Periodic point#Iterated functions

Given a mapping {{mvar|f}} from a set {{mvar|X}} into itself,

:$f:\; X\; \backslash to\; X,$

a point {{mvar|x}} in {{mvar|X}} is called periodic point if there exists an {{mvar|n}}>0 so that

:$\backslash \; f\_n(x)\; =\; x$

where {{mvar|f{{sub|n}}}} is the {{mvar|n}}th iterate of {{mvar|f}}. The smallest positive integer {{mvar|n}} satisfying the above is called the prime period or least period of the point {{mvar|x}}. If every point in {{mvar|X}} is a periodic point with the same period {{mvar|n}}, then {{mvar|f}} is called periodic with period {{mvar|n}} (this is not to be confused with the notion of a periodic function).

If there exist distinct {{mvar|n}} and {{mvar|m}} such that

:$f\_n(x)\; =\; f\_m(x)$

then {{mvar|x}} is called a preperiodic point. All periodic points are preperiodic.

If {{mvar|f}} is a diffeomorphism of a differentiable manifold, so that the derivative $f\_n^\backslash prime$ is defined, then one says that a periodic point is hyperbolic if

:$|f\_n^\backslash prime|\backslash ne\; 1,$

that it is attractive if

:

and it is repelling if

:$|f\_n^\backslash prime|>\; 1.$

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.

A period-one point is called a fixed point.

The logistic map

$$x\_\{t+1\}=rx\_t(1-x\_t),\; \backslash qquad\; 0\; \backslash leq\; x\_t\; \backslash leq\; 1,\; \backslash qquad\; 0\; \backslash leq\; r\; \backslash leq\; 4$$

exhibits periodicity for various values of the parameter {{mvar|r}}. For {{mvar|r}} between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence {{math|0, 0, 0, …,}} which attracts all orbits). For {{mvar|r}} between 1 and 3, the value 0 is still periodic but is not attracting, while the value $\backslash tfrac\{r-1\}\{r\}$ is an attracting periodic point of period 1. With {{mvar|r}} greater than 3 but less than {{tmath|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 $\backslash tfrac\{r-1\}\{r\}.$ As the value of parameter {{mvar|r}} rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of {{mvar|r}} one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

Given a real global dynamical system {{tmath|(\R, X, \Phi),}} with {{mvar|X}} the phase space and {{math|Φ}} the evolution function,

:$\backslash Phi:\; \backslash R\; \backslash times\; X\; \backslash to\; X$

a point {{mvar|x}} in {{mvar|X}} is called periodic with period {{mvar|T}} if

:$\backslash Phi(T,\; x)\; =\; x\backslash ,$

The smallest positive {{mvar|T}} with this property is called prime period of the point {{mvar|x}}.

• Given a periodic point {{mvar|x}} with period {{mvar|T}}, then $\backslash Phi(t,x)\; =\; \backslash Phi(t+T,x)$ for all {{mvar|t}} in {{tmath|\R.}}
• Given a periodic point {{mvar|x}} then all points on the orbit {{mvar|γ{{sub|x}}}} through {{mvar|x}} are periodic with the same prime period.

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

{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}

Category:Limit sets