When two or more simply behaving relaxation oscillators are coupled, new and complex qualitative behavior arises in many cases. This research aims at understanding these effects by exploring the geometry of the phase flow using previously developed analytic insights and estimates. In another direction, the investigator extends the theory of adiabatic invariance to slowly varying ergodic systems by using combinatorial and geometric ideas. When two simple oscillators such as an electric circuit or a biological cell, each behaving in a simple periodic fashion, are coupled, the resulting system can acquire totally new features such as chaotic behavior, bifurcations, etc., of which there seems to have been no hint in the individual components. The aim of this research is to describe and to explain phenomena of this sort in differential equations that model coupled biological, electric, or mechanical systems. In a different direction, the phenomenon of adiabatic invariance is of fundamental importance in the theory of dynamical systems; it arises when a system undergoes a slow change over a long period of time. Somewhat unexpectedly, a system may develop a "memory": some quantities, called the adiabatic invariants, change little over the course of the system's evolution. (The simplest example is a pendulum on a string whose length is, say, halved slowly; in this case the ratio of the energy to the frequency changes very little; this ratio is an adiabatic invariant). Adiabatic invariants arise in the study of elecromagnetically confined plasmas, in particle accelerators, in planetary motion and in many other cases. Although the subject is about 80 years old, there is no satisfactory analysis of this phenomenon for an important class of chaotic systems. This is a serious gap in the subject considering the fact that adiabatic invariance is a basic aspect of Hamiltonian dynamical systems. It is proposed to develop a rigorous theory of the phenomenon of adiabatic invariance for chaotic systems. Such a theory would be of fundamental significance for applications of dynamical systems as well. As an example, it would be a small but important building block in the long-open problem of providing a rigorous foundation for the theory of ideal gases -- modeling a gas by a system of many colliding hard spheres in a vessel of changing size and shape.
