Date of Award

2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Mustafa R. S. Kulenović

Abstract

In my first manuscript, I investigate the global character of the difference equation of the form

xn+1 = f(xn, xn-1), n = 0, 1, ... (1)

with several period-two solutions, where f is increasing in all its variables. I show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period two solutions. An application of my results give global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.

In this paper I consider Eq.(1) which has three equilibrium points and up to three minimal period-two solutions which are in North-East ordering. More precisely, I will give sufficient conditions for the precise description of the basins of attraction of different equilibrium points and period-two solutions. The results can be immediately extended to the case of any number of the equilibrium points and the period-two solutions by replicating my main results.

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