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11. Consider the equation dy/dt = ay - y³ = y(a − y²). (a) Again consider the cases a < 0, a = 0, and a > 0. In each case, find the critical points, draw the phase line, and determine whether each critical point is asymptotically stable, semistable, or unstable. (b) In each case, sketch several solutions of Eq. (ii) in the ty-plane. (c) Draw the bifurcation diagram for Eq. (iii), that is, plot the location of the critical points versus a. For Eq. (iii), the bifurcation point at a = 0 is called a pitchfork bifurcation; your diagram may suggest why this name is appropriate.

11. Consider the equation dy/dt = ay - y³ = y(a − y²). (a) Again consider the cases a < 0, a = 0, and a > 0. In each case, find the critical points, draw the phase line, and determine whether each critical point is asymptotically stable, semistable, or unstable. (b) In each case, sketch several solutions of Eq. (ii) in the ty-plane. (c) Draw the bifurcation diagram for Eq. (iii), that is, plot the location of the critical points versus a. For Eq. (iii), the bifurcation point at a = 0 is called a pitchfork bifurcation; your diagram may suggest why this name is appropriate.

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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11. Consider the equation dy/dt = ay - y³ = y(a − y²). (a) Again consider the cases a < 0, a = 0, and a > 0. In each case, find the critical points, draw the phase line, and determine whether each critical point is asymptotically stable, semistable, or unstable. (b) In each case, sketch several solutions of Eq. (ii) in the ty-plane. (c) Draw the bifurcation diagram for Eq. (iii), that is, plot the location of the critical points versus a. For Eq. (iii), the bifurcation point at a = 0 is called a pitchfork bifurcation; your diagram may suggest why this name is appropriate.
Transcribed Image Text:11. Consider the equation dy/dt = ay - y³ = y(a − y²). (a) Again consider the cases a < 0, a = 0, and a > 0. In each case, find the critical points, draw the phase line, and determine whether each critical point is asymptotically stable, semistable, or unstable. (b) In each case, sketch several solutions of Eq. (ii) in the ty-plane. (c) Draw the bifurcation diagram for Eq. (iii), that is, plot the location of the critical points versus a. For Eq. (iii), the bifurcation point at a = 0 is called a pitchfork bifurcation; your diagram may suggest why this name is appropriate.
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