Deterministic Chaos

6.F. Cycles

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

f(x1) = x2

f(x2) = x3

...

f(xn-1) = xn

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 x_n back to x₁.

Cycles and graphical iteration.

Cycles and fixed points.

Stability of cycles

A theoretical analysis of the logistic map cycles

Cycles exhibit the same types of stability as do fixed points.

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

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