Dynamics and Bifurcation for a Class of Second-Order Nonlinear Difference Equations

We investigate the dynamics of a class of second-order nonlinear iterative maps given by $x_{n+1}= a x_{n-1}^\alpha + b x_n^\alpha$, where the parameters $a, b$, and $\alpha >0$ govern the system's behavior. First, we establish the existence and stability of equilibrium points, showing that at $\alpha = 1$ a transcritical bifurcation occurs, producing an exchange of stability between the two equilibria. Next, via linear stability analysis, we derive necessary and sufficient conditions under which the prime period-two solution exists and is unstable. Furthermore, we identify a pseudo-subcritical flip bifurcation at a critical threshold where the system transitions from saddle-type behavior to repelling behavior, accompanied by the emergence of an unstable prime period-two solution. Our results provide a complete parameter-dependent characterization of the stability and dynamics of the solutions.