The dichotomy of omega-limit sets fails for cooperative standard CNNs

The paper investigates some basic aspects of the solution semiflow associated to a class of cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment pwl neuron activation. It is assumed that the SCNN neuron interconnection matrix is irreducible. By means of two counterexamples the following basic facts are shown: 1) in general the semiflow associated to the SCNN is not eventually strongly monotone; 2) in the general case also the fundamental property of the omega-limit set dichotomy fails. The consequences of these results are discussed in the context of the existing methods for addressing convergence of cooperative dynamical systems.