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Basin of Attraction through Invariant Curves and Dominant Functions
Al-Sharawi, Ziyad Al-Ghassani, Asma Amleh, Amal
Al-Sharawi, Ziyad
Al-Ghassani, Asma
Amleh, Amal
Date
2015
Advisor
Type
Peer-Reviewed
Article
Published version
Article
Published version
Degree
Description
Abstract
We study a second-order difference equation of the form ๐งโโโ= ๐งโ๐น(๐งโโโ) + โ, where both ๐น(๐ง) and ๐ง๐น(๐ง) are decreasing. We consider a set of invariant curves at โ = 1 and use it to characterize the behaviour of solutions when โ > 1 and when 0 < โ < 1.The case โ > 1 is related to the Y2K problem. For 0 < โ < 1, we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.