Some Chapters

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.

[PDF] Topological aspects of the structure of chaotic attractors in R^3, Tsvetelin D. Tsankov and Robert Gilmore

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 that 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 or period g-1, where g is the genus. We descreibe the structure of the global Poincare surface of section for an attractor enclosed by a genus-g bounding torus and determine the transition matrix for flows between the g-1 components of the Poincare surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.

[PDF] Entropy of Bounding Tori, J. Katriel and R. Gilmore

Branched manifolds that describe strange attractors in R^3 can be enclosed in, and are organized by, bounding tori. Tori of genus g are labeled by a symbol sequence, or ``periodic orbit'', of period g-1. We show that the number of distinct canonical bounding tori grow exponentially like N(g) \simeq e^{\lambda (g-1) }, with e^{\lambda}=3, so that the ``bounding tori entropy'' is log(3).

[PDF] Embeddings of a strange attractor into R^3, Tsvetelin D. Tsankov, Arunasri Nishtala, and Robert Gilmore,

The algorithm for determining a global Poincare surface of section is applied to a previously studied dynamical system on R^2 \times S^1 and a one-parameter family of embeddings of the strange attractor it generates in R^3. We find that the topological properties of the attractor are embedding-dependent to a limited extent. These embeddings rigidly preserve mechanism, which is a simple stretch-and-fold. The embeddings studied show three discrete topoloogical degrees of freedom: parity; global torsion; and braid type of the genus-one torus bounding the embedded attractor.

[PDF] Distinguishing between folding and tearing mechanisms in strange attractors, Greg Byrne, Robert Gilmore, and Christophe Letellier

We establish conditions for distinguishing between two topologically identical strange attractors that are enclosed by identical bounding tori, one of which is generated by a flow restricted to that torus, the other of which is generated by a flow in a different bounding torus and either imaged or lifted into the first bounding torus.

[PDF] Large-scale structural reorganization of Strange Attractors, T. D. Tsankov, C. Letellier, G. Byrne, and Robert Gilmore

Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems.

[PDF] All the Covers of the Horseshoe, Robert Gilmore, Arunasri Nishtala, and T. D. Tsankov

The Lorenz attractor can be mapped to a Rossler-like strange attractor by a local 2 \rightarrow 1 mapping. Conversaely, the Reossler attractor can be lifted to a Lorenz-like strange attractor by the inverse map. It can also be lifted to many other inequivalent covering strange attractors. We classify all possible lifts of Rossler-like strange attractors exploiting an analogy between dynamical systems theory and Lie group theory.

Last update: May 18, 2005