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Takens' Embedding Theorem asserts that when the states of a hidden dynamicalsystem are confined to a low-dimensional attractor, complete information aboutthe states can be preserved in the observed time-series output through thedelay coordinate map. However, the conditions for the theorem to hold ignorethe effects of noise and time-series analysis in practice requires a carefulempirical determination of the sampling time and number of delays resulting ina number of delay coordinates larger than the minimum prescribed by Takens'theorem. In this paper, we use tools and ideas in Compressed Sensing to providea first theoretical justification for the choice of the number of delays innoisy conditions. In particular, we show that under certain conditions on thedynamical system, measurement function, number of delays and sampling time, thedelay-coordinate map can be a stable embedding of the dynamical system'sattractor.

A First Analysis of the Stability of Takens' Embedding