Chaotic dynamics of a piecewise cubic map

An analytical study of the dynamics of a piecewise cubic map depending on two parameters is carried out in this paper. For this type of map, it is shown that a stable fixed point coexists with a chaotic attractor. The way in which the deterministic dynamics of the map undergoes chaos is not representable by means of the standardly proposed routes to chaos. It appears that in this case the chaos is initiated by the appearance of an unstable solution born out of a tangent bifurcation. The general geometric approach presented here in obtaining the analytical results does not make use of the Schwarzian curvature of the map. Moreover, it is shown that only the first few iterates essentially describe the global dynamics of the map. The methodology presented here could be used to good advantage for understanding the dynamics of other types of maps as well. Computational results are provided by the cell-mapping method to expose and confirm the analytical results given in this paper.