FILTERED NOISE CAN MIMIC LOW-DIMENSIONAL CHAOTIC ATTRACTORS

This contribution presents four results. First, calculations indicate that when examined by the Grassberger-Procaccia algorithm alone, filte red noise can mimic low-dimensional chaotic attractors. Given the ubiq uity Of signal filtering in experimental investigations, this is poten tially important. Second, a criterion is derived which provides an est imate of the minimum data accuracy needed to resolve the dimension of an attractor. Third, it is shown that a criterion derived by Eckmann a nd Ruelle [Physica D 56, 185 (1992)] to estimate the minimum number of data points required in a Grassberger-Procaccia calculation can be us ed to provide a further check on these dimension estimates. Fourth, it is shown that surrogate data techniques recently published by Theiler and his colleagues [in Nonlinear Modeling and Forecasting, edited by M. Casdagli and S. Eubanks (Addison Wesley, Reading, MA, 1992)] can su ccessfully distinguish between linearly correlated noise and nonlinear structure. These results, and most particularly the first, indicate t hat Grassberger-Procaccia results must be interpreted with far greater circumspection than has previously been the case, and that the algori thm should be used in combination with additional procedures such as c alculations with surrogate data. When filtered signals are examined by this algorithm alone, a finite noninteger value of D2 is consistent w ith low-dimensional chaotic behavior, but it is certainly not a defini tive diagnostic of chaos.