Strange Attractors are Classified by Bounding Tori

There is at present a doubly discrete classification for strange attractors of low dimension, $d_L<3$. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with $g$ holes that is dressed by a surface flow with $2(g-1)$ singular points. All known strange attractors in $R^3$ are classified by genus, $g$, and flow type.

Figure 1

Branched manifolds describing different stretch-tear-squeeze mechanisms. (a) Threefold cover of the Rossler system with $C_3$ symmetry; (b) twofold cover; and (c) fourfold cover of the Lorenz system with rotation symmetry about an axis through a focus.

Figure 2

Canonical representation of dressed tori with genus $g\le 6$. For each canonical form we provide a name and the index pair $(g,n)$.

Figure 3

Constraints on global perestroikas. (a) Two branches through different parts of a genus-$g$ torus merge near a branch line. As control parameters vary, one of the branches may fold. (b) Some folding configurations are allowed, (c) while others violate causality.