Attracting periodic point

Periodic point

In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which returns to itself 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 function with period n.

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,

and 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

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

See also

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