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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

Given an endomorphism f on a set X

f:X→Xdisplaystyle f:Xto X

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

 fn(x)=xdisplaystyle f_n(x)=x

where fndisplaystyle 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 exist distinct n and m such that

fn(x)=fm(x)displaystyle 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 fn′displaystyle f_n^prime is defined, then one says that a periodic point is hyperbolic if

|fn′|≠1,neq 1,

that it is attractive if

|fn′|<1,displaystyle

and it is repelling if

|fn′|>1.>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

A period-one point is called a fixed point.

The logistic map

xt+1=rxt(1−xt),0≤xt≤1,0≤r≤4displaystyle x_t+1=rx_t(1-x_t),qquad 0leq x_tleq 1,qquad 0leq rleq 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. 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 (R, X, Φ) with X the phase space and Φ the evolution function,

Φ:R×X→Xdisplaystyle Phi :mathbb R times Xto X

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

Φ(t,x)=xdisplaystyle 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 Φ(t,x)=Φ(t+p,x)displaystyle Phi (t,x)=Phi (t+p,x), for all t in R
  


• Given a periodic point x then all points on the orbit γxdisplaystyle gamma _x 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

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