Study of a one-dimensional map with multiple basins

The cubic iterative equation x/sub n/+1 = ax/sub n//sup 3/+(1-a)x/sub n/ has two critical points, and in the periodic regime it displays a dependence on the initial condition. This dependence results from the presence of two critical points and leads to two attractors and a ''split bifurcation'' not found in maps with one critical point. We determine the sequence and patterns of the periodic orbits; these differ from those observed for maps with a single critical point. We note that a conjugacy principle divides the periodic windows into two distinct categories. Also, we observe a correlation between crises of the attractors and the locations of unstable orbits.