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Bifurcation Diagram for a Generalized Logistic Map

​
leftmost value of λ,
λ
left
2
unimodality, z
2
number of iterations, n
280
number of iterations to be dropped,
n
drop
100
zoom-in level, ζ
3
This Demonstration shows a bifurcation diagram for a generalized logistic map,
f(
x
n
)=
x
n+1
=
λ
2
(1-|
x
n
z
|
)-1
[1–7]. This map is very well-suited for numerical analysis because:
1. The basin of attraction for an attracting set is strictly confined within
|x|≤1
for any initial value
x
0
and for any parameter values
λ>0
and
z>0
.
2. The basin of attraction for an attracting set abruptly vanishes at
λ=4
for any value of
z
, that is, "all boundary crises occur at
λ=4
."
3. Since the function
f(x)
is symmetric around
x=0
, this map is particularly convenient for renormalization group analysis.
The blue box on the left is the locator where the image inside the box is rescaled on the right in accordance with the zoom-in level
ζ
. By dragging the locator, you can change where to zoom in.
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