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Van der Pol oscillator + Hopf bifurcation

  1. Nov 27, 2007 #1


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    1. The problem statement, all variables and given/known data
    Consider the biased van der Pol oscillator: [tex]\frac{d^2x}{dt^2}=u (x^2-1)\frac{dx}{dt} + x = a[/tex]. Find the curves in (u,a) space at which Hopf bifurcations occur. (Strogatz 8.21)

    2. Relevant equations

    3. The attempt at a solution
    Not even sure where to start with this question, other than turning it into a 2-d system. Any advice on how to approach this?
  2. jcsd
  3. Jun 2, 2008 #2
    I'm attempting a very similar question. I've managed to turn it into a 2D system but haven't got much further.

    dx/dt = y

    dy/dt = u(x^2 -1)y + x

    A Hopf bifurcation is one where, as you vary u, the eigenvalues of the Jacobian change from having a negative real component to having a positive real component.

    I'll let you know if I make more progress.
  4. Jun 2, 2008 #3
    I've just looked in Strogatz at question 8.21 and I think you've copied out the question incorrectly:
    you wrote an equals sign between d^2x/dt^2 and u(x^2 -1)
    but Strogratz wrote a plus.

    Luckily, this means we're working on exactly the same question.
    The equations for the 2D system are not as written above; insteat they are:

    dx/dt = y

    dy/dt = a - x - u(x^2 -1)y
  5. Jun 2, 2008 #4
    I think I've answered the question. My method was as follows:

    I wrote down the Jacobian and got an expression for its eigenvalues. I then found the fixed points. I found what the eigenvalues are at the fixed points. It's then just a matter of inspecting your solution, looking at the definition of a Hopf bifurcation and seeing what values u and a need to take.

    Best of luck,
    quantum boy
  6. Apr 16, 2010 #5
    i took did the jacobian and got eigenvalues of 0 and u(a^2)-u.....cause i found a fixed point of (a,0) from nullclines....now what do I do...
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