Stability, Instability and Geometry in Applied Spectral Problems.

应用光谱问题中的稳定性、不稳定性和几何。

• 批准号: 1615418
• 负责人: Jared Bronski
• 金额: $ 28.99万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615418&HistoricalAwards=false
• 关键词: Stability; Instability; Geometry; Applied; Spectral

Many models studied in engineering and science admit some type of special solutions. These special solutions, including traveling waves and other coherent structures, are important phenomenon. They often correspond to behaviors observed in the real world, and as such it is important to understand them. One important question is that of robustness: if one takes initial conditions near one of these special solutions, one would like to predict if the solutions would have a similar behavior or do something different. This idea of robustness, known in mathematics as stability, governs which solutions are likely to be observed in practice. Since real systems are noisy and have uncertainty, one is only likely to observe special solutions that are stable. This research project is aimed at understanding and quantifying the stability of solutions to a number of mathematical models governing diverse phenomena, including the behavior of a large-scale electrical network and wave propagation in a channel. Results will include proving theorems that either show that these solutions are stable or, if they are unstable, by characterizing the unstable manifold, thereby giving some sense of the manner in which nearby solutions diverge from the solution in question. This project focuses on a number of models that arise in different areas of mathematics and the sciences, including the nonlinear Schrodinger equation, various shallow water models, the Kuramoto model for synchronization of oscillators, and various generalizations of the Kuramoto model that take into account phenomena such as neural plasticity and Hebbian interactions. After finding coherent structures such as fixed points, traveling waves, and periodic solutions, we study the spectrum of the linearization of the dynamics about the coherent structure. We are interested in establishing asymptotic or orbital stability of these solutions or, in the case where they are not stable, in counting the dimension of the unstable manifold. We approach the question on stability using geometric and topological arguments to count the dimension of the unstable manifold.

工程和科学领域研究的许多模型都承认某些类型的特殊解决方案。这些特殊的解决方案，包括行波和其他相干结构，是重要的现象。它们通常与现实世界中观察到的行为相对应，因此理解它们很重要。一个重要的问题是鲁棒性：如果我们在这些特殊解决方案之一附近采用初始条件，我们会想要预测这些解决方案是否会具有类似的行为或做不同的事情。这种鲁棒性的概念在数学中称为稳定性，它决定了在实践中可能观察到的解决方案。由于真实系统充满噪声且具有不确定性，因此人们只可能观察到稳定的特殊解决方案。该研究项目旨在理解和量化管理不同现象的许多数学模型解的稳定性，包括大规模电网的行为和通道中的波传播。结果将包括证明定理，这些定理要么表明这些解是稳定的，要么如果它们不稳定，则通过表征不稳定流形，从而给出附近解与所讨论解的分歧方式的某种意义。该项目重点研究数学和科学不同领域中出现的许多模型，包括非线性薛定谔方程、各种浅水模型、用于振荡器同步的Kuramoto模型以及考虑神经可塑性和赫布相互作用等现象的Kuramoto模型的各种推广。在找到不动点、行波和周期解等相干结构后，我们研究了相干结构动力学线性化的谱。我们感兴趣的是建立这些解的渐近或轨道稳定性，或者在它们不稳定的情况下，计算不稳定流形的维数。我们使用几何和拓扑参数来计算不稳定流形的维数来解决稳定性问题。