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Math Help - Equilibruim points poincare map

  1. #1
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    Equilibruim points poincare map

    Hello,
    I really need help with this problem! Got a big exam tomorrow and this type of problem is one I struggle with the most. Desperate for all any help.

    See attached for problems. A walk through with solutions would really make my year. Thanks to all in advance.
    Attached Thumbnails Attached Thumbnails Equilibruim points poincare map-1.jpg   Equilibruim points poincare map-2.jpg  
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  2. #2
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    I can kinda understand and can figure out the first part a)
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  3. #3
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    no one?
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  4. #4
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    (3). To find the equilibrium points, set:

    \begin{aligned} -y+x(\mu-x^2-y^2)&=0 \\<br />
x+y(\mu-x^2-y^2)&=0\end{aligned}

    Solving those simultaneously, I get the only equilibrium point is the origin. Now, I assume you can linearize that system and calculate the eigenvalues of the linear system. The eigenvalues will tell you if the origin is a sink and from that conclude if the origin is stable.

    Don't know about B.

    For C, solve the equations:

    x=x\sqrt{\frac{\mu}{\mu e^{-4\pi \mu}+x^2(1-e^{-4\pi \mu})}}

    and

    x=x\sqrt{\frac{1}{4\pi x^2+1}}

    If x is limited to the positive x-axis, then the only solution is \sqrt{\mu}

    It can only be periodic if the flow through the Poincare section (the x-axis) returns to the exact same point and that can only happen if some point on the x-axis is a fixed point.

    Also, try and get to Mathematica. In version 7, it has a very nice function called "StreamPlot" which plots effortlessly, the flow of these kinds of systems. For example, the code:

    Code:
    StreamPlot[{-y + x*(\[Mu] - x^2 - y^2), 
        x + y*(\[Mu] - x^2 - y^2)} /. \[Mu] -> 1, 
      {x, -2, 2}, {y, -2, 2}]
    produced the plot below and it suggests that when \mu=1 we have a limit cycle that is the unit circle.
    Attached Thumbnails Attached Thumbnails Equilibruim points poincare map-nonlinearflow.jpg  
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  5. #5
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    this came up in my exam though and i couldnt do it!

    Thanks anyway.
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