Just what do we mean when we say that that chaos is
indistinguishable from the output of a random process? To answer this question, we
consider the time evolution of a large number of nearby initial conditions in
the neighborhood of a chaotic attractor. The animation at the right
(Click HERE to Stop.)
illustrates the result in the case of the attractor discovered by Lorenz.
^{(1)}
At first, the points - there are one hundred thousand in the yellow
square - travel together. But with the passage of time, they are stretched out,
split apart and folded back on each other. After repeated episodes of
stretching and folding, the points are distributed over the entire attractor
so that the most one can say is that there is a probability distribution
which determines the likelihood that a trajectory will be in a given
neighborhood of the attractor at any particular time.
^{(2)}
This distribution, or "measure," is **invariant** in the sense that it
maps to itself under the action of the differential equations. Hence, one
often finds reference to an attractor's *invariant measure* in the
literature.
^{(3)}
In other words, the long-term state of the system can be predicted, but only
statistically, and we have the answer to our question.

Of course, the rate at which the invariant measure is approached is system-
and often times parameter-dependent. In the case of Lorenz' equations,
relaxation to the invariant measure is accelerated by the existence of a saddle
equilibirum of index 1 situated at the origin between the "wings" of the butterfly.
^{(4)}
Trajectories in the vicinity of the saddle are attracted to it in the direction
of by two-dimensional stable manifold, only to be shot off toward one
wing or the other along one of the branches of the one-dimensional unstable
manifold.

In fact, the manifolds wind about each other in an extraordinarily
complicated fashion forming what is called a *homoclinic tangle*.
^{(5)}
Among other things, the existence of the tangle implies chaos in a mathematical
sense, as well as the attractor's fractal geometry.
^{(6)}
More generally, there is every reason to believe that the attractor is simply
the closure of the unstable manifold of the saddle.

1. Lorenz, E. N. 1963. Deterministic nonperiodic flow.
*J. Atm. Phys.* **357**:130-141. The Lorenz equations are a
"three mode" approximation of the partial differential equation describing
convective flow in a fluid heated from below. The equations themselves may
be written as follows:

In the present case, the parameters, a, b and c, are asigned the following values:dx/dt = a(y-x)

dy/dt = bx - y - xz

dz/dt = xy - cz

a = 10.0;

b = 28.0;

c = 2.66667.

For a thorough and enjoyable review of the Lorenz equations and their
properties, see Sparrow, C. T. 1982. *The Lorenz Equations: Bifurcations,
Chaos and Strange Attractors.* Springer-Verlag, New York.

2. For pedagogic purposes, the initial points were distributed over a neighborhood of visible extent. In fact, we can make this neighborhood as small as we like without affecting the long-term result: relaxation to the invariant measure. Only the time required to reach this conclusion is affected.

3. For fixed parameter values, there is no requirement that there be a unique invariant measure. A trival counter-example is the case of coexisting stable points. In short, it is not unexpected that different initial conditions will be associated with different measures. On the other hand, any sensible definition of the concept of an "attractor" presupposes the existence of such a measure.

4. The index of a saddle is the number of unstable
directions as indicated by the number of positive eigenvalues. In the present
case, the system is three-dimensional and there are three equilibira, all of
which are saddles. One of these, the origin, is of index 1: two stable
directions (and hence a two-dimensional stable manifold) and one unstable
direction. The other two saddles are located at the "centers" of the wings.
These have a single stable direction and two unstable directions -
*i.e.,* trajectories spiral away from them on a two dimensional
unstable manifold. Hence their index is 2. The reader familiar with the
rudiments of local stability analysis can calculate the three equilibria and
verify that the origin has three real eigenvalues of which one is positive,
while the saddle-foci have a negative real eigenvalue and a pair of complex
eigenvalues, the real parts of which are positive. These results are parameter
dependent, and form part of the remarkable story (see the reference to Sparrow,
above) whereby globally stable dynamics give way to chaos as the parameters are
varied.

5. See, for example, Guckenheimer, J. and P. Holmes.
1983. *Nonlinear Oscillations, Dynamical Systems and Bifurcations of
Vector Fields.* Springer-Verlag. New York.

6. As pointed out, for example, by Shaw (Shaw, R. 1981.
Strange attractors, chaotic behavior and information flow. *Z. f.
Natürforsch.* **36a**: 80-112.), the fractal structure is not
visible in simulation. This is because the phase space is strongly contracting
with the consequence that the attractor in section is indistiguishable from a
one-dimensional curve. That the section **must** be composed of layers was
pointed out by Lorenz who observed that, were this not the case, the
fundamental existence and uniquess properties of differential would be violated.