Collaborative proposal: Computing Dynamics of Multiparameter Systems

在上个世纪发展起来的动力系统理论的语言和思想已经在应用科学中变得无处不在。虽然微分方程式和地图的分析语言仍然是大多数科学思想定量描述的基础，但目前的科学结果往往是基于不是从第一原理推导出来的模型获得的，其中许多基本参数还没有测量过，而且往往涉及随机项。该项目的主要目标是开发可扩展的计算技术，以在大范围的参数值范围内提供关于全球动力学的正确而稳健的信息。分歧理论表明，健壮性的代价是一个粗略的描述。然而，科学家和工程师正在使用唯象派生模型的数值模拟来加深他们对动态过程的理解，这一事实表明，这些技术提供的信息必须既是定量的，也是定性的。由于在可能参数值的大范围内对系统进行研究会产生大量信息，因此这些方法必须以有效的、可查询的方式组织这些信息，才有实际用途。我们期望这个项目中提出的工作将产生(1)可靠的计算工具，通过构建一个全球动力学以组合和代数结构编码的数据库，以及(2)查询数据库以识别感兴趣的动态结构和分叉的有效方法。这项工作将解决确定动态系统在变化参数上的全局分解的基本问题。全球动力学以数据库的形式存储，基于确定性系统的计算，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到的动力学行为的影响。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。将被考虑的生物学模型被用来解决生物学中的中心问题，包括空间环境在生态和进化中的作用以及信号转导/基因调控网络动力学的稳健性。这些活动将产生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潜在应用。

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.