Dynamical Systems with a View towards Applications

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.