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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.

Advanced Engineering Mathematics
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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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.
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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