THE LOCAL ACTIVITY CRITERIA FOR "DIFFERENCE-EQUATION" CNN

In this paper we use an exponential conformal mapping and a z-transform to "translate" the local activity criteria for continuous-time reaction–diffusion cellular nonlinear networks (CNN) to those for difference-equation CNNs. A difference-equation CNN is modeled by a set of difference equations with a constant sampling interval δt>0. Since a difference-equation CNN tends to a continuous-time CNN when δt→0, we can view the Laplace transform of a continuous-time CNN as the limit of the conformal-mapping z-transform of a corresponding difference-equation CNN. Based on the relation between Laplace transform and our conformal-mapping z-transform, we extend the local activity criteria from a continuous-time CNN to a difference-equation CNN. We have proved the rather surprising result that the class of all reaction–diffusion difference-equation CNNs with two state variables and one diffusion coefficient is locally active everywhere, i.e. its local passive region is empty. In particular, as δt→0, the local-passive region of a continuous-time CNN cell transforms into the "edge-of-chaos" region of a corresponding difference-equation CNN cell with δt>0. Remarkably, as δt→0 the locally active edge-of-chaos region degenerates into a locally passive region as the difference equation tends to a differential equation. These results highlight a fundamental difference between the qualitative properties of systems of nonlinear differential- and difference-equations.