9406022 LevI The goal of this research is to find and to explain new phenomena in mathematical models of physical systems. More specifically, I propose to work in two different directions: (1) Dynamics of coupled relaxation oscillators and (2) Persistence of invariant sets in Hamiltonian systems. The first of these directions aims at understanding what happens when simple oscillators are coupled together. Systems of this kind arise in numerous physical settings, particularly in biology and electric circuits. While an individual oscillator behaves in a simple and predictable way, several such simple "cells" put together can behave in a totally unexpected and mysterious way. I propose to study the "anatomy" of this interaction of simple "cells" which produces complex and rich behavior of the "organism". The second topic, in contrast to the first, deals with Hamiltonian systems which model many situation where energy dissipation can be neglected, such as motion of particles in particle accelerators, geometric optics and celestial mechanics. There are still deep fundamental unanswered questions in the theory of such systems due to their richness and complexity, although remarkable progress has been made in the 1960s (the Kolmogorov-Arnold-Moser theory) and in 1980's (the Aubry-Mather theory). It is proposed to address some of these questions on a particular class of systems (first introduced by Hedlund) which is more tractable than the general case on the one hand, but is sufficiently rich to give new insights and hints on the other. Two research directions are proposed: first, the dynamics of coupled relaxation oscillators, and second, persistence of invariant sets in Hamiltonian systems. In the first of these areas I propose to develop qualitative theory of coupled relaxation oscillators, extending the classical work of Cartwright--Littlewood as well as the more recent work on periodically forced relaxation oscillators. In the last couple of years two int eresting phenomena were discovered in this area, and I hope that there are more to be found. The second of the two directions deals with understanding the behavior of invariant sets in symplectic maps of R^4 under strong perturbations from the integrable case. I plan to consider a special class of systems, namely, geodesic flows on a 3D torus with a particular kind of Riemannian metric introduced by Hedlund. This is sufficiently simple to be tractable on the one hand and yet sufficiently nontrivial to give insight into the general situation.
