Periodic solution of a difference equation

Summary

A periodic solution of a difference equation is a solution that is a periodic sequence. For a first-order autonomous equation $x_{n+1} = f(x_n)$, it is the orbit of a periodic point of $f$. Equilibria are the periodic solutions of period $1$. Periodic solutions, their periods and their stability describe much of the long-run behaviour of nonlinear recursions. Their birth is studied through bifurcations such as period doubling. It is the discrete counterpart of a periodic solution of a differential equation.