Question 1 For the canonical form of the pitchfork bifurcation where μ is a real parameter d -x(t) = µx − x³, dt determine the local stability of any equilibria, and sketch the bifurcation diagram. Next, consider the function sin(x) f(x,a) = = a x First, set f(a, 0) = a - 1 and show that x→0 Then, consider the differential equation lim ƒ(a,x) = a − 1. - d dt³(t) = a. sin(y) - = f(y,a) y where a is a real parameter. Find the bifurcation point (y*, a*), sketch the bifurcation diagram, and show that the differential equation for y is similar to the differential equation for x by Taylor expanding f(y,a) near the bifurcation point.
Question 1 For the canonical form of the pitchfork bifurcation where μ is a real parameter d -x(t) = µx − x³, dt determine the local stability of any equilibria, and sketch the bifurcation diagram. Next, consider the function sin(x) f(x,a) = = a x First, set f(a, 0) = a - 1 and show that x→0 Then, consider the differential equation lim ƒ(a,x) = a − 1. - d dt³(t) = a. sin(y) - = f(y,a) y where a is a real parameter. Find the bifurcation point (y*, a*), sketch the bifurcation diagram, and show that the differential equation for y is similar to the differential equation for x by Taylor expanding f(y,a) near the bifurcation point.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 54E: Plant Growth Researchers have found that the probability P that a plant will grow to radius R can be...
Question
![Question 1
For the canonical form of the pitchfork bifurcation where μ is a real parameter
d
-x(t) = µx − x³,
dt
determine the local stability of any equilibria, and sketch the bifurcation diagram.
Next, consider the function
sin(x)
f(x,a) =
= a
x
First, set f(a, 0) = a - 1 and show that
x→0
Then, consider the differential equation
lim ƒ(a,x) = a − 1.
-
d
dt³(t) = a.
sin(y)
-
= f(y,a)
y
where a is a real parameter.
Find the bifurcation point (y*, a*), sketch the bifurcation diagram, and show that the
differential equation for y is similar to the differential equation for x by Taylor expanding
f(y,a) near the bifurcation point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d8b8c35-fcea-418b-8926-d475acc1096c%2F143d1182-e61b-47c1-8d3f-5690d81701d2%2Fj8lva5e_processed.png&w=3840&q=75)
Transcribed Image Text:Question 1
For the canonical form of the pitchfork bifurcation where μ is a real parameter
d
-x(t) = µx − x³,
dt
determine the local stability of any equilibria, and sketch the bifurcation diagram.
Next, consider the function
sin(x)
f(x,a) =
= a
x
First, set f(a, 0) = a - 1 and show that
x→0
Then, consider the differential equation
lim ƒ(a,x) = a − 1.
-
d
dt³(t) = a.
sin(y)
-
= f(y,a)
y
where a is a real parameter.
Find the bifurcation point (y*, a*), sketch the bifurcation diagram, and show that the
differential equation for y is similar to the differential equation for x by Taylor expanding
f(y,a) near the bifurcation point.
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