A shadowing trajectory is a true trajectory of a map or differential equation, that closely tracks a computed approximate solution. For systems exhibiting hyperbolic and near-hyperbolic chaos, long shadowing trajectories can be shown to exist, meaning that trajectories constructed in the presence of one-step errors still represent true system behavior. We will report on recent studies of strongly nonhyperbolic systems, in the case where finite-time Lyapunov exponents fluctuate about zero, for which long shadowing trajectories apparently do not exist. Scaling laws govern the length of shadowing trajectories in terms of one-step error and the statistics of the finite-time Lyapunov exponents.
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