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：异质和随机系统中的非线性动力学Abstract0072311 Kuskethe项目涵盖了四个研究领域，这些领域都使用了渐近学和计算的组合。第一个目标是了解小噪声如何改变非线性动力学中的过渡，重点是界面动力学，毛毛的过渡以及反馈延迟的激光器中的振荡。 这项研究还有助于对非线性系统中噪声的影响的广泛表征，可用于有效建模。 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。本地化的强度可以取决于以不均匀的方式竞争效果。第三个目标是使用包络方程近似和完整模型的计算，了解异质性对反应扩散问题模式动力学的影响。 第四个目标是开发定价金融衍生品的新方法。为了给各种选择定价，我们将偏微分方程中的边界的方法与奇异的扰动方法结合在一起。该项目的总体目标是开发渐近理论和方法的数学模型。 此类方法通常与计算结合在一起，以便完全分析这些模型。 在数学模型中包括混乱或不均匀性时，我们会尝试回答以下问题：异质性是否会导致系统行为的质量变化？ 该模型对物理变化或计算错误高度敏感吗？ 模型是否现实，并且类似模型是否可以用作“现实世界”系统的准确描述或预测指标？ 该模型的缺陷是什么？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。