Differentiating between colored random noise and deterministic chaos with the root mean squared deviation

Abstract

A method for distinguishing between data from a strange attractor and data from colored random noise is presented. For both types of data the apparent dimension, as measured by the scaling of the correlation function with length, is finite and noninteger for certain length scales. This would seem to indicate that such measurements by themselves are insufficient for concluding that the dynamics of the underlying system are low-dimensional. To distinguish these two types of data, we have developed the variance growth test. The test looks for the increase in the variance, as measured by the average squared deviation, of a subset of data as the length of that subset is increased. For strange attractor data the variance saturates once the length of the subset exceeds the characteristic first return time. In contrast, the variance of colored random noise continues to increase with increasing subset length indefinitely, with a scaling law that is related to the apparent correlation dimension. Application of the method to the Bargatze data set shows that the AL index behaves like a deterministic dynamical system. Copyright 2001 by the American Geophysical Union.