It is known that symmetric standard CNNs are completely stable ^[1], i.e., each trajectory converges towards some equilibrium point. However, in any practical neural network realization it is not possible to obtain perfect symmetric neuron interconnections, so that the robustness of complete stability with respect to small perturbations of the nominal symmetric neuron interconnections is an extremely important issue. This chapter is devoted to the investigation of such a fundamental issue. The main result obtained is that symmetry of the interconnection matrix is not sufficient in the general case to ensure robustness of complete stability. More precisely, a number of standard CNN configurations are discussed showing that it is possible to prove the existence of stable limit cycles and more complicated attractors, as close to the symmetry condition as one pleases.

13.1 Introduction

The standard Cellular Neural Networks (CNNs) introduced by Chua and Yang in 1988 ^[1], are nowadays one of the most investigated paradigms for neural information processing ^[2]. CNNs feature a number of crucial advantages for the electronic implementation, with respect to more traditional neural architectures as the Hopfield neural networks ^[3]. Such advantages mainly derive from the essentially local and spatial-invariant interconnecting structure, which is inherent to the definition as a CNN array ^[1].

In view of the practical applications for solving signal...