Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control

We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold λ. 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.