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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):

Figure: CONTAINMENT.

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.

Next: Refinement of a numerical Up: Shadowing Previous: Tutorial

Wayne Hayes
Fri Dec 27 17:41:39 EST 1996

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