Testing for Local Activity and Edge of Chaos

In this paper we study the local activity, local passivity and edge of chaos of continuous-time reaction–diffusion cellular nonlinear networks (CNN) with one-port first-order, one-port secondorder, two-port second-order, two-port third-order and three-port third-order cells. We prove that the local passive regions determined by cell impedance ZQ(s) and cell admittance YQ(s) for firstand second-order cells are equivalent to each other. We also present an efficient procedure to determine the edge-of-chaos parameter region by combining the local active regions derived from YQ(s) and the pole locations of ZQ(s). In order to characterize the fundamental limitations of local passivity on the emergence of complexity we study the local active property from a parameter space spanned by both the cell parameters and the external excitations called the cell’s port currents (in view of its interpretations from classical circuit theory). Analytical results of locally passive, restricted locally passive, edge-of-chaos and locally active parameter regions for CNN cells modeled by cubic nonlinearities are presented. We illustrate our results by analyzing CNN cells modeled by Chua’s circuits with a cubic nonlinearity. We find that the morphology of the edge-of-chaos and the local active parameter regions have a close connection to the pattern formation behaviors of CNNs. Simulation results are presented to verify our theoretical results.