Dynamics and Bifurcation of xn+1=α+βxn−2A+Bxn+Cxn−2xn+1=α+βxn−2A+Bxn+Cxn−2

Abstract

In this paper, we study dynamics and bifurcation of the third order rational difference equation x n + 1 = α + β x n − 2 A + B x n + C x n − 2 ,

n = 0 , 1 , 2 , … with positive parameters α , β , A , B , C and non-negative initial conditions { x − k , x − k + 1 , … , x 0 } . We study the dynamic behavior, the sufficient conditions for the existence of the Neimark-Sacker bifurcation, and the direction of the Neimark-Sacker bifurcation. Then, we give numerical examples with figures to support our results.