# Periodic point

Short description: Point which a function/system returns to after some time or iterations

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 a mapping f from a set X into itself,

$\displaystyle{ f: X \to X, }$

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

$\displaystyle{ \ f_n(x) = x }$

where fn 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

$\displaystyle{ f_n(x) = f_m(x) }$

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

$\displaystyle{ |f_n^\prime|\ne 1, }$

that it is attractive if

$\displaystyle{ |f_n^\prime|\lt 1, }$

and it is repelling if

$\displaystyle{ |f_n^\prime|\gt 1. }$

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.

### Examples

A period-one point is called a fixed point.

The logistic map

$\displaystyle{ x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4 }$

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 $\displaystyle{ \tfrac{r-1}{r} }$ is an attracting periodic point of period 1. With r greater than 3 but less than $\displaystyle{ 1 + \sqrt 6, }$ 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 $\displaystyle{ \tfrac{r-1}{r}. }$ 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).

## Dynamical system

Given a real global dynamical system $\displaystyle{ (\R, X, \Phi), }$ with X the phase space and Φ the evolution function,

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

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

$\displaystyle{ \Phi(T, x) = x\, }$

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

### Properties

• Given a periodic point x with period T, then $\displaystyle{ \Phi(t,x) = \Phi(t+T,x) }$ for all t in $\displaystyle{ \R. }$
• Given a periodic point x then all points on the orbit γx through x are periodic with the same prime period.