WOLFRAM|DEMONSTRATIONS PROJECT

High-Precision Newton Algorithm for Generalized Logistic Maps with Unimodality z

​
unimodality control, z
1
leftmost value of λ,
λ
L
2.
number of points for each value of λ, N
50
desired bifurcation order,
r
max
13
precision-control for Newton algorithm, p
40
This Demonstration shows a table of superstable parameter values of a period-doubling periodic attractor. The test map is defined as
f
GLM
(
x
i
)=
x
i+1
=λ(1-|
x
i
z
|
)2-1
, which generalizes the well-known logistic map
f
LM
(
x
i
)=
x
i+1
=λ
x
i
(1-
x
i
)
. Here
i
is an iteration number,
-1≤
x
i
≤1
, and
z
is the unimodality (or the degree) of the local maximum of
f
GLM
(x)
. The superstable parameter values are used for the renormalization group analysis of many low-dimensional dynamical systems with chaotic behavior. See the references [1–4].
The algorithm used is a high-precision Newton algorithm with fixed precision arithmetic. For
z=2
, superstable parameter values for
f
GLM
[x]
are exactly the same as those for
f
LM
[x]
. In this Demonstration, the required number types and their interval ranges are as follows:
1.
z
is a rational number between 1 and 4.
2.
1≤
λ
L
<4
.
3.
N≥0
is an integer, with 0 for the fastest calculation and the poorest visualization; 50 for moderate speed and moderate visualization; and 500 for the slowest speed with good visualization.
(4)
r
max
is an integer,
2≤
r
max
≤13
. You can select higher values manually, but at your own risk. For example, with
r
max
=31
, the author's computer calculated the super-stable parameter values for
z=2
for one week! The minimum precision length used for this calculation was 40.
(5)
p
is an integer used in the fixed precision arithmetic. The Newton algorithm for finding superstable parameter values with a low precision may result in wrong answers. High precision is good but it requires a long calculation time. This Demonstration is designed for
16≤p≤40
.