CHAOTIC DYNAMICS OF FLOWS, MAPS, AND SUSPENSIONS

Dynamical systems with chaotic properties have been studied in both discrete and continuous time domains. Both approaches have advantages in demonstrating the mechanisms which generate chaotic motion. In this paper we discuss an example of each type and how they can be connected through smooth suspensions. We focus on the similarity in the underlying dynamics of a class of continuous flows in three-space and a class of iterated mappings in the plane.

In particular, we examine the geometrical similarities between the dynamics of a certain class of suspended Cremona transformations and the dynamics of Rössler’s Band. They both are characterized by Rössler’s ”walking-stick diffeomorphisms”.

As the first step on his ladder to higher chaos, Rössler shows how the walking-stick diffeomorphism can be naturally extended to the three-dimensional folded-towel diffeomorphism. We present a version of the folded-towel diffeomorphism for which the different stages of the transformation can be studied explicitly. With the help of interactive computer graphics, we have analyzed the manifold structure as well as local transformations close to the attractors of these chaotic dynamical systems …