Periodic point

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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[edit]

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.

Examples[edit]

Dynamical system[edit]

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[edit]

  • 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[edit]

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).

See also[edit]

This article incorporates material from hyperbolic fixed point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.