Period-adding and the Farey tree structure in a class of one-dimensional discontinuous nonlinear maps

In this paper, we consider the dynamics of a class of one-dimensional discontinuous nonlinear maps which is linear on one side of the phase space and nonlinear on the other side. The period-adding phenomena of the map are investigated by studying the period-m orbit which has iterations on the linear side and one iteration on the nonlinear side, where is an integer. Analytical conditions for the existence and stability of such kind of periodic orbits are found. Our numerical simulations suggest that the stable period-m orbit of this type plays important roles in the dynamics of the map. As the bifurcation parameter varies, such kind of periodic orbits form finite or infinite period-adding sequences. Between those "primary" periodic orbits, there are alternative series of other periodic orbits organized by the Farey tree structure or chaotic orbits.