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of the attractor is repeated on successively smaller scales. Thus the Hénon attractor
displays scale-invariant Cantor-set-like structure transverse to the local linear structure
of the attractor. Ott concludes that because of this self-similar structure the attractor is
probably strange. In fact it has been verified by direct calculation that initially nearby
points exponentially separate with iteration number, thereby coinciding with at least one
definition of a strange attractor.
Rössler's dynamical intuition of chaos involves the geometric operation of stretching
and folding, much like a baker kneading dough for baking bread [
40
]. The conceptual
baker in this mathematical example takes some dough and rolls it out on a floured bread
board. When the dough has been rolled out thin enough he folds it back and rolls it
out again. This operation is repeated over and over again. In this analogy the dough
corresponds to a set of initial conditions that are taken sufficiently near to one another
that they appear to form a continuous distribution of states. The stretching and folding
operation represents the dynamical process undergone by each initial state. In this way
the baker's transformation represents a mapping of the distribution of initial states onto
a distribution of final states.
Two initially nearby orbits cannot separate forever on an attractor of finite size, there-
fore the attractor must fold over onto itself. Once folded, the attractor is again stretched
and folded again. This process is repeated over and over, yielding an attractor struc-
ture with an infinite number of layers to be traversed by the various trajectories. The
infinite richness of the attractor structure affords ample opportunity for trajectories to
diverge and follow increasingly different paths. The finite size of the attractor insures
that these diverging trajectories eventually pass close to one another again, but they do
so on different layers of the attractor. Crutchfield
et al
.[
17
] visualized these orbits on a
chaotic attractor as being shuffled by this process, much as a deck of cards is shuffled
by a dealer. The randomness of the chaotic process is therefore a consequence of this
shuffling; this stretching and folding creates folds within folds
ad infinitum
, resulting
in a fractal structure in phase space, just as Hénon found for his attractor. The essential
fractal feature of interest is that the greater the magnification of a region of the attractor,
the greater the degree of detail that is uncovered. The geometric structure of the attractor
itself is self-similar and its fractal character is manifest in the generated time series.
We have discussed the properties of mappings as mathematical objects. It is also
of interest to see how these maps arise in classical mechanics, that branch of physics
describing the motion of material objects. It is through the dynamics of physical phe-
nomena that we can learn how chaos may influence our expectations regarding other
complex situations.
4.4
Integrable and non-integrable Hamiltonians
We know that a Hamiltonian web has constant energy and that this conserved quan-
tity determines the web's dynamics. Other symmetries play a similar role so that
other constants of the motion simplify the dynamics because the phase-space trajectory
{
P
(
t
),
Q
(
t
)
} would be forced to follow the hypersurface
F
(
p
,
q
)
=
constant. Usually,
