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 "f" from a set "X" into itself,
a point "x" in "X" is called periodic point if there exists an "n" so that
where formula_3 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" with period "n" (this is not to be confused with the notion of a periodic function).
If there exist distinct "n" and "m" such that
then "x" is called a preperiodic point. All periodic points are preperiodic.
If "f" is a diffeomorphism of a differentiable manifold, so that the derivative formula_5 is defined, then one says that a periodic point is "hyperbolic" if
that it is "attractive" if
and it is "repelling" if
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 
exhibits periodicity for various values of the parameter "r". For "r" between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which attracts all orbits). For "r" between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With "r" greater than 3 but less than 1 + , 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 . As the value of parameter "r" rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of "r" 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 (R, "X", Φ) with "X" the phase space and Φ the evolution function,
a point "x" in "X" is called "periodic" with "period" "t" if there exists a "t" > 0 so that
The smallest positive "t" with this property is called "prime period" of the point "x".