@MISC{Norway09shadowingby, author = {Henrik Bernt Håkonsen Bergen Norway}, title = {Shadowing by High-Precision Optimization}, year = {2009} }

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Abstract

Abstract. We use high-precision optimization to obtain explicit shadows for machine-precision trajectories of several sensitive dynamical systems: the quadratic map r xH1- xL, a billiard problem from the SIAM 100-Digit Challenge, the Duffing differential equation and the Ikeda delay differential equation. The main tools are the numerical optimization algorithms as implemented in Mathematica, good estimates of the working precision required to get reliable trajectories, and a way of estimating the sensitivity of the system so as to be able to feed a Jacobian to the optimization algorithm. In these examples one can get a sense of how the optimal shadowing error grows as the length of the noisy trajectory grows. Something quite special happens in the case of the quadratic map: because the system is bounded, it must eventually become periodic and by computing the periodic part explicitly we can apply ideas of Chow and Palmer to show that for any N-term trajectory there is a good shadow (error less than 10-9), regardless of how large N is. 1.