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 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 .
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. 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.

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.