或晶格微分方程的形式。 该项目通过开发广泛适用的技术来研究这些现象，模型参数变化的鲁棒性以及它们对扰动的稳定性，从而有助于奇异扰动理论。 该项目的一部分包括针对本科生的研究经验机会。具体的研究目标分为受不同应用领域启发的三个部分。 通过这些应用程序的共同线程是，它们在概念上是通过奇异扰动的微分方程来描述的，在局部奇异的分叉现象与解决方案的全球行为之间发生微妙的相互作用。 第一部分涉及半干旱地区倾斜地形上植被条纹图案的形成和弹性。 该过程是由反应扩散 - 辅助方程建模的，重点是当对流统治时，在极限的存在和稳定性分析上。 第二部分与晶格微分方程中的周期性流动波有关，这些方程模拟了骨髓神经纤维中的脉冲传播模型，其中空间离散的fitzhugh-nagumo方程是一个原型示例。 这需要扩展在普通微分方程的研究中使用的技术，在这种研究中，正常双曲线的丧失发生到晶格微分方程的无限尺寸设置。 第三部分与神经内分泌细胞模型中破裂溶液之间的峰值添加过渡有关。 这些过渡的构建涉及考虑双曲线和非遗传动力学以及将解决方案的局部和全球行为联系起来，提供了一个框架，在该框架中可以在其他系统中理解复杂的分叉。 项目目标需要扩展到几何奇异扰动理论，几何降低和同型/异斜分叉理论领域的现有理论方法。 该项目的一部分包括研究本科生的研究经验机会。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

项目成果

Pulse Replication and Accumulation of Eigenvalues

• DOI: 10.1137/20m1340113
• 发表时间: 2020-05
• 期刊: SIAM J. Math. Anal.
• 作者: P. Carter;J. Rademacher;Bjorn Sandstede

Existence of Transonic Solutions in the Stellar Wind Problem with Viscosity and Heat Conduction

具有粘性和热传导的恒星风问题跨音速解的存在性

• DOI: 10.1137/20m1314240
• 发表时间: 2021
• 期刊: SIAM Journal on Applied Dynamical Systems
• 影响因子: 2.1
• 作者: Bauer, Adam;Carter, Paul
• 通讯作者: Carter, Paul

Spike-Adding Canard Explosion in a Class of Square-Wave Bursters

一类方波爆破中的尖峰添加鸭式爆炸

• DOI: 10.1007/s00332-020-09631-y
• 发表时间: 2020
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: Carter, Paul