Stability of a three cell cellular neural network

Complete stability of a cellular neural network (CNN) is a strong form of stability where almost all solution curves of the associated differential equations tend to a stable equilibrium point. For the three cell system considered here the state space is the three-dimensional Euclidean space R³ which allows following the evolution of trajectories geometrically. The author carries out a stability analysis by studying the vector field associated with the state equations. Specifically he notes the directions of the vector field in certain convex, compact subregions of the state space, capitalizing on the fact that the differential equations are piecewise-linear and actually linear in the regions considered. A three cell CNN with an opposite-sign template is described by differential equations. Complete stability of the opposite-sign cellular neural network is established for a certain parameter range. The proof is geometric in nature and provides an example of a qualitative analysis of a nonlinear differential equation