The resulting iterated map is thus
= 5.97811605717192 tanh(1.42668906710772xn) - 5.32641819604181xn
Note that the linear growth rate f
'(0) = ab
= 3.20249462462648... in
contrast to the logistic map for which f
'(0) = 4.
The computer program that was used to calculate these values is
available in PowerBASIC source
code, and the resulting map (in cyan) is compared with the logistic map
(in black) below:
The system has a Lyapunov exponent (base e
) of 0.6907063..., which is
slightly smaller than the logistic map for which the Lyapunov exponent
is ln(2) = 0.693147181... but larger than the sine map for which the
Lyapunov exponent is 0.689067... (see J. C. Sprott, Chaos and Time-Series Analysis
, Oxford, 2003).
A least squares fit of the four coefficients to 1000 iterates of the
logistic map gives the slightly different mapping:
= 5.821 tanh(1.487xn
- 23.942 tanh(0.2223xn
with a mean square error of 3.249 × 10-4, although it
is still slowly converging as the second term approaches ever more
closely a linear function.