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与应用之间的差距。动力系统理论处于几个数学领域的交叉点，在工程和其他科学学科中具有天然的应用。在这个项目中，首席研究员将把相关的动力学结果从有限维的情况推广到无限维的情况。这些结果包括由演化偏微分方程生成的动态半流的结果。这样的方程可以模拟各种物理现象。该项目的第二个组成部分是利用首席研究员在跨学科研究方面的专业知识来识别生物科学中自然出现的反复出现的主题和新兴现象，从而将新的现象学纳入现代动力系统理论。除了这些科学进步之外，拟议的项目还为学生和博士后提供了充足的培训机会。该项目以四条研究线为中心。前两条线试图将有限维现象扩展到无限维。在第一个项目中，首席研究员旨在证明，在随机力存在的情况下，对大时间轨道分布的统一描述比目前已知的要普遍得多。第二个项目试图提取低维结构和嵌入高维的动态现象。具体来说，PI将旨在表明剪切引起的混沌是包括纳维-斯托克斯系统在内的物理模型不稳定的来源。剩下的两个项目研究一类与生物学相关的反应网络。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“耗竭”的新概念，这是一种适合数学分析的分岔现象。从应用的角度来看，耗竭在几种情况下自然发生，并可能产生可怕的生物学后果。最后一个项目试图使用可扩展的反应网络作为模型来回答耗散动力系统的一个基本重要问题，即，从观察的角度来看，哪些不变措施是可见的？该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.