Dynamical Behavior and Bifurcations Analysis for Second-Order Rational Difference Equation

In this paper, we explore the dynamical behavior of a novel rational difference equation. The asymptotic stability of the equilibrium points is analyzed using a nonlinear stability criterion and supported by numerical simulations. The existence of periodic solutions is also discussed. Furthermore, we investigate the codimension-1 bifurcations of the equation. Specifically, we demonstrate the occurrence of transcritical, flip, and Neimark–Sacker bifurcations. For each type, the corresponding topological normal form is computed to provide deeper insight into the system’s local dynamics. To validate our theoretical findings, numerical simulations and bifurcation analyzes are performed using MATLAB. Finally, to control the system’s chaotic behavior, the OGY (Ott–Grebogi–Yorke) method is employed as an effective chaos control strategy.