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Homework Help: Need help with Bifurcation problem!

  1. Apr 6, 2008 #1
    Need urgent help with Bifurcation problem!

    1. The problem statement, all variables and given/known data
    I'm stuck at part (ii) of this question! The question is as follows:

    The differential equation is dy/dt = y^2 + (µ + 1)*y.
    (i) Using µ = 0 sketch the phase line. Repeat for µ = -1 and µ = -2.
    (ii) Calculate the position & type of equilibria as a function of µ and hence sketch the bifurcation diagram.


    2. Relevant equations



    3. The attempt at a solution

    For µ = 0, I got 2 equil. solutions at y = 0 and y = -1 where y = 0 is a source and y = -1 is a sink.
    For µ = -1, I got 1 equil. solution at y = 0 which is a source.
    For µ = -2, I got 2 equil. solutions at y = 0 which is a sink and y = 1 which is a source.

    The phase lines for both µ = 0 and µ = -2 are identical so I know that there is a qualitative change at µ = -1 (between µ = 0 and µ = -2)

    dy/dt = y^2 + (µ+1)*y

    taking the RHS to 0
    y^2 + (µ+1)*y = 0
    y(y+µ+1) = 0
    y = 0 or y = -1-µ

    I am not sure about the y values that I've got above (y=0 and y=-1-µ) and I am pretty much stuck at this point. I have no idea as what to do next after this. Please help. Thanks!
     
  2. jcsd
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