Basins of attraction of equilibrium points of second order difference equations

We investigate the basins of attraction of equilibrium points and period-two solutions of the difference equation of the form x n + 1 = f ( x n , x n − 1 ) , n = 0 , 1 , … , where f is decreasing in the first and increasing in the second variable. We show that the boundaries of the basins of attraction of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.