Deterministic Chaos


How can we locate the points belonging to an n-cycle?
For example, consider the 2-cycle consisting of the points x = a and x = b.
From the graph it is easy to see T(a) = b and T(b) = a. Consequently,
T(T(a)) = T(b) = a   and   T(T(b)) = T(a) = b
That is,
T2(a) = a   and   T2(b) = b
In other words, if x = a and x = b belong to a 2-cycle for T(x), both are fixed points for T2(x) = T(T(x)).
This is useful, because we know how to locate graphically the fixed points of any function: look for the intersections of the graph of T2(x) (purple below) with the diagonal.
Click the picture to animate.
Note the fixed points of T(x) also are fixed points of T2(x).
This is hardly a surprise:
if T(c) = c, then T2(c) = T(T(c)) = T(c) = c.
This observation has obvious generalizations. For example, the points of a 4-cycle of T(x) are fixed points of T4(x). But the fixed points of T4(x) include also fixed points of T(x) and the points of 2-cycles of T(x).
Do you see the general pattern?

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