Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: unbounded chaotic sets

In this work we consider a class of generalized piecewise smooth maps, proposed in the study of engineering models. It is a class of one-dimensional discontinuous maps, with a linear branch and a nonlinear one, characterized by a power function with a term and a vertical asymptote. The bifurcation structures occurring in the family of maps are classified according to the invertibility or non-invertibility of the map, depending on the parameters characterizing the two branches. When the map is non-invertible we prove the persistence of chaos. In particular, the existence of robust unbounded chaotic attractors. The parameter space is characterized by intermingled regions of attracting cycles born by smooth fold bifurcations, issuing from codimension-two bifurcation points. The main result is related to the description of the relationship between two types of bifurcations, smooth fold bifurcations and border collision bifurcations (BCBs). We describe the particular role of codimension-two bifurcation points associated with these bifurcations related to cycles with the same symbolic sequences. We show that they exist related to the border collision of any admissible cycle. We prove that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs. We prove that in the considered range all the unstable cycles are always homoclinic, and that an unbounded chaotic set always exists, either in an invariant set of zero measure or of full measure.