Nonlinear Dynamics in Heterogeneous and Random Systems

数学科学：异构和随机系统中的非线性动力学[摘要]该项目涵盖四个研究领域，均采用渐近与计算相结合的方法。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，重点关注界面动力学，爆裂器中的跃迁以及延迟反馈激光器中的振荡。该研究还有助于对非线性系统中噪声影响的广泛表征，有助于有效建模。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。局部化的强度可能以一种不均匀的方式依赖于竞争效应。第三个目标是利用包络方程近似和完整模型的计算，了解非均质性对反应扩散问题中模式动力学的影响。第四个目标是开发金融衍生品定价的新方法。为了给各种选项定价，我们将偏微分方程的移动边界方法与奇异摄动方法结合起来。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，以便全面分析这些模型。当在数学模型中包含无序或非均匀性时，我们试图回答以下问题：异质性是否会导致系统行为的质变？模型是否对物理变化或计算不准确高度敏感？该模型是否真实，类似的模型是否可以用作“真实世界”系统的准确描述或预测器？该模型有哪些不足之处，该如何改进？在这个项目中，我们专注于非均质性发挥重要作用的数学模型，包括激光动力学、化学反应、细胞内钙浓度、神经元动力学、工程结构和金融产品的模型。

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.