In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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,
:
a point in is called periodic point if there exists an >0 so that
:
where is the th
iterate of . The smallest positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
).
If there exist distinct and such that
:
then is called a preperiodic point. All periodic points are preperiodic.
If is a
diffeomorphism of a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
, so that the
derivative is defined, then one says that a periodic point is ''hyperbolic'' if
:
that it is ''
attractive'' if
:
and it is ''repelling'' if
:
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
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
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
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,
:
a point in is called ''periodic'' with ''period'' if
:
The smallest positive with this property is called ''prime period'' of the point .
Properties
* Given a periodic point with period , then
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