On Tori and Circles

Abstract

In this chapter, as a generalization of the ideas from Section 4.3, we show that if a 1-periodic nonautonomous differential equation is also periodic in x, then it gives rise to a differential equation on a torus (the surface of a doughnut). The dynamics of such equations are explored most conveniently in terms of their Poincaré maps, which happen to be maps on a circle. Accordingly, in the spirit of Chapter 3, we include a brief discussion of such maps and study a landmark example, the standard circle map. Poincaré, in conjunction with his work on classical mechanics, was the first to study vigorously the subject of differential equations on a torus, in particular circle maps. Since his days, a deep analytical theory of circle maps has emerged. The purpose of this chapter is merely to point out a few rudimentary facts and some highlights. We will return to this subject in Part IV and explore several seminal examples from the theory of oscillations and Hamiltonian mechanics, where tori are naturally omnipresent.