Deterministic Chaos

6.F. Cycles

An n-cycle for a function f(x) is a collection of n numbers, x1, x2, ..., xn related in this way

f(x1) = x2,   f(x2) = x3,   ...,   f(xn-1) = xn,   and   f(xn) = x1

The reason for calling this a cycle should be clear: iteration takes us from x1 to x2, from x2 to x3, ..., and from xn back to x1.

Cycles and graphical iteration. Cycles and fixed points. Stability of cycles A theoretical analysis of the logistic map cycles

Here is an application of counting cycles to a problem in number theory.

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