Collaborative Research: Computing Dynamics of Multiparameter Systems

合作研究：多参数系统的计算动力学

• 批准号: 0914995
• 负责人: William Kalies
• 金额: $ 25.22万
• 依托单位: Florida Atlantic University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0914995&HistoricalAwards=false
• 关键词: Collaborative; Research; Computing; Dynamics; Multiparameter

The language and ideas of dynamical systems theory that have been developed over the last century have become ubiquitous in the applied sciences. While the analytic language of differential equations and maps is still the basis for most quantitative descriptions of scientific ideas, current scientific results are often obtained based on models which are not derived from first principles, for which many of the essential parameters have not been measured, and which often involve stochastic terms. The key objective of this project is to develop scalable computational techniques to provide correct robust information about global dynamics over large ranges of parameter values. Bifurcation theory implies that the cost for robustness is a coarse description. However, the fact that scientists and engineers are using numerical simulations of phenomenologically derived models to further their understanding of dynamic processes indicates that the information these techniques provide must be both quantitative and qualitative. Since the study of systems over broad ranges of possible parameter values produces considerable information, to be of practical use these methods must organize this information in an efficient, queriable manner. We expect the work proposed in this project will produce (1) reliable computational tools for global decompositions of dynamical systems by constructing a database in which the global dynamics is encoded in combinatorial and algebraic structures and (2) efficient methods for querying the database to identify dynamical structures and bifurcations of interest. This work will address the fundamental question of determining global decompositions of dynamical systems over varying parameters. The global dynamics is stored in the form of a database based on calculations for deterministic systems, but within the framework of these computations we will also explore how to predict the effects of noise on the observable dynamical behavior. These computational techniques will be tested on and applied to a variety of problems from mathematical biology. The biological models which will be considered are used to address central questions in biology including the role of the spatial environment in ecology and evolution and the robustness of the dynamics of signal transduction/gene regulatory networks. These activities will produce computational tools for global decompositions of dynamical systems, which will be made available to scientists and engineers for potential applications in a wide variety of disciplines.

在上个世纪发展起来的动力系统理论的语言和思想在应用科学中已经变得无处不在。虽然微分方程和地图的分析语言仍然是大多数科学思想的定量描述的基础，但当前的科学结果通常是基于不是从第一原理导出的模型获得的，其中许多基本参数尚未测量，并且通常涉及随机项。这个项目的主要目标是开发可扩展的计算技术，以提供正确的强大的全球动态信息在大范围的参数值。分叉理论意味着鲁棒性的代价是一个粗略的描述。然而，事实上，科学家和工程师正在使用数值模拟的现象学衍生模型，以进一步了解动态过程表明，这些技术提供的信息必须是定量和定性的。由于在广泛的范围内的可能的参数值的系统的研究产生了相当多的信息，实际使用这些方法必须组织这些信息在一个有效的，queriable的方式。我们希望在这个项目中提出的工作将产生（1）可靠的计算工具，通过构建一个数据库，其中的全球动态编码在组合和代数结构的动力系统的全局分解和（2）查询数据库，以确定感兴趣的动力结构和分叉的有效方法。这项工作将解决的基本问题，确定全球分解动力系统在不同的参数。全局动力学以基于确定性系统计算的数据库的形式存储，但在这些计算的框架内，我们还将探索如何预测噪声对可观察动力学行为的影响。这些计算技术将在数学生物学的各种问题上进行测试和应用。将被考虑的生物模型用于解决生物学中的核心问题，包括空间环境在生态学和进化中的作用以及信号转导/基因调控网络动态的鲁棒性。这些活动将产生用于动力系统全球分解的计算工具，供科学家和工程师在各种学科中应用。