Topological aspects of the structure of chaotic attractors in R3

Strange attractors with Lyapunov dimension d(L) <3 can be classified by branched manifolds. They can also be classified by the bounding tori that enclose them. Bounding tori organize branched manifolds (classes of strange attractors) in the same way as the branched manifolds organize the periodic orbits in a strange attractor. We describe how bounding tori are constructed and expressed in a useful canonical form. We present the properties of these canonical forms and show that they can be uniquely coded by analogs of periodic orbits of period g-1, where g is the genus. We describe the structure of the global Poincaré surface of section for an attractor enclosed by a genus- g torus and determine the transition matrix for flows between the g-1 components of the Poincaré surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.