In mathematics, in the study of iterated functions and dynamical systems, a **periodic point** of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.

Given a mapping f from a set X into itself,

a point x in X is called periodic point if there exists an n so that

where f_{n} is the nth iterate of f. The smallest positive integer n satisfying the above is called the *prime period* or *least period* of the point x. If every point in X is a periodic point with the same period n, then f is called *periodic* with period n (this is not to be confused with the notion of a periodic function).

If there exist distinct n and m such that

then x is called a **preperiodic point**. All periodic points are preperiodic.

If f is a diffeomorphism of a differentiable manifold, so that the derivative is defined, then one says that a periodic point is *hyperbolic* if

that it is *attractive* if

and it is *repelling* if

If the dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a *source*; if the dimension of its unstable manifold is zero, it is called a *sink*; and if both the stable and unstable manifold have nonzero dimension, it is called a *saddle* or saddle point.

A period-one point is called a fixed point.

The logistic map

exhibits periodicity for various values of the parameter r. For r between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, …, which attracts all orbits). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With r greater than 3 but less than there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and As the value of parameter r rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of r one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

Given a real global dynamical system with X the phase space and Φ the evolution function,

a point x in X is called *periodic* with *period* T if

The smallest positive T with this property is called *prime period* of the point x.

- Given a periodic point x with period T, then for all t in
- Given a periodic point x then all points on the orbit γ
_{x}through x are periodic with the same prime period.