# 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 an endomorphism *f* on a set *X*

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 *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*.

If there exists distinct *n* and *m* such that

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 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.

## Dynamical system

Given a real global dynamical system (**R**, *X*, Φ) with *X* the phase space and Φ the evolution function,

a point *x* in *X* is called *periodic* with *period* *t* if there exists a *t* > 0 so that

The smallest positive *t* with this property is called *prime period* of the point *x*.

### Properties

- Given a periodic point
*x*with period*p*, then for all*t*in**R** - Given a periodic point
*x*then all points on the orbit through*x*are periodic with the same prime period.

### Examples

The logistic map

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 (*r*-1)/*r* is an attracting periodic point of period 1. With *r* greater than 3 but less than 1 + √6, 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 (*r*-1)/*r* and a non-attracting period-2 cycle between two periodic points. 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).

## See also

- Limit cycle
- Limit set
- Stable set
- Sharkovsky's theorem
- Stationary point
- Periodic points of complex quadratic mappings

*This article incorporates material from hyperbolic fixed point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*