Flip bifurcations of two systems of difference equations

This paper investigates the bifurcations of the following difference equations xn+1=axn+byne−xn,yn+1=cyn+dxne−yn,xn+1=ayn+bxne−yn,yn+1=cxn+dyne−xn, where a,b,c, and d are positive constants and the initial conditions x0 and y0 are positive numbers. Psarros, Papaschinopoulos, and Schinas (Math. Methods Appl. Sci., 2016, 39: 5216–5222) presented the semistability of the fixed point (0,0) when one eigenvalue is equal to 1 and the other eigenvalue has absolute value less than 1. In this paper, we consider another case: one eigenvalue is equal to −1. With the aid of the center manifold reduction theorem, we rigorously show that these two systems undergo flip bifurcations or generalized flip bifurcations. Moreover, the stability of the fixed point (0,0) and the existence of period‐two cycles are also given.