EXTENDING LOCAL PASSIVITY THEORY AND HOPF BIFURCATION AT THE EDGE OF CHAOS IN OREGONATOR CNN

Abstract

The fundamental local passivity theory asserts that a wide spectrum of complex behaviors may exist if the cells in the reaction–diffusion are not locally passive. This local passivity principle has provided a powerful tool for studying the complexity in a homogeneous lattice formed by coupled cells. In this paper, the complexity matrix YQ(s), which is the tool for testing the local passivity theory, is modified based on the characteristic polynomial AQ(λ). Then, the local passivity theory is applied to the study of the Oregonator CNN to judge if the cell parameters of a CNN are chosen at the edge of chaos. Analysis of the bifurcation and the numerical simulations show that nonzero diffusion term in Oregonator CNN may cause a reaction–diffusion equation oscillating under the appropriate choice of diffusion coefficient if the local passivity theory is not satisfied. That is, if the cell parameters of a CNN are chosen at the edge of chaos, the system is potentially unstable.