Bifurcation analysis and chaos control of a second-order exponential difference equation

The aim of this article is to study the local stability of equilibria, investigation related to the parametric conditions for transcritical bifurcation, period-doubling bifurcation and Neimark-Sacker bifurcation of the following second-order difference equation x_n+1 = αx_n + βx_n−1 exp(−σx_n−1) where the initial conditions x₋₁, x₀ are the arbitrary positive real numbers and α, β and σ are positive constants. Moreover, chaos control method is implemented for controlling chaotic behavior under the influence of Neimark-Sacker bifurcation and period-doubling bifurcation. Numerical simulations are provided to show effectiveness of theoretical discussion.