Tongues of periodicity in a family of two-dimensional discontinuous maps of real Möbius type

In this paper we consider a two-dimensional piecewise-smooth discontinuous map representing the so-called “relative dynamics” of an Hicksian business cycle model. The main features of the dynamics occur in the parameter region in which no fixed points at finite distance exist, but we may have attracting cycles of any periods. The bifurcations associated with the periodicity tongues of the map are studied making use of the first-return map on a suitable segment of the phase plane. The bifurcation curves bounding the periodicity tongues in the parameter plane are related with saddle-node and border-collision bifurcations of the first-return map. Moreover, the particular “sausages structure” of the bifurcation tongues is also explained.