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Shadowing Lemmas, and theoretical results

The GHYS method for rigorously proving that a shadow exists is constructive. That is, one explicitly constructs a subspace around the numerical orbit which must contain a true orbit. This is essentially an interval arithmetic, or enclosure method (see, eg.,  [59]). In contrast, shadowing lemmas are capable of proving the existence of a shadow non-constructively [14, 15, 16, 17, 18, 60, 80]. However, they always involve computing the Jacobian of the map, or solving the variational equations of the ODE, and then estimating hyperbolicity, so they are all roughly equal in computational expense.

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

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