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 from a set into itself,
a point in is called periodic point if there exists an >0 so that
where is the th iterate of . The smallest positive integer satisfying the above is called the prime period or least period of the point . If every point in is a periodic point with the same period , then is called periodic with period (this is not to be confused with the notion of a periodic function).
If there exist distinct and such that
then is called a preperiodic point. All periodic points are preperiodic.
If is a diffeomorphism of a differentiable manifold, so that the derivative 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 . For between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence which attracts all orbits). For 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 greater than 3 but less than 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 rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of 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 with the phase space and the evolution function,
a point in is called periodic with period if
The smallest positive with this property is called prime period of the point .
Properties Given a periodic point with period , then for all in Given a periodic point then all points on the orbit through 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