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Math Help - Help needed understanding the steps in this Poincare map example

  1. #1
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    Help needed understanding the steps in this Poincare map example

    Take a system of equations:
    \dot{r}=r(1-r^2)
    \dot{\theta}=1

    Let S be the positive x-axis and r_0 is an initial condition on S.

    Since \dot{\theta}=1 the first return to S occurs at t=2\pi.
    Then r_1=P(r_0) where r_1 satisfies
    \int_{r_0}^{r_1}\frac{dr}{r(1-r^2)} = \int_0^{2\pi}dt = 2\pi \Rightarrow r_1

    Hence P(r)=[1+e^{-4\pi}(r^{-2}-1)]^{-\frac{1}{2}}

    I don't understand how the "hence" bit is derived. I've spent a fair amount of time googling, and I believe partial fractions may hold part of the solution, but honestly I'm a little lost...

    Many thanks!!!
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  2. #2
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    The way a Poincare map works is by taking pictures, once per "period", of phase space. In this case, what you really have is the following relation:

    \int_{r_{n}}^{r_{n+1}}\frac{dr}{r(1-r^{2})}=2\pi.

    Moreover, the Poincare map looks like the following: r_{n+1}=P(r_{n}).

    So, perform the integral I just mentioned, and solve the equation for r_{n+1} as a function of r_{n}. That, I think, will get you the "hence" part.
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  3. #3
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    Thanks! That was very helpful.

    I've managed to work through the problem and get the result given - or rather, I worked forward from the integral (I had to "cheat" and use Maple - clearly a bit of calculus revision is in order), and back from the "Hence P(r)=" bit, and met in the middle. But at least I now can see the method even if my integration skills are rusty, and that's better than being stuck on a page of notes thinking "what the hell?!?" I'm not easily able to leave something I don't understand and just continue, and this had me stuck for a day and a half.
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