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Homoclinic bifurcation -- Saddle-saddle connection, separatrix

  1. Oct 24, 2015 #1
    1. The problem statement, all variables and given/known data
    The equation:
    eq.png
    where A is the parameter.
    1) Find the value of parameter A at which the homoclinic bifurcation occurs.
    2)Find the separatrices
    3)Calculate the first integral and find the saddle-saddle connection equation.

    2. Relevant equations


    3. The attempt at a solution
    First order system:
    eq2.png
    There are two saddles at (-1,0), (1,0) and a spiral at (0,0). When parameter A is decreased below 0, a stable limit cycle occurs. The limit cycle ceases to exist at approx. A=-0.34. The analytical result turns out to be A=-12/35.

    However, I'm having trouble finding the separatrices. I'm using forward and backward integration to plot many of the solutions in MATLAB and it is possible to notice where the separatrix should be, but I don't know how to plot the exact separatrix. Still working on that.

    Lastly, I tried computing the first integral of the system by dividing (dx/dt) / (dy/dt). After integrating and substituting the saddle values into the equation, I get that the constant of integration equals (-1/4). I then insert this value of the constant into the equation and get this equation:
    eq3.png
    which I'm pretty sure is wrong. Since saddle connecton should happen at around A=-(12/35), I insert this value, but the graphical plot looks like nonsense to me.

    I kindly await any help with questions 2) and 3). Thank you
     
  2. jcsd
  3. Oct 29, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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