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.
a point x in X is called periodic point if there exists an n so that
where 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 with period n.
If there exists distinct n and m such that
then x is called a preperiodic point. All periodic points are preperiodic.
and 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.
- A period-one point is called a fixed point.
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.
- Given a periodic point x with period p, then for all s in R
- Given a periodic point x then all points on the orbit through x are periodic with the same prime period.
- Limit cycle
- Limit set
- Stable set
- Sharkovsky's theorem
- Stationary point
- Periodic points of complex quadratic mappings