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Shadowing non-hyperbolic systems

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 tex2html_wrap_inline2490 around each point tex2html_wrap_inline2260 of the refined numerical trajectory such that two sides tex2html_wrap_inline2494 are parallel to the expanding direction, while the other two sides tex2html_wrap_inline2496 are parallel to the contracting direction. In order to prove the existence of a shadow, the image under the map f of tex2html_wrap_inline2490 must map over tex2html_wrap_inline2502 such that tex2html_wrap_inline2504 makes a ``plus sign'' with tex2html_wrap_inline2502 (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 tex2html_wrap_inline2510 in tex2html_wrap_inline2512 from one contracting edge to the other, i.e.,  roughly parallel with the expanding direction. Its image tex2html_wrap_inline2514 is then stretched such that there is a section of tex2html_wrap_inline2514 lying wholly within tex2html_wrap_inline2518 , and in particular tex2html_wrap_inline2514 leaves tex2html_wrap_inline2518 through the contracting sides at both ends. Let tex2html_wrap_inline2524 be the section of tex2html_wrap_inline2514 lying wholly within tex2html_wrap_inline2518 . Now look at tex2html_wrap_inline2530 in tex2html_wrap_inline2532 . Repeat this process along the orbit, producing tex2html_wrap_inline2534 lying wholly within the final parallelogram tex2html_wrap_inline2536 . Then any point lying along tex2html_wrap_inline2534 , traced backwards, represents a true trajectory that stays within tex2html_wrap_inline2540 , 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 tex2html_wrap_inline2504 doesn't make a ``plus sign'' with tex2html_wrap_inline2502 . 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 up previous
Next: Refinement of a numerical Up: Shadowing Previous: Tutorial

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

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