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.
