#

Some Chapters

## Prof.
Robert Gilmore

Physics Department

Email: robert.gilmore@drexel.edu

Below please find some of the papers we are working on that
are currently in porgress.

**[PDF]
Strange attractors are classified by Bounding Tori,
Tsvetelin D. Tsankov and and Robert Gilmore**

####
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