Binary Coding Functions for Generalized Logistic Maps with zUnimodality
Binary Coding Functions for Generalized Logistic Maps with zUnimodality
This Demonstration shows binary coding functions for onedimensional iterative maps with unimodality [1]. The test map, ()==λ(1)21, generalizes the classic logistic map, ()==λ(1) [12–15]. Here is the iteration number, is the iterate of () starting from the initial value (i.e. =∘∘⋯∘()=()), is the main control parameter, and is the subcontrol parameter ( determines the unimodality, the degree of the local maximum of ()).
z
f
GLM
x
n
x
n+1
x
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z

f
LM
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x
n+1
x
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x
n
n
x
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th
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GLM
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0
x
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GLM
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GLM
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GLM
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(n)
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GLM
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0
λ
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z
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GLM
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n
This Demonstration uses one of three coding functions
1
C
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0
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∑
i=0
A
σ
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i
(i+1)
2
2
C
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0
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∑
i=0
A
σ
i
∏
j=0
B
σ
x
j
(i+1)
2
3
C
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0
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∑
i=0
A
σ
i
(1)
i
∏
j=0
B
σ
x
j
(i+1)
2
where (x) is a unitstep function satisfying
A
σ
A
σ
0  if  x<0 
1  if  x≥0 
and (x) is another step function (a sign function) satisfying
B
σ
B
σ
A
σ
1  if  x<0 
+1  if  x≥0 
The blue box on the left is a Locator that you can drag; the image inside the box is rescaled on the right in accordance with the zoomin level . The binary coding function () (, , or ) acts on 251 (or 2001) equally spaced initial condition for both plots.
ζ
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C
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0
k=1
2
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0