Nonlinear Dynamics with Applications to Physical Systems

The proposed research aims at discovering new phenomena of fundamental mathematical interest as well as physical relevance, and at explaining them in the simplest possible way. The three main proposed projects are (i) study of the robustness or fragility of resonances arising in many physical applications, e.g. in solid state physics, (ii) explaining the geometrical mechanism of the recently discovered ponderomotive magnetism (observed in many physical experiments) and developing a connection between averaging theory and differential geometry – two seemingly unrelated fields, and (iii) exploiting the recently discovered connection between Hill's equation on the one hand and the tire track problem on the other, thus unifying two seemingly unrelated areas. Some results of this project are likely to give a new and simpler understanding of some phenomena of fundamental interest, and may make their way into textbooks. The project consists of three parts, unified by the desire to discover and understand new phenomena in dynamics. Some years ago J. B. Keller and V. Arnold discovered and analyzed a fundamental feature of resonances in two different classes of problems: circle maps and Mathieu-type equations. The goal of the first part of the project is to add a yet one more class to the two studied by Arnold and Keller, namely the area-preserving cylinder maps. Besides of its basic interest, the latter example comes up in solid-state physics among many other settings. The second part of the project aims at understanding the somewhat mysterious mechanism of ponderomotive magnetism, and also aims to explore a connection between two fundamental objects, one from mechanics (the gyroscopic effect) and the other from differential geometry (Jacobi fields). The third project proposes to exploit the recently found connection between two seemingly unrelated objects: (i) Hill's equation, studied extensively over a couple of centuries, and (ii) tire tracks, a more recently studied object. It is hoped that the results from one area will give new insights into the other. 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.