Bifurcations and oscillatory behavior in a class of competitive Cellular Neural Networks

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some station ary state. In this paper it is shown that the interconnection symmetry, tho ugh ensuring complete stability, is not in the general case sufficient to g uarantee that complete stability is robust with respect to sufficiently sma ll perturbations of the interconnections. To this end, a class of third-ord er CNN's with competitive (inhibitory) interconnections between distinct ne urons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, ev en if the interconnection matrix is arbitrarily close to some symmetric mat rix. This result is of obvious relevance in view of CNN's implementation, s ince perfect interconnection symmetry in unattainable in]hardware (e.g. VLS I) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leon ard model of the voting paradox, a special Volterra-Lotka model of three co mpeting species. Finally, it is shown that the results in this paper can al so be viewed as an extension of previous results by Zou and Nossek for a tw o-cell CNN with opposite-sign interconnections between distinct neurons. Su ch an extension has a significant interpretation in the framework of a gene ral theorem by Smale for competitive dynamical systems.