Abstract
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.
Highlights
Research ArticleWe consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold λ
⎨1, x ∈ 0, λ, fλ x ⎩0, x ∈ −∞, 0 ∪ λ, ∞, 1.2 in which the positive number λ can be regarded as a threshold bifurcation parameter
It is our intention to derive a complete asymptotic and bifurcation analysis for our new equation and show that, among other things, our expectation is not quite true and perhaps such discrepancy is due to the nonlinear nature of our model at hand
Summary
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. 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. 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
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