Superstable 3-Cycle in the Logistic Map

J. C. Sprott

Department of Physics, University of Wisconsin, Madison, WI 53706, USA
September 18, 2008

In the logistic map

Xn+1 = f(Xn) = AXn(1 - Xn),

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) = A7 - 8A6 + 16A5 + 16A4 - 64A3 + 128 = 0,

which can be solved by Newton's method

An+1 = An - g(An)/g'(An),

where

g'(A) = dg(A)/dA = 7A6 - 48A5 + 80A4 + 80A3- 192A2.

Starting from a first guess of A1 = 3.8, the solution converges to A = 3.83187405528331557... after about 80 iterations. The PowerBASIC source and executable code are available.



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