Limit cycle structure for block-sequential threshold systems

Abstract

This paper analyzes the possible limit set structures for the standard threshold block-sequential finite dynamical systems. As a special case of their work on Neural Networks (generalized threshold functions), Goles and Olivos (1981 [2]) showed that for the single block case (parallel update) one may only have fixed points and 2-cycles as ω-limit sets. Barrett et al (2006 [1]), but also Goles et al (1990 [3]) as a special case, proved that for the case with n blocks (sequential update) the only ω-limit sets are fixed points. This paper generalizes and unifies these results to standard threshold systems with block-sequential update schemes. © 2012 Springer-Verlag Berlin Heidelberg.