In
mathematics, in the study of
iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function ...
s and
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, 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,
:
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
:
where
is the ''n''th
iterate of ''f''. The smallest positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''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
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
).
If there exist distinct ''n'' and ''m'' such that
:
then ''x'' is called a preperiodic point. All periodic points are preperiodic.
If ''f'' is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
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 ma ...
, so that the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is defined, then one says that a periodic point is ''hyperbolic'' if
:
that it is ''
attractive
Attraction may refer to:
* Interpersonal attraction, the attraction between people which leads to friendships, platonic and romantic relationships
** Physical attractiveness, attraction on the basis of beauty
** Sexual attraction
* Object or event ...
'' if
:
and it is ''repelling'' if
:
If the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repello ...
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
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popula ...
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
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
and Φ the
evolution function,
:
a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''T'' if
:
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
for all ''t'' in R.
* Given a periodic point ''x'' then all points on the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...
through ''x'' are periodic with the same prime period.
See also
*
Limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinit ...
*
Limit set
*
Stable set
*
Sharkovsky's theorem
*
Stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" i ...
*
Periodic points of complex quadratic mappings
{{PlanetMath attribution, id=4516, title=hyperbolic fixed point
Limit sets