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 function In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some ...

s and

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

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 from a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 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 :

f_n(x) = f_m(x)

then is called a preperiodic point. All periodic points are preperiodic. If is a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

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 In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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 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 coo ...

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 In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

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 The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

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 In celestial mechanics, an orbit (also known as orbital revolution) 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 ...

through are periodic with the same prime period.

Iterated functions

Examples

Dynamical system

Properties

See also