Bifurcations of a second-order difference equation related to a class of reaction-diffusion equations

In this article, we consider a second-order difference equation which was related to a class of reaction-diffusion equations. Firstly, we discuss the topological types of its fixed point in order to investigate bifurcations. Secondly, by applying centre manifold reduction theorem, we study local codimension 1 bifurcations, such as transcritical bifurcation, pitchfork bifurcation and flip bifurcation. Finally, by computing Poincare-Birkhoff normal forms we investigate a generalized Neimark-Sacker bifurcation. More concretely, we give the conditions of existence of two invariant cycles, those of only one and those of none.