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 an endomorphism f on a set X

$f: X \to X$

a point x in X is called periodic point if there exists an n so that

$\ f_n(x) = x$

where $f_n$ 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

$f_n(x) = f_m(x)$

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 $f_n^\prime$ is defined, then one says that a periodic point is hyperbolic if

$|f_n^\prime|\ne 1,$

that it is attractive if

$|f_n^\prime|< 1,$

and it is repelling if

$|f_n^\prime|> 1.$

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.

Dynamical system

Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function,

$\Phi: \mathbb{R} \times X \to X$

a point x in X is called periodic with period t if there exists a t > 0 so that

$\Phi(t, x) = x\,$

The smallest positive t with this property is called prime period of the point x.

Properties

• Given a periodic point x with period p, then $\Phi(t,x) = \Phi(t+p,x)\,$ for all t in R
• Given a periodic point x then all points on the orbit $\gamma_x$ through x are periodic with the same prime period.

Examples

The logistic map

$x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4$

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 (r-1)/r is an attracting periodic point of period 1. With r greater than 3 but less than 1 + √6, 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 (r-1)/r and a non-attracting period-2 cycle between two periodic points. 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).