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 a mapping Template:Mvar from a set Template:Mvar into itself,

$f:X\to X,$

a point Template:Mvar in Template:Mvar is called periodic point if there exists an Template:Mvar>0 so that

$f_n(x)=x$

where Template:Mvar is the Template:Mvarth iterate of Template:Mvar. The smallest positive integer Template:Mvar satisfying the above is called the prime period or least period of the point Template:Mvar. If every point in Template:Mvar is a periodic point with the same period Template:Mvar, then Template:Mvar is called periodic with period Template:Mvar (this is not to be confused with the notion of a periodic function).

If there exist distinct Template:Mvar and Template:Mvar such that

$f_n(x)=f_m(x)$

then Template:Mvar is called a preperiodic point. All periodic points are preperiodic.

If Template:Mvar is a diffeomorphism of a differentiable manifold, so that the derivative $f_n^{\prime }$ 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 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.

The logistic map

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

exhibits periodicity for various values of the parameter Template:Mvar. For Template:Mvar between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence Template:Math which attracts all orbits). For Template:Mvar between 1 and 3, the value 0 is still periodic but is not attracting, while the value $\frac{r-1}{r}$ is an attracting periodic point of period 1. With Template:Mvar greater than 3 but less than Template:Tmath 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 $\frac{r-1}{r}.$ As the value of parameter Template:Mvar rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of Template:Mvar 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 Template:Tmath with Template:Mvar the phase space and Template:Math the evolution function,

$\mathrm{\Phi }:ℝ\times X\to X$

a point Template:Mvar in Template:Mvar is called periodic with period Template:Mvar if

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

The smallest positive Template:Mvar with this property is called prime period of the point Template:Mvar.

Properties

• Given a periodic point Template:Mvar with period Template:Mvar, then $\mathrm{\Phi }(t,x)=\mathrm{\Phi }(t+T,x)$ for all Template:Mvar in Template:Tmath
• Given a periodic point Template:Mvar then all points on the orbit Template:Mvar through Template:Mvar are periodic with the same prime period.

See also

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