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 $fn$ 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


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

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.

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 $0, 0, 0, …,$ 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 $(\R, X, \Phi),$ with $X$ the phase space and $Φ$ the evolution function,
 * $$\Phi: \R \times X \to X$$

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 $\R.$
 * Given a periodic point $x$ then all points on the orbit $&gamma;x$ through $x$ are periodic with the same prime period.