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Closed Orbit of a Flow on a Manifold

  1. Mar 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Let γ be a closed orbit of the flow φ on the manifold M and suppose there exists T>0 and X0 є γ such that φT(X0) = X0. Prove that φT(X) = X for every X0 є γ. Furthermore locate two closed orbits γ1 and γ2 and positive periods T1 and T2 for the flow of r ̇=r(r-1)(r-2); θ ̇=r^2?



    2. Relevant equations

    Just flow formulae and such

    3. The attempt at a solution

    My thought was that given the closed orbit, we are going to be restricted to the plane of the orbit thus our solution must be some oscillating point for our given T. Thus for any given X subed into our group action (the flow), we are going to yield a result that will as well lie within the plane. I am struggling to understand how to exactly show that the result we get will be the same as X. Ie is φT(X0) = X0 similar to solving a general solution? And for the second part of the problem, with our polar coordinate system, I am completely lost as to how to determine two distinct points. given our proof in the first part my though is that we would have infinitely many parts.....but what do I know..

    Any help would be appreciated THX!!
     
  2. jcsd
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