Collapsing of Chaos in One Dimensional Maps

In their numerical investigation of the family of one dimensional maps f`(x) = 1 − 2|x|`, where ` > 2, Diamond et al. [P. Diamond et al., Physica D 86 (1999) 559–571] have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at −1, a repelling fixed point. This is a numerical artifact because the continuous maps are chaotic and almost every (true) trajectory can be shown to be dense in [−1, 1]. The goal of this paper is to extend and resolve this obvious contradiction. We model the numerical simulation with a randomly selected map. While they used 27 bit precision in computing f`, we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision. The fraction of initial points eventually winding up at −1 remains bounded away from 0 for every numerical precision. ©2000 Elsevier Science B.V. All rights reserved.