---
tag: 3W358BSRN2
kind: concept
name: Periodic solution of a difference equation
msc: [39A23, 37C25]
specializes: [Q3B6K9HYRG]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_point
---

## Summary

A periodic solution of a difference equation is a solution that is a [[periodic sequence|Q3B6K9HYRG]].
For a first-order autonomous equation $x_{n+1} = f(x_n)$, it is the orbit of a [[periodic point|T75CZ5H47N]] 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|FHNDX22WMK]] of a differential equation.

## Explanations

### L1 human

A periodic solution is a sequence produced by the recursion that comes back to the same state after a fixed number of steps.
From then on it repeats the same cycle forever.
For $x_{n+1} = 1 - x_n$ with $x_0 = 0$, the solution is $0, 1, 0, 1, \dots$, a cycle of period $2$.
In the logistic map $x_{n+1} = r x_n (1 - x_n)$ with $r$ a little above $3$, solutions settle near a cycle of period $2$.