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Locate any bifurcation in 2D system

  1. May 10, 2014 #1
    1. The problem statement, all variables and given/known data

    bifurcation for the following 2D system:

    2. Relevant equations

    x'=ux−y+x^3,y′=bx−y

    3. The attempt at a solution
    I have got ux−y+x^3=0, y=bx, then x=0 and ±sqrt(b−u).

    But I don't how to continue to find the bifurcation?
     
  2. jcsd
  3. May 10, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    What happens when [itex]b - u < 0[/itex]?

    In general, you are looking for values of the parameters for which at least one eigenvalue of [tex]
    \begin{pmatrix}
    \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \\
    \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y}
    \end{pmatrix}
    [/tex] at a fixed point has zero real part.
     
  4. May 10, 2014 #3
    if b-u<0,no critical points exist.
     
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