We consider three dimensional systems of differential equations
that possess a chaotic attractor.
As an example, we consider the Rossler equations:
In this example, we use 
, 
, and 
.
A single (numerically computed)
trajectory is shown in Figure 4.
This trajectory provides a good approximation to the attractor.
Figure 4: A single trajectory in the Rossler system.
Figure 5 shows (approximate) periodic orbits that were extracted from a single chaotic trajectory. Any two periodic orbits form a knot, and a collection of intertwined periodic orbits is called a braid. One key idea of braid analysis is to use the topological properties of a braid made up of a few periodic orbits to infer the existence of other periodic orbits. We saw that in the one dimensional maps, the existence of a given periodic orbit can imply the existence of others. There is an analogous result for two dimensional maps, which can be applied to a Poincaré section in a three dimensional flow.
Figure 5: Some periodic orbits in the Rossler attractor.
One goal of this analysis is to identify an organizing template. A template is a branched manifold into which the periodic orbits can be placed in a way that preserves their topological structure in the flow. (See Tufillaro [5], Chapter 5, and the references therein.)
These ideas can be used to analyze experimental time series data.
This type of analysis is often called
topological time series analysis.
The steps involved include identifying approximate periodic
orbits in the data, embedding the data in
(by using, for example, time-delay coordinates), and finding the
topological relations among the periodic orbits.
One may then be able to predict the organizing template of the
experimental system.
One example of topological time series analysis is the work of Mindlin, et al [9], and Tufillaro [1], who have applied these methods to experimental data taken from the Belousov-Zhabotinskii reaction.
Another example is the analysis of the variations in the amplitude of a forced vibrating string. An elastic ``string'' (actually a metal wire) is stretched between two fixed points. The wire is subjected to a periodic force by running an alternating current through the wire and placing it in a magnetic field. When the string is forced near its fundamental frequency, the resulting vibrations have a fairly large amplitude. Some string models predict that the amplitude will vary chaotically in certain parameter ranges.
Molteno and Tufillaro [2] conducted experiments in which they observed a sequence of bifurcations and chaotic vibrations. O'Reilly and Holmes [3] have also observed chaotic vibrations. Tufillaro, et al [4] applied the methods of topological times series analysis to measurements of the transverse displacement of the wire at a single point. They extracted periodic orbits from the data, and they were able to identify an organizing template for the periodic orbits. In fact, they found that the dynamics could be described with a one dimensional unimodal map.
A similar result can be seen in the Rossler system.
Figures 4, 6 and
7 show pictorially how
the dynamics in the Rossler attractor can be reduced
(at least approximately) to
a one dimensional unimodal map.
First, we consider a Poincaré cross section 
, as shown in
Figure 4.
A plot of the crossings through
is given in Figure 6.
The set of crossings appears to be one-dimensional.
This leads us to consider the return map of just one variable,
say x.
In Figure 7, we plot 
vs. 
, where
the 
are the x coordinates of the successive crossings of
.
We see that this one dimensional map is unimodal,
with a critical point at 
.
Figure 6: Cross section of the Rossler attractor.
Figure 7: Plot of vs. for the cross section
of the Rossler attractor.
This plots suggests that this map is
well approximated by a one-dimensional unimodal map.
The vertical dotted line indicates the location of the
critical point of the map.