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Abstract

Chaotic dynamical systems exhibit sensitive dependence on initial conditions. Round-off errors introduced in computer simulations may cause a computed orbit to quickly diverge from the true orbit. One may therefore question the validity of these computations. Shadowing provides a means for studying the validity of a computation. If we are able to show that a true solution with a different initial condition, called a shadowing orbit, stays close to a computed solution, which we call a pseudo-orbit, then perhaps the computed solution has some validity. We will investigate two methods for proving the existence of a shadowing orbit. The first method comes from two theorems by Chow and Palmer. These theorems provide us with sufficient conditions for when a pseudo-orbit is shadowed by a true orbit and provide us with a bound on the shadowing distance of a true orbit. The second method is by Grebogi, Hammel, Yorke and Sauer. Their method contains a pseudo-orbit in a carefully constructed sequence of parallelograms which helps to prove the existence of a shadowing orbit. These two methods will then be used to prove the existence of a shadowing orbit for examples in one and two dimensions. Finally we will discuss a refinement technique which takes a 'noisy' orbit and produces a less 'noisy' orbit which shadows the original orbit. This technique will then be applied to some examples to find a numerical shadow for a pseudo-orbit.