Failure of dimension analysis in a simple five-dimensional system

Abstract

Dynamic systems may be characterized by their fractal dimension. The classical Grassberger-Procaccia algorithm is widely used to analyze time series. However, if this method is used beyond its intrinsic limitations it may cause incorrect classification of systems. We found that a simple deterministic five-dimensional system leads to erroneous dimension values around 5.5 if the following methods are used uncritically: The classical Grassberger-Procaccia algorithm, a pointwise correlation dimension algorithm, and an algorithm for calculation of the information dimension yielded this erroneous result for a wide range of numbers of data points (N=30 000–${10}^6$) and various delay times. Estimates of dimensions are only reliable if long plateaus of the local slope of the correlation integrals exist for small distances; these were not found in our example. This example suggests that a correlation dimension of 5 is too high to be recognized using even one million noise-free data points.