For non-hyperbolic systems, we may have to be satisfied with
finite-length shadows. The first studies of shadows for non-hyperbolic
systems appear to be Beyn [8] and Hammel *et al. * [31].
Grebogi, Hammel, Yorke, and Sauer [32, 28] provide the first
rigorous proof of the existence of a shadow for a non-hyperbolic system over
a non-trivial length of time. For their method to work, the system does
not need to be uniformly hyperbolic, but only
``strongly'' hyperbolic. Their method consists of two parts. First,
they *refine* a noisy trajectory using an iterative method that
produces a nearby trajectory with less noise. This procedure will
be discussed in detail below. When refinement
converges to the point that the noise is of order the machine
precision, they invoke *containment*, which can rigorously prove
the existence of a nearby true trajectory.

The containment process in two dimensions consists of building
a parallelogram around each point of the refined numerical
trajectory such that
two sides are parallel to the expanding direction, while the
other two sides are parallel to the contracting direction.
In order to prove the existence of a shadow, the image under the map *f*
of must map over such that makes a ``plus sign''
with (Figure 2):

To ensure this occurs, a bound on the second derivative of *f* is needed,
and the expansion and contraction amounts need to be resolvable by the
machine precision. The proof of the existence of a true orbit then
relies on the following argument. Draw a curve in from
one contracting edge to the other, *i.e., * roughly parallel with the
expanding direction. Its image is then stretched such that
there is a section of lying wholly within , and in
particular leaves through the contracting sides at
both ends. Let be the section of lying wholly
within . Now look at in .
Repeat this process along the orbit, producing lying wholly
within the final parallelogram . Then any point lying along
, traced backwards, represents a true trajectory that stays
within , and we are done.

With this picture, there is a nice geometric interpretation of the requirement that the angle between the stable and unstable directions be bounded away from 0: if the angle gets too small, then the parallelogram essentially loses a dimension, and doesn't make a ``plus sign'' with . Practically speaking, this occurs when the angle becomes comparable with the noise amplitude of the refined orbit. Hence, the more accurate the orbit, the longer it can be shadowed.

Fri Dec 27 17:41:39 EST 1996

Access count (updated once a day) since 1 Jan 1997: 8689