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Proving that three closed orbits must contain a fixed point

  1. Oct 25, 2016 #1
    A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C1 and C2 lie inside the third cycle C3. However C1 does not lie inside C2, nor vice-versa.

    What is the configuration of the orbits?
    Show that there must be at least one fixed point bounded by C1, C2, and C3.


    I'm having trouble with this problem. I know that any closed orbit must enclose a number of fixed points such that the total index is 1 (also known as winding number). Therefore:

    I1 = I2 = I3 = 1

    where Ik is the index of Ck.

    I = (2π)-1∫dθ

    I don't exactly see how there can be a fixed point that is bounded by all *three* of the fixed points. Since C1 and C2 are not contained with in each other, the only way a fixed points could be bounded by both of them would be if they overlapped somewhat like a Venn Diagram, but I didn't think it was permissible for orbits to cross each other like that (?).
     
  2. jcsd
  3. Oct 31, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Oct 31, 2016 #3

    mfb

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    Staff: Mentor

    I don't see that either. Maybe it is three separate problems?

    Show that there is at least one fixed point bounded by C1.
    Show that there is at least one fixed point bounded by C2.
    Show that there is at least one fixed point bounded by C3.

    That would be trivial, but what about this?

    Show that there is at least one fixed point bounded by C3 but not bounded by C1 or C2.
     
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