с 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.
с 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.
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 (c) 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%2F20b872d3-0f1e-49da-a2a8-aa6123148fbf%2Fd8db2bc3-ef19-49b3-b0ef-3301c1864782%2Fq4zhqni_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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