Bifurcation Analysis for Nonlinear Recurrence Relations with Threshold Control and 2k-Periodic Coefficients

Abstract

A nonlinear recurrence involving a piecewise constant McCulloch-Pitts function and 2k-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0 to ∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. Among other things, we show that each solution tends towards one of four different limits. Furthermore, the accompanying initial regions for each type of solutions can be determined. It is hoped that our analysis will provide motivation for further results for recurrent McCulloch-Pitts type neural networks.