Periodic Oscillations in Weakly Connected Cellular Nonlinear Networks

Oscillatory nonlinear networks represent a circuit architecture for image and information processing. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of weakly connected cellular networks of oscillators is investigated. It is assumed that each cell admits of a Lur´e description. In case of weak coupling the main dynamic features of the network are revealed by the phase deviation equation (i.e., the equation that describes the phase deviation due to the weak coupling). Firstly a very accurate analytic expression of the phase deviation equation is derived via the joint application of the describing function technique and of Malkin´s Theorem. Then a complete analysis of the phase-deviation equation is carried out for 1-D arrays of oscillators. It is shown that the total number of periodic limit cycles with their stability properties can be estimated. Finally, in order to show the accuracy of the proposed approach, two networks containing second-order and third-order oscillators, respectively, are studied in detail.