Collaborative proposal: Computing Dynamics of Multiparameter Systems

动力系统理论的语言和思想在上个世纪发展起来，在应用科学中无处不在。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性原理推导出来的模型，其中许多基本参数尚未测量，并且经常涉及随机项。该项目的关键目标是开发可扩展的计算技术，以提供关于大范围参数值的全球动力学的正确可靠信息。分岔理论认为鲁棒性代价是一个粗糙的描述。然而，事实上，科学家和工程师正在使用现象学衍生模型的数值模拟来进一步理解动态过程，这表明这些技术提供的信息必须是定量和定性的。由于在广泛的可能参数值范围内研究系统会产生大量的信息，为了实际应用，这些方法必须以一种有效的、可查询的方式组织这些信息。我们期望在这个项目中提出的工作将产生(1)通过构建一个以组合和代数结构编码的全局动力学的数据库，为动态系统的全局分解提供可靠的计算工具；(2)查询数据库以识别动态结构和感兴趣的分支的有效方法。这项工作将解决在不同参数上确定动力系统全局分解的基本问题。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的动力学行为的影响。这些计算技术将被测试并应用于数学生物学的各种问题。将考虑的生物学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/基因调控网络的动态稳健性。这些活动将产生用于动力系统全局分解的计算工具，这些工具将提供给科学家和工程师，用于各种学科的潜在应用。

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.