- #1
Questioneer
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Homework Statement
How do I find periodic points of a given function? I'm looking at discrete cases only (iterations of the function).
Homework Equations
A point is defined to be a periodic point of period n if f^n(x)=x, where f^n(x) is defined recursively as f(f^n-1(x)). [If this is not clear, Wolfram has a great description of fixed points, which is highly connected to this discussion].
The Attempt at a Solution
You can always look at intersections of f(x)=id(x) <-- the identity function to find the fixed points. The easiest way to find periodic points is to find intersections of f^2(x)=id(x) for all of period 2. obviously, I can't graph f^n(x)=id(x) and calculate an intersection.
As a good example to see where this comes from, Devaney pg. 38 #3
Sketch the graph of the tent map
T_2(x)={ 2x 0<=x<=1/2
{ 2-2x 1/2<=x<=1
on the unit interval. Use the graph of (T_2)^n(x) to conclude that T_2 has exactly 2^n periodic points of period n.
Thanks in advance.