Dynamic Behavior and Bifurcation Analysis of a Difference Equation Including Exponential Terms

Abstract

In this paper, the local stability, the boundedness, rate of convergence and the conditions of Neimark-Sacker bifurcation concerning difference equationxn+1=α1xn+α2xn-1+β1xnexp(-x2n-1)+β2xn-1exp(-x2n-1), n=0,1, ...,are investigated. We focus on the Neimark-Sacker bifurcations of the discrete model. The center manifold theorem and bifurcation theory are explicitly applied to reach conclusions about the occurrence and stability of the Neimark-Sacker bifurcation. Many numerical simulations that confirm the existence of the Neimark-Sacker bifurcation are also provided.