Basin of Attraction through Invariant Curves and Dominant Functions

Ziyad Alsharawi,Asma Al-Ghassani,A M Amleh

doi:10.1155/2015/160672

Abstract

We study a second-order difference equation of the formzn+1=znF(zn-1)+h, where bothF(z)andzF(z)are decreasing. We consider a set of invariant curves ath=1and use it to characterize the behaviour of solutions whenh>1and when0<h<1. The caseh>1is related to the Y2K problem. For0<h<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.

Highlights

IntroductionSecond-order difference equations of the form zn+1 = znF (zn−1) , n ∈ N := Z+ ∪ {0} , (1)
Second-order difference equations of the form zn+1 = znF, n ∈ N := Z+ ∪ {0}, (1)where F(z) is a continuous function, have been widely used in applications [1,2,3,4]
Our ultimate goal is to reach a general theory for this type of difference equation, we find it interesting to consider F(z) = b/(−1+z) as a prototype, and so we focus this work on the dynamics of the equation zn+1 = znF + h, F (z)

Summary

Introduction

Second-order difference equations of the form zn+1 = znF (zn−1) , n ∈ N := Z+ ∪ {0} , (1). A well-known example is Pielou’s difference equation which has been suggested to model the growth of a single species with delayed-density dependence [4, 6]. Adding or subtracting a constant h from (1) can be interpreted mathematically as a perturbation of the model, or biologically as constant stocking or constant yield harvesting [7,8,9]. These meaningful applications motivate investigating the dynamics of the difference equation zn+1 = znF (zn−1) ± h, n ∈ N, h > 0

Objectives

Conclusion