Convergence of a Class of Cooperative Standard Cellular Neural Network Arrays

This paper considers a nonsymmetric standard (S) cellular neural network (CNN) array with cooperative (nonnegative) interconnections between neurons and a typical three-segment piecewise-linear (PL) neuron activation. The CNN is defined by a one-dimensional cell-linking (irreducible) cloning template with nearest-neighbor interconnections and has periodic boundary conditions. The flow generated by the SCNN is monotone but, due to the squashing effect of the horizontal segments in the PL activations, is not eventually strongly monotone (ESM). A new method for addressing convergence of the cooperative SCNN array is developed, which is based on the two main tools: (1) the concept of a frozen saddle, i.e., an unstable saddle-type equilibrium point (EP) enjoying certain dynamical properties that hold also for an asymptotically stable EP (a sink); (2) the analysis of the order relations satisfied by the sinks and saddle-type EPs. The analysis permits to show a fundamental result according to which any pair of ordered EPs of the SCNN contains at least a sink or a frozen saddle. On this basis it is shown that the flow generated by the SCNN enjoys a Limit Set Dichotomy and convergence properties analogous to those valid for ESM flows. Such results hold in the case where the SCNN displays either a local diffusion or a global propagation behavior.