Inmathematics, in the study ofiterated functions anddynamical systems, aperiodic point of afunction is a point which the system returns to after a certain number of function iterations or a certain amount of time.

Given amappingf from asetX into itself,

${\displaystyle f:X\to X,}$

a pointx inX is called periodic point if there exists ann>0 so that

${\displaystyle f_n(x)=x}$

wheref_n is thenthiterate off. The smallest positiveintegern satisfying the above is called theprime period orleast period of the pointx. If every point inX is a periodic point with the same periodn, thenf is calledperiodic with periodn (this is not to be confused with the notion of aperiodic function).

If there exist distinctn andm such that

${\displaystyle f_n(x)=f_m(x)}$

thenx is called apreperiodic point. All periodic points are preperiodic.

Iff is adiffeomorphism of adifferentiable manifold, so that thederivative${\displaystyle f_n^{\mathrm{\prime }}}$ is defined, then one says that a periodic point ishyperbolic if

${\displaystyle |f_n^{\mathrm{\prime }}|\ne 1,}$

that it isattractive if

and it isrepelling if

${\displaystyle |f_n^{\mathrm{\prime }}|>1.}$

If thedimension of thestable manifold of a periodic point or fixed point is zero, the point is called asource; if the dimension of itsunstable manifold is zero, it is called asink; and if both the stable and unstable manifold have nonzero dimension, it is called asaddle orsaddle point.

A period-one point is called afixed point.

Thelogistic map

$${\displaystyle x_{t+1}=r{x}_t(1-x_t),0\le x_t\le 1,0\le r\le 4}$$

exhibits periodicity for various values of the parameterr. Forr between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence0, 0, 0, …, whichattracts all orbits). Forr between 1 and 3, the value 0 is still periodic but is not attracting, while the value${\displaystyle \frac{r-1}{r}}$ is an attracting periodic point of period 1. Withr greater than 3 but less than⁠${\displaystyle 1+\sqrt{6},}$⁠ 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${\displaystyle \frac{r-1}{r}.}$ As the value of parameterr rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values ofr one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

Given areal global dynamical system⁠${\displaystyle (\mathbb{R},X,\mathrm{\Phi }),}$⁠ withX thephase space andΦ theevolution function,

${\displaystyle \mathrm{\Phi }:\mathbb{R}\times X\to X}$

a pointx inX is calledperiodic withperiodT if

${\displaystyle \mathrm{\Phi }(T,x)=x}$

The smallest positiveT with this property is calledprime period of the pointx.

• Given a periodic pointx with periodT, then${\displaystyle \mathrm{\Phi }(t,x)=\mathrm{\Phi }(t+T,x)}$ for allt in⁠${\displaystyle \mathbb{R}.}$⁠
• Given a periodic pointx then all points on theorbitγ_x throughx are periodic with the same prime period.

• 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 onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Limit sets

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