A prominent idea in the theory of chaos is that of shadowing, which says that, in many cases, the numerical results one sees after accuracy is lost are not total nonsense, but are in fact very close to the exact trajectory for an initial value that is near the one used. Using high-precision computation, I have researched the use of optimization as a way of finding exact shadows for several chaotic systems, such as the quadratic map r x (1 - x) and a billiard problem from the SIAM 100-Digit Challenge.
Haakonsen, Henrik, "Shadowing Chaos Via Optimization" (2009). Honors Projects. Paper 14.
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