Global Dynamics of Some Discrete Dynamical Systems With Applications

Author

Arzu Bilgin, University of Rhode IslandFollow

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

x_n+1 = f(x_n, x_n-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.

Recommended Citation

Bilgin, Arzu, "Global Dynamics of Some Discrete Dynamical Systems With Applications" (2016). Open Access Dissertations. Paper 524.
https://digitalcommons.uri.edu/oa_diss/524