Periodic point
4941621
145201583
2007-07-17T11:22:13Z
Adam majewski
441224
/* Iterated functions */
In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which returns to itself after a certain number of function iterations or a certain amount of time.
== Iterated functions ==
Given an [[endomorphism]] ''f'' on a [[set (mathematics)|set]] ''X''
:<math>f: X \to X</math>
a point ''x'' in ''X'' is called '''periodic point''' if there exists an ''n'' so that
:<math>\ f^n(x) = x</math>
where <math>f^n</math> is the ''n''th [[iterated function|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 function]]''' with period ''n''.
If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>(f^n)^\prime</math> is defined, then one says that a periodic point is '''hyperbolic''' if
:<math>|(f^n)^\prime|\ne 1,</math>
and that it is '''attractive''' if
:<math>|(f^n)^\prime|< 1</math>
and it is '''repelling''' if
:<math>|(f^n)^\prime|> 1.</math>
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 (mathematics)|fixed point]].
== Dynamical system ==
Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],
:<math>\Phi: \mathbb{R} \times X \to X</math>
a point ''x'' in ''X'' is called '''periodic''' with '''period''' ''t'' if there exists a ''t'' ≥ 0 so that
:<math>\Phi(t, x) = x\,</math>
The smallest positive ''t'' with this property is called '''prime period''' of the point ''x''.
=== Properties ===
* Given a periodic point ''x'' with period ''t'', then <math>\Phi(s, x) = \Phi(s + t, x)\,</math> for all ''s'' in '''R'''
* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.
==See also==
* [[Limit cycle]]
* [[Limit set]]
* [[Stable manifold|Stable set]]
* [[Sharkovsky's theorem]]
* [[Stationary point]]
*[[Periodic points of complex quadratic mappings]]
{{planetmath|id=4516|title=hyperbolic fixed point}}
[[Category:Limit sets]]