On the effect of boundary conditions on CNN dynamics: stability and instability, bifurcation processes and chaotic phenomena

The effect of the boundary conditions on the global dynamics of cellular neural networks (CNNs) is investigated. As a case study one dimensional template CNNs are considered. It is shown that if off-diagonal template elements have the same sign (i.e. the CNN is completely stable), then the qualitative dynamics is not influenced by the boundary conditions. If the off-diagonal template elements have opposite sign, then the boundary conditions behave as bifurcation parameters and can give rise to a very complex dynamic behavior. In particular they determine the equilibrium point patterns, the transition from stability to instability and the occurrence of several bifurcation phenomena leading to strange and/or chaotic attractor.