# Periodic point

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.

## Iterated functions

Given an endomorphism f on a set X

$f: X \to X$

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

$\ f_n(x) = x$

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.

If there exists distinct n and m such that

$f_n(x) = f_m(x)$

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 $f_n^\prime$ is defined, then one says that a periodic point is hyperbolic if

$|f_n^\prime|\ne 1,$

that it is attractive if

$|f_n^\prime|< 1,$

and it is repelling if

$|f_n^\prime|> 1.$

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.

## Dynamical system

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

$\Phi: \mathbb{R} \times X \to X$

a point x in X is called periodic with period t if there exists a t ≥ 0 so that

$\Phi(t, x) = x\,$

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

### Properties

• Given a periodic point x with period p, then $\Phi(t,x) = \Phi(t+p,x)\,$ for all s in R
• Given a periodic point x then all points on the orbit $\gamma_x$ through x are periodic with the same prime period.