Advances in Difference Equations
Volume 2010 (2010), Article ID 143849, 19 pages
doi:10.1155/2010/143849
Complete asymptotic analysis of a nonlinear recurrence relation with threshold control
Qi Ge1
, Chengmin Hou1
and Sui Sun Cheng3
1Department of Mathematics, Yanbian University, Yanji 133002, China
3Department of Mathematics, Tsing Hua University, Taiwan 30043, Taiwan
Abstract
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.