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 a mapping $f$ from a set $X$ into itself,

f:X\to X,

a point $x$ in $X$ is called periodic point if there exists an $n$ >0 so that

\ f_{n}(x)=x

where $f n$ is the $n$ th 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

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 }|\neq 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.

Examples

A period-one point is called a fixed point.

The logistic map

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

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 $0, 0, 0, \dots,$ which attracts all orbits). For $r$ between 1 and 3, the value 0 is still periodic but is not attracting, while the value ${\tfrac {r-1}{r}}$ is an attracting periodic point of period 1. With $r$ greater than 3 but less than ⁠ $1+{\sqrt {6}},$ ⁠ 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 ${\tfrac {r-1}{r}}.$ 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 ⁠ $(\mathbb {R} ,X,\Phi ),$ ⁠ 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

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