Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control
<p/> <p>We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold <inline-formula><graphic file="1687-1847-2010-542073-i1.gif"/></inline-formula>. Treating the threshold as a bi...
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2010-01-01
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Series: | Advances in Difference Equations |
Online Access: | http://www.advancesindifferenceequations.com/content/2010/542073 |
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doaj-c760bd7b462942549e2afaa2feb0ec5f2020-11-24T23:28:39ZengSpringerOpenAdvances in Difference Equations1687-18391687-18472010-01-0120101542073Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant ControlCheng SuiSunHou ChengminHan Lili<p/> <p>We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold <inline-formula><graphic file="1687-1847-2010-542073-i1.gif"/></inline-formula>. Treating the threshold as a bifurcation parameter that varies between 0 and <inline-formula><graphic file="1687-1847-2010-542073-i2.gif"/></inline-formula>, 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.</p> http://www.advancesindifferenceequations.com/content/2010/542073 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Cheng SuiSun Hou Chengmin Han Lili |
spellingShingle |
Cheng SuiSun Hou Chengmin Han Lili Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control Advances in Difference Equations |
author_facet |
Cheng SuiSun Hou Chengmin Han Lili |
author_sort |
Cheng SuiSun |
title |
Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control |
title_short |
Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control |
title_full |
Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control |
title_fullStr |
Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control |
title_full_unstemmed |
Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control |
title_sort |
complete asymptotic and bifurcation analysis for a difference equation with piecewise constant control |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1839 1687-1847 |
publishDate |
2010-01-01 |
description |
<p/> <p>We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold <inline-formula><graphic file="1687-1847-2010-542073-i1.gif"/></inline-formula>. Treating the threshold as a bifurcation parameter that varies between 0 and <inline-formula><graphic file="1687-1847-2010-542073-i2.gif"/></inline-formula>, 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.</p> |
url |
http://www.advancesindifferenceequations.com/content/2010/542073 |
work_keys_str_mv |
AT chengsuisun completeasymptoticandbifurcationanalysisforadifferenceequationwithpiecewiseconstantcontrol AT houchengmin completeasymptoticandbifurcationanalysisforadifferenceequationwithpiecewiseconstantcontrol AT hanlili completeasymptoticandbifurcationanalysisforadifferenceequationwithpiecewiseconstantcontrol |
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1725548510906941440 |