Chapter 4: The Architecture of Chaotic Attractors

With the studies of Lorenz, Ruelle and Ueda it began to be appreciated that a set of simple differential equations could produce chaotic behavior: at the same time this type of behavior began to be discovered in several very simple systems starting to the late 1970s. However, in order to become a real scientific discipline, it was necessary to identify the invariant structures underlying chaotic attractors. For dissipative systems there were several ideas that allowed a discussion of the architecture of chaotic behavior. To identify them, it was useful to study a system as simple as possible in order to identify in detail the ingredients that were necessary to produce chaotic behavior. This chapter is somewhat more technical than the previous chapters. Even read rapidly, it will give an idea of the concepts that are used to characterize chaos. The reader will note, however, that (s)he can read most of this book without mastering the contents of this chapter.