# The Global Dynamics of Some Rational Difference Equations Exhibiting Neimark-Sacker Bifurcation

2023

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

Mustafa R.S. Kulenovic

## Abstract

This thesis is presented in manuscript format. The first chapter will introduce preliminary definitions and theorems that will be used in the succeeding chapters.

The second chapter will consider the dynamics of a second-order sigmoid Beverton-Holt equation

xn+1 = x2n / ax2n + x2n-1 + d

where the parameters a and d are positive numbers and the initial conditions are non-negative numbers. For this equation, we will begin by giving a description of the local dynamics of the equation, and then will also examine the global dynamics, including an investigation into the basins of attraction of the zero and greater positive equilibriums. Furthermore, we will prove the occurence of Neimark-Sacker bifurcation and give an asymptotic approximation for the resultant invariant manifold produced. This approximation will be computed according to the process developed by K. Murakami. Lastly, we will give asymptotic approximations for the local stable and unstable manifolds of the lesser positive equilibrium and investigate the rates of convergence to the attracting equilibriums.

The third chapter will consist of an investigation into the dynamics of two special cases of the second-order fractional difference equation

xn+1 = βxnxnγx2n-1 + δxn / Bxnxn-1 + Dxn

where the parameters β, γ, δ, B, C, D are non-negative and the initial conditions are arbitrary non-negative numbers. Further we assume that B xnxn-1 + Cx2n-1 + Dxn > 0 for all n ≥ 0. In particular we consider the cases where β = D = 0 and γ = D = 0. We prove that in the first case, the equation exhibits supercritical Neimark-Sacker bifurcation. In the second case, we prove that the equation exhibits both supercritical and subcritical Neimark-Sacker bifurcation. In this case, the system probably exhibits Chenciner bifurcation. Again in both cases, we will give an asymptotic approximation for the invariant curve produced by the occurence of the Neimark-Sacker bifurcation computed according to the method developed by K. Murakami.

The fourth chapter will investigate the behavior of an evolutionary system as a discrete time model of population dynamics. This case study will be based on a sigmoid Beverto-Holt model and will encorporation the dynamics of a single phenotypic trait subject to Darwinian evolution. The system studied in this manuscript will be constructed according to the canonical way to model Darwinian evolution using the dynamics of the system

xn+1 = b(un)| v = unxn

un+1 = un + σ2F(xn, v, un) / ∂v | v = un

For our case study, we will look specifically at the system

xn+1 = b(un) x2n/1 + x2n

un+1 = un + σ2 b’ (un)/b(un) = F(un) n = 0, 1,…

We will study the local and global dynamics of this system and generalize the global results obtained. Lastly, we will give some simulations of this system for different definitions of the function b(u).

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