Classifying 2-Cycles -- Answers

1. To find 3-cycles of the dynamical system

   p(n + 1) = f(p(n))

   look at solutions of the equation

   p = f(f(f(p))) = p

   that are not equilibrium points.

   To find out if a 3-cycle is attracting look at the derivative of the function

   g(p) = f(f(f(p)))

   at a point in the 3-cycle. If the derivative is less than one in absolute value then the 3-cycle is attracting.

2. To find 4-cycles of the dynamical system

   p(n + 1) = f(p(n))

   look at solutions of the equation

   p = f(f(f(f(p)))) = p

   that are not equilibrium points and are not 2-cycles.

   To find out if a 4-cycle is attracting look at the derivative of the function

   g(p) = f(f(f(f(p))))

   at a point in the 4-cycle. If the derivative is less than one in absolute value then the 4-cycle is attracting.

3. The solution is worked out in the CAS files below.