In the logistic map

X_{n+1} = f(X_{n}) = AX_{n}(1
- X_{n}),

a superstable three-cycle occurs for f(f(f(X))) = X and df(f(f(X)))/dX = 0. A bit of messy algebra leads to the polynomial equation

g(A) = A^{7} - 8A^{6} + 16A^{5} + 16A^{4} - 64A^{3} + 128 = 0,

which can be solved by Newton's method

which can be solved by Newton's method

A_{n+1} = A_{n} - g(A_{n})/g'(A_{n}),

where

Starting from a first guess of A_{1}
= 3.8, the solution converges to A =
3.83187405528331557... after about
80 iterations. The PowerBASIC source and executable
code are available.

where

g'(A) = dg(A)/dA = 7A^{6} - 48A^{5} + 80A^{4} + 80A^{3}- 192A^{2}.

Starting from a first guess of A

Back to Sprott's Technical Notes