Dynamical Systems with a View towards Applications

该项目将扩大现有动力系统数学理论的范围，并将有助于弥合理论和应用之间的差距。动力系统理论位于数学的几个领域的十字路口，在工程学和其他科学学科中有着自然的应用。在这个项目中，首席研究者将相关的动力学结果从有限维情形推广到无限维情形。这些结果包括关于发展偏微分方程所产生的动力半流的结果。这样的方程模拟了各种物理现象。该项目的第二个组成部分是利用首席研究人员在跨学科研究方面的专门知识来确定生物科学中反复出现的主题和自然出现的现象，从而将新的现象学纳入现代动力系统理论。除了这些科学进步，拟议的项目还为学生和博士后提供了充足的培训机会。该项目围绕四个研究方向展开。前两行试图将有限维现象扩展到无限维。在第一个项目中，首席研究员的目标是证明，在随机力存在的情况下，大时间轨道分布的统一描述比目前所知的要普遍得多。第二个项目旨在提取低维结构和嵌入高维的动力学现象。具体地说，PI将旨在表明剪切诱导的混沌是包括Navier-Stokes系统在内的物理模型中不稳定的来源。剩下的两个项目调查了一类与生物学相关的反应网络。我们将研究可扩展网络的大时间行为的平均场方法。该项目还旨在研究“耗尽”这一新概念，这是一种可以进行数学分析的分叉现象。从应用的角度来看，枯竭在几种情况下是自然发生的，并可能产生可怕的生物后果。最后一个项目试图使用可伸缩的反应网络作为模型来回答一个对耗散动力系统至关重要的问题，即从观测的角度来看，哪些不变的量度是可见的？这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project will broaden the reach of the existing mathematical theory of dynamical systems, and will contribute to bridging the gap between theory and application. The theory of dynamical systems lies at the crossroads of several areas of mathematics, and has natural applications to engineering and to other scientific disciplines. In this project, the principal investigator will extend relevant dynamical results from the finite dimensional case to the infinite dimensional case. These include results about dynamical semi-flows generated by evolutionary partial differential equations. Such equations model a variety of physical phenomena. A second component of the project consists in leveraging the principal investigator’s expertise in interdisciplinary research to identify recurrent themes and emergent phenomena arising naturally in the biological sciences, thereby incorporating new phenomenology into a modern theory of dynamical systems. In addition to these scientific advances, the proposed projects offer ample training opportunities for students and postdocs.The project centers on four lines of research. The first two lines seek to extend finite-dimensional phenomena to infinite dimensions. In the first project, the principal investigator aims to show that in the presence of random forces, a unifying description of large-time orbit distribution holds much more generally than is currently known. The second project seeks to extract low dimensional structures and dynamical phenomena embedded in high dimensions. Specifically, the PI will aim to show that shear-induced chaos is a source of instability in physical models including the Navier-Stokes system. The remaining two projects investigate a class of reaction networks of relevance to biology. Mean-field approaches to the large-time behavior of scalable networks will be investigated. The project also aims to study the novel concept of `depletion’, a bifurcation phenomenon amenable to mathematical analysis. From the viewpoint of applications, depletion occurs naturally in several contexts and potentially has dire biological consequences. The final project seeks to use scalable reaction networks as a model to answer a question of fundamental importance for dissipative dynamical systems, namely, which invariant measures are visible from an observational viewpoint?This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.