Opuscula Math. 30, no. 3 (2010), 349-360
Bifurcation in a nonlinear steady state system
Sui Sun Cheng
Abstract. The steady state solutions of a nonlinear digital cellular neural network with \(\omega\) neural units and a nonnegative variable parameter \(\lambda\) are sought. We show that \(\lambda = 1\) is a critical value such that the qualitative behavior of our network changes. More specifically, when \(\omega\) is odd, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda \in [1,\infty)\), steady states cannot exist; while when \(\omega\) is even, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda = 1\), there are no nontrivial steady states, and for \(\lambda \in (1,\infty)\), there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers.
Keywords: bifurcation, cellular neural network, steady state, Krasnoselsky fixed point theorem.
Mathematics Subject Classification: 39A11, 65H10, 93C55.