Advances in Difference Equations
Volume 2010 (2010), Article ID 143849, 19 pages
Research Article

Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control

1Department of Mathematics, Yanbian University, Yanji 133002, China
2Department of Mathematics, Tsing Hua University, Taiwan 30043, Taiwan

Received 23 November 2009; Accepted 10 January 2010

Academic Editor: Toka Diagana

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 λ. We work out a complete asymptotic analysis for all solutions of this equation when the threshold varies from 0+ 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 λ 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.