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.

在工程和科学中研究的许多模型都承认某些类型的特解。这些特殊的解，包括行波和其他相干结构，是重要的现象。它们通常对应于在真实的世界中观察到的行为，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之一附近取初始条件，我们就想预测这些解是否会有类似的行为或做一些不同的事情。这种鲁棒性的思想，在数学中被称为稳定性，决定了哪些解决方案在实践中可能被观察到。由于真实的系统是有噪声的，并且具有不确定性，所以人们只可能观察到稳定的特殊解。该研究项目旨在了解和量化一些数学模型的解决方案的稳定性，这些数学模型管理各种现象，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，要么表明，这些解决方案是稳定的，或者，如果他们是不稳定的，通过表征不稳定的流形，从而给一些意义上的方式，附近的解决方案偏离的解决方案的问题。该项目的重点是在数学和科学的不同领域出现的一些模型，包括非线性薛定谔方程，各种浅水模型，用于振荡器同步的仓本模型，以及考虑到神经可塑性和赫布相互作用等现象的仓本模型的各种推广。在找到相干结构如不动点、行波和周期解之后，我们研究了关于相干结构的动力学线性化的谱。我们感兴趣的是建立这些解决方案的渐近或轨道稳定性，或者在它们不稳定的情况下，在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，从而解决稳定性问题。