Periodic Mapping

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 from a set into itself,
:$f: X \to X,$
a point in is called periodic point if there exists an >0 so that
:$\ f_n(x) = x$
where is the th iterate of . The smallest positive integer satisfying the above is called the ''prime period'' or ''least period'' of the point . If every point in is a periodic point with the same period , then is called ''periodic'' with period (this is not to be confused with the notion of a periodic function).

If there exist distinct and such that
:$f_n(x) = f_m(x)$
then is called a preperiodic point. All periodic points are preperiodic.

If 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

$x_=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4$

exhibits periodicity for various values of the parameter . For between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence which attracts all orbits). For between 1 and 3, the value 0 is still periodic but is not attracting, while the value $\tfrac$ is an attracting periodic point of period 1. With greater than 3 but less than 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 $\tfrac.$ As the value of parameter rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of 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 with the phase space and the evolution function,
:$\Phi: \R \times X \to X$
a point in is called ''periodic'' with ''period'' if
:$\Phi(T, x) = x\,$
The smallest positive with this property is called ''prime period'' of the point .

Properties

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

See also

* Limit cycle
* Limit set
* Stable set
* Sharkovsky's theorem
* Stationary point
* Periodic points of complex quadratic mappings

{{PlanetMath attribution, id=4516, title=hyperbolic fixed point

Limit sets