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Math Help - Periodic solution

  1. #1
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    Periodic solution

    Use the Poincaré-Bendixon theorem to show that the following differential equations
    x_1' = 2x_1 - 2x_2 -x_1(x_1^2+x_2^2)
    x_2' = 2x_1 + 2x_2 -x_2(x_1^2+x_2^2)


    have a nontrivial periodic solution.

    Actually I would liek to solve the problem myself, but I don't really understand how this works. Anyone has any hints? And can someone summarize this theorem in reall easy words?
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  2. #2
    Super Member Deadstar's Avatar
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    Quote Originally Posted by EinStone View Post
    Use the Poincaré-Bendixon theorem to show that the following differential equations
    x_1' = 2x_1 - 2x_2 -x_1(x_1^2+x_2^2)
    x_2' = 2x_1 + 2x_2 -x_2(x_1^2+x_2^2)


    have a nontrivial periodic solution.

    Actually I would liek to solve the problem myself, but I don't really understand how this works. Anyone has any hints? And can someone summarize this theorem in reall easy words?
    Quick summarize...

    You first have to show that there is a trapping region. Do this by setting r^2 = x_1^2 + x_2^2.

    => 2rr' = 2x_1x_1' + 2x_2x_2' then sub in x_1' and x_2'. Show that a disk of radius r is a trapping region by showing that r' <= 0...

    Then the theorem says if there are no critical points within this trapping region (or if there is either an unstable spiral or unstable node) there will be a limit cycle (periodic solution). So just show one of those is true.
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