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Dynamical systems

  1. Oct 20, 2008 #1
    (1) Show that any C1 vector Field on S2 (the torus) possesses at least one singularity.

    (2)Show that any isolated periodic orbit T of a C1
    planar vector field X is a limit cycle.

    Any help/suggestions are appreciated.
     
  2. jcsd
  3. Aug 21, 2009 #2

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    (1) There are C^1 vector fields on the torus without singularities. You must be omitting something.

    (2) Since the orbit is periodic it is a cycle, and since it is isolated it must be a limit cycle.
     
  4. Aug 21, 2009 #3
    S2 usually denotes a 2-sphere rather than a torus.
     
  5. Aug 21, 2009 #4

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    You're right... Johnson must have taken the liberty of denoting the Cartesian product [tex]S^2 := S\times S[/tex] for the torus, which is OK set-theoretically, but goes against standard notation.
     
    Last edited: Aug 21, 2009
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