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Poincare-Bendixson in n-dimensions

  1. Dec 8, 2013 #1
    Suppose that [itex]\dot{x}[/itex] = f(x) in [itex]\Re[/itex][itex]^{n}[/itex]. There exists a bounded 2D invariant manifold M for this system. There are no critical points in M. Does it follow that there is a periodic orbit in M?

    I've realized that this has it's similarities to Poincare-Bendixon's theorem but I don't believe that it holds for a system in n dimensions. Would I be correct in thinking that even with the strict conditions above we could end up with chaos? If I am correct here, does anybody know of any counterexamples for the above statement? Thanks.
     
  2. jcsd
  3. Dec 8, 2013 #2

    pasmith

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    No, as for example if [itex]M = S^1 \times S^1[/itex] (the 2-torus) and the reduction of the system on [itex]M[/itex] is (after rescaling time)
    [tex]
    (\dot \theta_1, \dot \theta_2) = (1, \alpha)
    [/tex]
    with irrational [itex]\alpha[/itex]. All trajectories are then dense in [itex]M[/itex], and there are no periodic orbits.

    No: three dimensions are necessary for chaos, so the behaviour of the system on M cannot be chaotic.
     
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