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.
Given a mapping f from a set X into itself, a point x in X is called periodic point if there exists an n>0 so that where fn 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 there exist distinct n and m such that then x is called a preperiodic point.
If f 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.
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).
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).
Given a real global dynamical system
with X the phase space and Φ the evolution function, a point x in X is called periodic with period T if The smallest positive T with this property is called prime period of the point x.
This article incorporates material from hyperbolic fixed point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.