We describe some recent results on product dynamics involving homoclinic and heteroclinic cycles and networks. Previously, in joint work with Peter Ashwin (Exeter), we proved results on products for which at least one factor was a homoclinic attractor. The analysis involved some surprisingly subtle connections with metric number theory. We now have a complete resolution of all the outstanding problems raised in the earlier work. The analysis is surprisingly delicate.

This research is joint with Alexandre Rodrigues (Porto) and Nikita Agarwal (University of Houston).