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
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
- 8
A
6
+ 16
A
5
+ 16
A
4
- 64
A
3
+ 128 = 0,
which can be solved by Newton's method
A
n
+1
=
A
n
-
g
(
A
n
)/
g
'(
A
n
),
where
g
'(
A
) =
dg
(
A
)/
dA
= 7
A
6
- 48
A
5
+ 80
A
4
+ 80
A
3
- 192
A
2
.
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.
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