Dynamics of a rational map: unbounded cycles, unbounded chaotic intervals and organizing centres

A one-dimensional rational map f(x)=(x2−a)/(x2−b) depending on the two parameters a and b is considered. Sequences of bifurcations peculiar of rational maps are evidenced, as those occurring due to unbounded cycles (that is, periodic orbits having one point at infinity, related to the vertical asymptotes) that are superstable, as well as to unbounded chaotic intervals. Moreover, two particular bifurcation points, having the role of organizing centres in the (a,b)-parameter plane, are studied. Each point is related to a pair of conditions, which allow us to consider them as the bifurcation points of codimension-2, as it is usual for this kind of organizing centres. However, the two conditions are related not to bifurcations but to degeneracies in the graph of the function. The sequences of bifurcations leading to attracting cycles associated with these particular points are investigated, analytically and numerically, making use of particular properties of the rational map.