Dynamics and bifurcation of xn+1=α+βxn−kA+Bxn+Cxn−k,k= 1,2

Abstract

The main goal of this thesis is to study the bifurcation of second and third order rational difference equations. We consider the sufficient conditions for the existence of the bifurcation. We study the second order rational difference equation xn+1 = α + βxn−1 A + Bxn + Cxn−1 , n = 0, 1, 2, . . . with positive parameters α, β, A, B, C and non-negative initial conditions {x−k, x−k+1, . . . , x0}. We study the dynamic behavior and the direction of the bifurcation of the periodtwo cycle. Then, we give numerical discussion with figures to support our results. Also we study the third order rational difference equation xn+1 = α + βxn−2 A + Bxn + Cxn−2 , n = 0, 1, 2, . . . with positive parameters α, β, A, B, C and non-negative initial conditions {x−k, x−k+1, . . . , x0}. We study the dynamic behavior and the direction of the Neimark-Sacker bifurcation. Then, we give numerical discussion with figures to support our results.