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

Abstract

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 $$m-1$$ iterations on the linear side and one iteration on the nonlinear side, where $$m\ge 2$$ 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.