Limit set dichotomy and multistability for a class of cooperative neural networks with delays

Recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points (EPs) (multistability) for neural networks (NNs) with nonsymmetric cooperative (nonnegative) interconnections and neuron activations modeled by piecewise linear (PL) functions. One basic difficulty is that the semiflows generated by such NNs are monotone but, due to the horizontal segments in the PL functions, are not eventually strongly monotone (ESM). This notwithstanding, it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows, although they are not ESM, enjoy convergence properties similar to those of ESM semiflows. The results obtained so far concern the case of cooperative NNs without delays. The goal of this paper is to extend some of the existing results to the relevant case of NNs with delays. More specifically, this paper considers a class of NNs with PL neuron activations, concentrated delays, and a nonsymmetric cooperative interconnection matrix A and delay interconnection matrix Aτ. The main result is that when A+Aτ satisfies a full interconnection condition, then the generated semiflows, which are monotone but not ESM, satisfy a limit set dichotomy analogous to that valid for ESM semiflows. It follows that there is an open and dense set of initial conditions, in the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the general case where the NNs possess multiple EPs, i.e., is a result on multistability, and is valid for any constant value of the delays.