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.




Contents






  • 1 Iterated functions


    • 1.1 Examples




  • 2 Dynamical system


    • 2.1 Properties




  • 3 See also





Iterated functions


Given an endomorphism f on a set X


f:X→X{displaystyle f:Xto X}f:Xto X

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


 fn(x)=x{displaystyle f_{n}(x)=x} f_{n}(x)=x

where fn{displaystyle f_{n}}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)}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 }}f_{n}^{prime } is defined, then one says that a periodic point is hyperbolic if


|fn′|≠1,{displaystyle |f_{n}^{prime }|neq 1,}|f_{n}^{prime }|neq 1,

that it is attractive if


|fn′|<1,{displaystyle |f_{n}^{prime }|<1,}|f_{n}^{prime }|<1,

and it is repelling if


|fn′|>1.{displaystyle |f_{n}^{prime }|>1.}|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


A period-one point is called a fixed point.


The logistic map


xt+1=rxt(1−xt),0≤xt≤1,0≤r≤4{displaystyle x_{t+1}=rx_{t}(1-x_{t}),qquad 0leq x_{t}leq 1,qquad 0leq rleq 4}x_{{t+1}}=rx_{t}(1-x_{t}),qquad 0leq x_{t}leq 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→X{displaystyle Phi :mathbb {R} times Xto X}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)=x{displaystyle Phi (t,x)=x,}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),}Phi (t,x)=Phi (t+p,x), for all t in R

  • Given a periodic point x then all points on the orbit γx{displaystyle gamma _{x}}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.







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