Examples of attracting sets of birkhoff type

We discuss an explicit example of a map of the plane R ² with a nontrivial attracting set. In particular, we are concerned with the concept of rotation number introduced by Poincaré for maps of the circle and its subsequent extension by Birkhoff to maps of the annulus. The use of rotation number allows nontrivial attractors to be distinguished. The map we discuss has an attracting set containing a set of orbits with infinitely many different rotation numbers. We obtain the map by considering an Euler iteration of a family of vector fields originally described by Arnold and find that the resulting Euler map undergoes some bifurcations which are analogous to those of the family of vector fields. Specifically, there are Hopf bifurcations where changes of stability of a fixed point result in the creation of an attracting circle. The circle which grows from the fixed point is then shown to undergo structural changes giving nontrivial attracting sets. This arises from Euler map behaviour for which the corresponding vector field behaviour is a heteroclinic saddle connection. It is possible to give an explicit trapping region for the Euler map which contains the attracting set and to describe some of its properties. Finally, an analogy is drawn with attracting sets which arise for forced oscillators.