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Consider the equation u' = (1 – b)u – au³ where a and b are fixed positive constants and 1 is a parameter that is allowed to vary. %3D (a) Show that if 1 < b then there is a single equilibrium that is asymptotically stable. (b) In the case where A > b, find all equilibria and determine their stability. (c) Sketch the bifurcation diagram.

Consider the equation u' = (1 – b)u – au³ where a and b are fixed positive constants and 1 is a parameter that is allowed to vary. %3D (a) Show that if 1 < b then there is a single equilibrium that is asymptotically stable. (b) In the case where A > b, find all equilibria and determine their stability. (c) Sketch the bifurcation diagram.

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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Answer all parts of this DE problem please

Consider the equation u' = (1 – b)u – au where a and b are fixed positive constants and A is a parameter that is allowed to vary. - (a) Show that if 1 < b then there is a single equilibrium that is asymptotically stable. (b) In the case where A > b, find all equilibria and determine their stability. (c) Sketch the bifurcation diagram.
Transcribed Image Text:Consider the equation u' = (1 – b)u – au where a and b are fixed positive constants and A is a parameter that is allowed to vary. - (a) Show that if 1 < b then there is a single equilibrium that is asymptotically stable. (b) In the case where A > b, find all equilibria and determine their stability. (c) Sketch the bifurcation diagram.
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