ESTIMATING CORRELATION DIMENSION FROM A CHAOTIC TIME-SERIES - WHEN DOES PLATEAU ONSET OCCUR

Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D-2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m-dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate is relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D-2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil(D-2), where Ceil(D-2), standing for ceiling of D-2, is the smallest integer greater than or equal to D-2. We then show numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil(D-2).