Global attraction in the sigma―delta modulator piecewise isometry

摘要

This paper presents a geometrical proof of a conjecture by Ashwin et al., that the map that describes the behaviour of the sigma-delta modulator has a global, polygonal attractor. The map in question is given in equation (1), and is a two-dimensional piecewise isometry, that is, a map whose action is to cut the phase plane into several pieces, each one of which is then rigidly rotated within the plane. The lines along which the cuts are made are the discontinuities of the map, and global attraction takes place for all points in the plane except those whose iterates eventually fall on a discontinuity.