Abstract
The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fαare discussed. We show that for a large class of parameter values the dynamics of fαcan be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uαcorresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uαcorresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fαusing symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.
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