Global saddle-type dynamics for convex second-order difference equations

A complete, elementary analysis is presented for second-order difference equations xn=g(xn-1,xn-2) where g is strictly monotone and convex in the first quadrant. It is shown that the dynamics of any such equation partitions the phase space into two basins of attraction, one of which is bounded and convex (possibly empty). In the case of two non-empty basins, each point on their common boundary corresponds to an asymptotically 2-periodic solution. The results and examples presented complement previous studies of second-order equations in the literature.