Nonlinear Dynamics in Heterogeneous and Random Systems

异构和随机系统中的非线性动力学

• 批准号: 0072311
• 负责人: Rachel Kuske
• 金额: $ 9.9万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072311&HistoricalAwards=false
• 关键词: Nonlinear; Dynamics; Heterogeneous; Random; Systems

NSF Award Abstract - DMS-0072311Mathematical Sciences: Nonlinear Dynamics in Heterogeneous and Random SystemsAbstract0072311 KuskeThe project covers four areas of research, which all use a combination of asymptotics and computation. The first goal is to understand how small noise alters transitions in nonlinear dynamics, focusing on interface dynamics, transitions in bursters, and oscillations in lasers with delayed feedback. This research also contributes to a broad characterization of the effects of noise in nonlinear systems, useful for effective modeling. The second objective is to predict and understand localized behavior in coupled oscillators, coupled laser arrays, calcium oscillations, and structural dynamics. The strength of the localization can depend on competing effects in a non-uniform way. The third objective is to understand the effects of heterogeneity on the dynamics of patterns in reaction-diffusion problems, using both envelope equation approximations and computations of the full model. The fourth goal is developing new methods for pricing financial derivatives. To price a variety of options, we combine methods for moving boundaries in partial differential equations with singular perturbation methods.The overall goal of the project is to develop asymptotic theory and methods for mathematical models of heterogeneous complex systems. Such methods are often combined with computations in order to completely analyze these models. When including disorder or inhomogeneities in a mathematical model, we try to answer the following questions: Does the heterogeneity cause qualitative changes in the behavior of the system? Is the model highly sensitive to physical variations or computational inaccuracies? Is the model realistic, and can similar models be used as accurate descriptions or predictors for the "real world" system? What are the deficiencies of the model, and how might it be improved? In this project we focus on mathematical models in which heterogeneity plays a significant role, including models of laser dynamics, chemical reactions, calcium concentrations in cells, neuronal dynamics, engineering structures, and financial products.

NSF奖摘要- DMS-0072311数学科学：非均匀随机系统中的非线性动力学 KuskeThe项目涵盖了四个研究领域，它们都使用了渐近和计算的结合。第一个目标是了解小噪声如何改变非线性动力学中的跃迁，重点是界面动力学，猝发器中的跃迁和延迟反馈激光器中的振荡。 这项研究也有助于在非线性系统中的噪声的影响，有效的建模有用的广泛的表征。 第二个目标是预测和理解耦合振荡器，耦合激光阵列，钙振荡和结构动力学中的局部行为。局部化的强度可以以非均匀的方式取决于竞争效应。第三个目标是了解异质性对反应扩散问题中模式动态的影响，使用包络方程近似和完整模型的计算。 第四个目标是开发金融衍生品定价的新方法。为了给各种期权定价，我们将偏微分方程中的移动边界方法与奇异摄动方法相结合。该项目的总体目标是发展非均质复杂系统数学模型的渐近理论和方法。 这些方法通常与计算相结合，以便完整地分析这些模型。 当在数学模型中包含无序或不均匀性时，我们试图回答以下问题：不均匀性是否会导致系统行为的质的变化？ 该模型是否对物理变化或计算误差高度敏感？ 模型是否真实，类似的模型能否用作“真实的世界”系统的准确描述或预测器？ 该模型有哪些不足之处，如何改进？ 在这个项目中，我们专注于数学模型，其中异质性起着重要的作用，包括激光动力学，化学反应，细胞中的钙浓度，神经元动力学，工程结构和金融产品的模型。