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>0 so that

${\displaystyle \ f_{n}(x)=x}$

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 (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

${\displaystyle |f_{n}^{\prime }|\neq 1,}$

that it is attractive if

${\displaystyle |f_{n}^{\prime }|<1,}$

and it is repelling if

${\displaystyle |f_{n}^{\prime }|>1.}$

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.

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 ${\displaystyle (\mathbb {R} ,X,\Phi ),}$ with X the phase space and Φ the evolution function,

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

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 ${\displaystyle \mathbb {R} .}$
• Given a periodic point x then all points on the orbit γ_x through x are periodic with the same prime period.