Iterations by odd functions with two extrema

Abstract

Let I = [−1, 1] and fI → I be continuous, piecewise monotone and odd with two extrema. A periodic orbit is called symmetric if − x is in the orbit when x is in the orbit. A periodic orbit which is not symmetric is called asymmetric. The first result of this paper proves an ordering of the periods for the symmetric orbits. There are two possibilities depending on how f behaves in a neighbourhood of 0. The second result of this paper proves that for a one-parameter family of odd functions with negative Schwarzian derivative there are three different types of nondegenerate bifurcations: saddle node, period-doubling pitchfork and period-preserving pitchfork. The last type of bifurcation occurs exactly when a symmetric orbit bifurcates to two asymmetric orbits.