Branched manifolds for the three types of unimodal maps

Abstract

Branched manifold is certainly the finest description of the structure of chaotic attractors, characterizing how the unstable periodic orbits are knotted. Many chaotic attractors produced by strongly dissipative systems were thus topologically described. In spite of this, the different possibilities for the branched manifolds which may be constructed from unimodal maps were never exhaustively listed. This is the task of the present work, starting from the folded (Logistic) map, the torn (Lorenz) map, and the less known torn away map introduced by Rössler in 1979. The case of the “reverse” horseshoe map is also discussed.