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.

Fri Dec 27 17:41:39 EST 1996

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