mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 is ...

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. Examples include the mathematical models that describe the swinging of a ...

s, a periodic point of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

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 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 ...

''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 Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

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 desc ...

). 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 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 two m ...

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. F ...

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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of the

stable 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 ...

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 repe ...

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 on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...

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 popular ...

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 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 in a ...

(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 usually ...

and Φ the

evolution function 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 in a ...

, :

\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 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 as a p ...

\gamma_x

through ''x'' are periodic with the same prime period.

Iterated functions

Examples

Dynamical system

Properties

See also