Global Dynamics of Some Quadratic Difference Equations

Author

Mark Dipippo, University of Rhode IslandFollow

Date of Award

2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Mustafa Kulenović

Abstract

Consider the difference equation

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where all parameters α, β, a_i, b_i, a_ij, b_ij, i, j, = 0, 1, …, k and the initial conditions χ_i, i ∈ {-k, …, 0} are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

We investigate the global dynamics of several anti-competitive systems of rational difference equations which are special cases of general linear fractional system of the form

[Mathematical equations can not be displayed here, refer to PDF]

where all parameters and the initial conditions χ₀, y0 are arbitrary nonnegative numbers such that both denominators are positive. We find the basins of attraction of all attractors of these systems.

We investigate global dynamics of the equation

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where the parameters a, c and f are nonnegative numbers with condition a + e + f > 0 and the initial conditions χ_-1, χ₀ are arbritary nonnegative numbers such that χ_-1 + χ₀ > 0.

Recommended Citation

Dipippo, Mark, "Global Dynamics of Some Quadratic Difference Equations" (2016). Open Access Dissertations. Paper 476.
https://digitalcommons.uri.edu/oa_diss/476