Ω-GLOBALLY ATTRACTIVE EQUILIBRIUM POINTS OF THE CNN

The dynamic behavior of the standard Cellular Neural Network (CNN) was studied via explicit solutions of the CNN vector state equations for certain piecewise linear domains Ω. The concepts of Ω-globally attractive domain, Ω-globally attractive equilibrium point, and Ω-globally expelling domain for the CNN state equations are presented. The terminology "Ω-global" indicates that the property is global relative to a domain Ω. It is shown that every "saturated domain" D (i.e. for every vector X∈ D, each corresponding component of X has the same sign as all others and its absolute value is greater than or equal to 1), is either Ω-globally attractive or Ω-globally expelling. Attractive equilibrium points in another special domain were also studied. Sufficient conditions for convergence to such equilibria are given, along with concrete new computer simulation examples which demonstrate this theory. Finally, several definitions and simulation results are given to explicate the complex dynamical relationships which were observed between initial states in the linear domain Ω (0) (i.e. every vector X∈Ω(0) has the property that the absolute value of each component is less that 1) and Ω-globally attractive equilibrium points in the saturated domain.