Periodic point explained

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 from a set into itself,

f: X \to X, a point in is called periodic point if there exists an >0 so thatf _{n(x)} = x where is the th iterate of . The smallest positive integer satisfying the above is called the prime period or least period of the point . If every point in is a periodic point with the same period, then is called periodic with period (this is not to be confused with the notion of a periodic function).

If there exist distinct and such that

f _{n(x)} = f _{m(x)} then is called a preperiodic point. All periodic points are preperiodic.f ^{\prime} _{n} is defined, then one says that a periodic point is hyperbolic if|f ^{\prime|\ne} _{n} 1 ,

that it is attractive if

|f ^{\prime|<} _{n} 1 ,

and it is repelling if

|f ^{\prime|>} _{n} 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.

$$x\_=rx\_t(1-x\_t),\; \backslash qquad\; 0\; \backslash leq\; x\_t\; \backslash leq\; 1,\; \backslash qquad\; 0\; \backslash leq\; r\; \backslash leq\; 4$$

exhibits periodicity for various values of the parameter . For between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence which attracts all orbits). For between 1 and 3, the value 0 is still periodic but is not attracting, while the value

\tfrac{r -1 }{r} is an attracting periodic point of period 1. With greater than 3 but less than 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 \tfrac{r -1 }{r}. As the value of parameter rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of 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 with the phase space and the evolution function,

\Phi: \R x X \to X a point in is called periodic with period if\Phi(T, x) = x The smallest positive with this property is called prime period of the point .

Properties

• Given a periodic point with period, then

\Phi(t,x) = \Phi(t + T,x) for all in

• Given a periodic point then all points on the orbit through 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