Invisible Errors in Dimension Calculations: Geometric and Systematic Effects

Abstract

We use box-counting methods to attempt to reliably calculate the generalized dimensions (including box-counting dimension, i.e., capacity dimension, and information dimension) for the Henon attractor (a = 1.4, b = 0.3). In order to investigate possible errors arising in more general situations, we have analyzed the asymptotic behavior of the cover of the attractor as the number of iterates considered approaches infinity. The error in estimating the box-counting dimension depends in part on the geometric shape of the “boxes” used, and we give a heuristic derivation of the rate of approach. We introduce the use of disks rather than squares to minimize errors in estimates of the number of “boxes” required. The resulting dimension estimates have very small fitting errors: the points in a log-log plot are quite well fit by a straight line. However, what would happen for even smaller box sizes cannot be estimated.