So far we’ve concentrated on the equation x f x ( ), which we visualized as a vector field on the line. Now it’s time to consider a new kind of differential equation and its corresponding phase space. This equation,

R R f ( ),

corresponds to a vector field on the circle. Here R is a point on the circle and R is the velocity vector at that point, determined by the rule R R f ( ). Like the line, the circle is one-dimensional, but it has an important new property: by flowing in one direction, a particle can eventually return to its starting place (Figure 4.0.1). Thus periodic solutions become possible for the first time in this book! To put it another

way, vector fields on the circle provide the most basic model of systems that can oscillate.