Border Collision and Smooth Bifurcations in a Family of Linear-Power Maps

In this work we describe some properties and bifurcations which occur in a family of linear-power maps typical in Nordmark' systems. The continuous case has been investigated by many authors since a few years, while the discontinuous case has been considered only recently. In particular, having a vertical asymptote, it gives rise to new kinds of bifurcations. Organizing centers related to codimension-two bifurcation points, due to the intersection of a border collision bifurcation and a smooth fold bifurcation of cycles having a different symbolic sequence are evidenced. It is shown the relevant role played by a codimension-two point existing on any border collision bifurcation curve, and related to the smooth fold bifurcation of cycles with the same symbolic sequence. We recall some of the properties proved up to now, evidencing the rich structure which is still to be understood.