Bifurcation structures in a family of one-dimensional linear-power discontinuous maps

In this work we consider a class of generalized piecewise smooth maps, proposed in the study of applied 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 x and a vertical asymptote. The bifurcation structures occurring in the family of maps are classi...ed according to the invertibility or non-invertibility of the map, depending on two parameters characterizing the two branches, together with the value of the power in the nonlinear term, that is 0 < < 1; = 1 and > 1. When the map is noninvertible we prove the persistence of chaos. In particular, the existence of robust unbounded chaotic attractors is proved. The parameter space is characterized by intermingled regions of possible stable cycles born by smooth fold bifurcations, issuing from codimension-two bifurcation points. The main result is related to the description of the bifurcation structure which involves both types of bifurcations: smooth fold bifurcations and border collision bifurcations. The particular role of codimension-two bifurcation points associated with the interaction between border collision bifurcation and smooth bifurcations is described, in two di¤erent cases: (i) when related to cycles with di¤erent symbolic sequence, in which case they act as organizing centers and are issuing points of in...nitely many families of bifurcation curves, both of fold type and of border collision type; (ii) when related to cycles with the same symbolic sequence, and are limit sets of in...nite families of border collision bifurcation curves. Moreover, the transition invertible/noninvertible is commented.