b Verify that a transformation of the independent variable from a to y, defined by (1-cos(Ty)), x = for the case a = 4 is a bijective relation [0, 1[↔ [0, 1[ that transforms the logistic map to the tent map, 0 ≤ y ≤ 1/2, 2-2y 1/2 ≤y<1. 0 3+ y² = {20
b Verify that a transformation of the independent variable from a to y, defined by (1-cos(Ty)), x = for the case a = 4 is a bijective relation [0, 1[↔ [0, 1[ that transforms the logistic map to the tent map, 0 ≤ y ≤ 1/2, 2-2y 1/2 ≤y<1. 0 3+ y² = {20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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plz solve question (b) with explanation within 30-40 mins and get multiple upvotes
![2.2
Logistic map
map [0, 1[-> [0, 1(, х ня ' %3D ах(1 — х),
a € [0, 4], is called the logistic map.
The
a
Find the fixed points of this map as a
function of the parameter a and calculate
the corresponding Lyapunov exponents.
a
4
b
Verify that a transformation of the independent
variable from to y, defined by
1
(1–cos(7y)),
x =
1 x
for the case a = 4 is a bijective relation [0, 1[→ [0, 1[ that transforms the logistic map to
the tent map,
S 2y
(2 – 2y 1/2 < y < 1.
0<y< 1/2,
Y + y' =
Using the result of part b, calculate the invariant measure for the logistic map, i.e., the
probability density function p(x) that remains unchanged under the action of the map,
p(x'(x)) = p(x). Hint: The invariant density p(y) for the tent map is simple and easily
guessed. Then apply the transformation y + x to this density.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F207efe14-1e58-40f9-9899-be2d1f036ec0%2F815e6065-f238-4a5f-bc9a-295944999e1b%2Fss0h9fp_processed.png&w=3840&q=75)
Transcribed Image Text:2.2
Logistic map
map [0, 1[-> [0, 1(, х ня ' %3D ах(1 — х),
a € [0, 4], is called the logistic map.
The
a
Find the fixed points of this map as a
function of the parameter a and calculate
the corresponding Lyapunov exponents.
a
4
b
Verify that a transformation of the independent
variable from to y, defined by
1
(1–cos(7y)),
x =
1 x
for the case a = 4 is a bijective relation [0, 1[→ [0, 1[ that transforms the logistic map to
the tent map,
S 2y
(2 – 2y 1/2 < y < 1.
0<y< 1/2,
Y + y' =
Using the result of part b, calculate the invariant measure for the logistic map, i.e., the
probability density function p(x) that remains unchanged under the action of the map,
p(x'(x)) = p(x). Hint: The invariant density p(y) for the tent map is simple and easily
guessed. Then apply the transformation y + x to this density.
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