On attracting basins of multiple equilibria of a class of cellular neural networks

In this paper, we study the distribution of attraction basins of multiple equilibrium points of cellular neural networks (CNNs). Under several conditions, the boundaries of the attracting basins of the stable equilibria of a completely stable CNN system are composed of the closures of the stable manifolds of unstable equilibria of (n - 1) dimensions. As demonstrations of this idea, under the conditions proposed in the literature which depicts stable and unstable equilibria, we identify the attraction basin of each stable equilibrium of which the boundary is composed of the stable manifolds of the unstable equilibria precisely. We also investigate the attracting basins of a simple class of symmetric 1-D CNNs via identifying the unstable equilibria of which the stable manifold is (n - 1) dimensional and the completely stable asymmetric CNNs with stable equilibria less than 2(n).