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 f from a set X into itself,
- [math]\displaystyle{ f: X \to X, }[/math]
a point x in X is called periodic point if there exists an n>0 so that
- [math]\displaystyle{ \ f_n(x) = x }[/math]
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 every point in X is a periodic point with the same period n, then f is called periodic with period n (this is not to be confused with the notion of a periodic function).
If there exist distinct n and m such that
- [math]\displaystyle{ f_n(x) = f_m(x) }[/math]
then x is called a preperiodic point. All periodic points are preperiodic.
If f is a diffeomorphism of a differentiable manifold, so that the derivative [math]\displaystyle{ f_n^\prime }[/math] is defined, then one says that a periodic point is hyperbolic if
- [math]\displaystyle{ |f_n^\prime|\ne 1, }[/math]
that it is attractive if
- [math]\displaystyle{ |f_n^\prime|\lt 1, }[/math]
and it is repelling if
- [math]\displaystyle{ |f_n^\prime|\gt 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.
The logistic map
[math]\displaystyle{ x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4 }[/math]
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). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value [math]\displaystyle{ \tfrac{r-1}{r} }[/math] is an attracting periodic point of period 1. With r greater than 3 but less than [math]\displaystyle{ 1 + \sqrt 6, }[/math] 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 [math]\displaystyle{ \tfrac{r-1}{r}. }[/math] 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).
Dynamical system
Given a real global dynamical system [math]\displaystyle{ (\R, X, \Phi), }[/math] with X the phase space and Φ the evolution function,
- [math]\displaystyle{ \Phi: \R \times X \to X }[/math]
a point x in X is called periodic with period T if
- [math]\displaystyle{ \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]\displaystyle{ \Phi(t,x) = \Phi(t+T,x) }[/math] for all t in [math]\displaystyle{ \R. }[/math]
- Given a periodic point x then all points on the orbit γx through x are periodic with the same prime period.
See also
- Limit cycle
- Limit set
- Stable set
- Sharkovsky's theorem
- Stationary point
- Periodic points of complex quadratic mappings
 |  |
