Copyright © 2010 Qi Ge et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a three-term nonlinear recurrence relation involving a nonlinear
filtering function with a positive threshold . We work out a complete asymptotic analysis for all
solutions of this equation when the threshold varies from to . It is found that all solutions
either tends to 0, a limit 1-cycle, or a limit 2-cycle, depending on whether the parameter is
smaller than, equal to, or greater than a critical value. It is hoped that techniques in this paper
may be useful in explaining natural bifurcation phenomena and in the investigation of neural
networks in which each neural unit is inherently governed by our nonlinear relation.