PROJECT TITLE :

Limit Set Dichotomy and Multistability for a Class of Cooperative Neural Networks With Delays

ABSTRACT:

Recent papers have got wind the interest to check convergence within 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, because of the horizontal segments in the PL functions, don't seem to be eventually strongly monotone (ESM). This notwithstanding, it's been shown that there are subclasses of irreducible interconnection matrices for that the semiflows, though they are not ESM, fancy convergence properties the same as those of ESM semiflows. The results obtained therefore way concern the case of cooperative NNs without delays. The goal of this paper is to extend a number of the present results to the relevant case of NNs with delays. Additional 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 A^τ. The most result's that when A+A^τ satisfies a full interconnection condition, then the generated semiflows, that are monotone however 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, within the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the final case where the NNs possess multiple EPs, i.e., may be a result on multistability, and is valid for any constant worth of the delays.