Degenerate Neimark–Sacker bifurcation of a second-order rational difference equation

In this paper, we investigate the fold bifurcation, flip bifurcation and degenerate Neimark–Sacker bifurcation of a second-order rational difference equation. As we know, many scholars used the first-order Poincaré-Lyapunov constant σ to determine the type of Neimark–Sacker bifurcation. However, by computing, we find that σ=0 which means this system undergoes a degenerate Neimark–Sacker bifurcation. Therefore, using the centre manifold theorem, the Normal Form theory and the bifurcation theory, we calculate the fifth-order term of this system and give the conditions of the degenerate Neimark–Sacker bifurcation. Simultaneously, the Neimark–Sacker bifurcation curve is not a traditional parabolic shape, but an unbounded extended line. Finally, the numerical simulations are provided to illustrate theoretical results.