A note on the dichotomy of limit sets for cooperative CNNs with delays

The paper considers a class of delayed standard (S) cellular neural networks (CNNs) with non-negative interconnections between distinct neurons and a typical three-segment pwl neuron activation. It is also assumed that such cooperative SCNNs satisfy an irreducibility condition on the interconnection and delayed interconnection matrix. By means of a counterexample it is shown that the solution semiflow associated to such SCNNs in the general case does not satisfy the fundamental property of the omega-limit set dichotomy and is not eventually strongly monotone. The consequences of this result are discussed in the context of the existing methods for addressing convergence of monotone semiflows defined by delayed cooperative dynamical systems.