On the structure of attractors for discrete, periodically forced systems with applications to population models

This work discusses the effects of periodic forcing on attracting cycles and more complicated attractors for autonomous systems of nonlinear difference equations. Results indicate that an attractor for a periodically forced dynamical system may inherit structure from an attractor of the autonomous (unforced) system and also from the periodicity of the forcing. In particular, a method is presented which shows that if the amplitude of the k-periodic forcing is small enough then the attractor for the forced system is the union of k homeomorphic subsets. Examples from population biology and genetics indicate that each subset is also homeomorphic to the attractor of the original autonomous dynamical system.