Richard P. McGehee Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions - each subregion corresponding to maps having a constant number of periodic orbits. This study leads to a simple method for constructing families of maps with bifurcation features such as flames and swallowtails.
Results apply both to circle maps and forced oscillator maps.