Copyright © 2010 Chengmin Hou 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 difference equation involving three parameters and a piecewise
constant control function with an additional positive threshold . Treating the threshold as a
bifurcation parameter that varies between 0 and , we work out a complete asymptotic and
bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending
on the size of the threshold. In particular, we show that when the threshold is either small or
large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that
the results obtained here will be useful in understanding interacting network models involving
piecewise constant control functions.