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'' &ge; 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]]