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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

$f: X to X$
a point x in X is called periodic point if there exists an n so that
$f^n\left(x\right) = x$
where $f^n$ is the nth 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 $\left(f^n\right)^prime$ is defined, then one says that a periodic point is hyperbolic if

$|\left(f^n\right)^prime|ne 1,$

and that it is attractive if

$|\left(f^n\right)^prime|< 1$

and it is repelling if

$|\left(f^n\right)^prime|> 1.$

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.

Dynamical system

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

$Phi: mathbb\left\{R\right\} times X to X$
a point x in X is called periodic with period t if there exists a t ≥ 0 so that
$Phi\left(t, x\right) = x,$
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 $Phi\left(s, x\right) = Phi\left(s + t, x\right),$ 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.