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 equation
x_n+1=α₁x_n+α₂x_n-1+β₁x_nexp(-x²_n-1)+β₂x_n-1exp(-x²_n-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.