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ODE system. Limit cycle; Hopf bifurcation.

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter [itex]a[/itex]. What is the nature of the Hopf bifurcation that occurs at the critical value of [itex]a[/itex] and state what the critical value is.

    2. Relevant equations

    [itex]\dot{x}=-y+x(a+x^2+(3/2)y^2)[/itex]

    [itex]\dot{y}=x+y(a+x^2+(3/2)y^2)[/itex]

    3. The attempt at a solution

    By setting each equation to zero, i found the only equilibrium point to be [itex](0,0)[/itex].

    For the Jacobian matrix at [itex](0,0)[/itex], I have:

    [itex]J(0,0)=\left( \begin{array}{cc}
    a & -1\\
    1 & a\end{array} \right)[/itex]

    So:

    [itex]\tau=2a[/itex]

    [itex]\delta=a^2+1>0[/itex]

    [itex]\bigtriangleup=4a^2-4a^2-4=-4<0[/itex]

    which gives:

    [itex]a<0[/itex]: [itex](0,0)[/itex] is an attractor spiral.

    [itex]a>0[/itex]: [itex](0,0)[/itex] is a repellor spiral.

    [itex]a=0[/itex]: [itex](0,0)[/itex] is a center.

    Does this mean that paths spiral into [itex](0,0)[/itex] for negative [itex]a[/itex], and then spiral out towards a stable limit cycle. And, for positive [itex]a[/itex], beyond the limit cycle, paths spiral in towards it.

    How do I find the critical value?
     
  2. jcsd
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