Periodic point

In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which returns to itself after a certain number of function iterations or a certain amount of time.

Iterated functions

Given an endomorphism "f" on a set "X":$f:\; X\; o\; X$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 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 function with period "n".

If "f" 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

:$|(f^n)^prime|\; e\; 1,$

and that it is attractive if

:

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.

Dynamical system

Given a real global dynamical system (R, "X", Φ) with "X" the phase space and Φ the evolution function, :$Phi:\; mathbb\{R\}\; imes\; X\; o\; X$a point "x" in "X" is called periodic with period "t" if there exists a "t" ≥ 0 so that:$Phi(t,\; x)\; =\; x,$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 $Phi(s,\; x)\; =\; Phi(s\; +\; t,\; x),$ for all "s" in R
* Given a periodic point "x" then all points on the orbit $gamma\_x$ through "x" are periodic with the same prime period.

ee also

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