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'' 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

:$, f_n^\prime, \ne 1,$

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 ''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 is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + , 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 . 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'', Φ) with ''X'' the phase space and Φ the evolution function,
:$\Phi: \mathbb \times X \to X$
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.

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