1-periodic, bounded vector fields on the plane
in [Math]
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From: LordBeotian on 12 May 2010 04:12 I think that the only 1-periodic, C^infty and uniformly bounded vector field on the plane is the zero vector field. By 1-periodic I mean that all the orbit returns to the starting point after time 1. While trying to prove this statement I'm not really satisfied of my argumets, so I hope something more linear can be found as a proof. My Idea is the following: suppose there is a point with nonzero vector field, then there is a closed orbit C starting from it. Take a reference point p inside the orbit, I want to prove that any external point x has an orbit which has a winding number wrt p which is the same as C (this wuould prove the statement since imply the existence of arbitrary long orbit which is impossible for a 1-periodic and bounded vector field). To prove that the winding number is the same as C I take a segment which connects C with x parametrized with f(t), t in [0,1]. The winding number of the orbit starting from f(t) must be constant unless there is a first point f(t*) where the vector field vanishes. If the vector field vanishes on f(p) then I take a neighbourhood B of f(t*) and I consider the time T(y) needed to exit the neighbourhood for the orbit which starts from y in B. T is well defined for f(t) and t<t*. The function T(f(t*)) have to diverge for t- >t* (actually I need an argument for this but it seems reasonable) but this is impossible because the vector field is 1-periodic. Is there a more strightforward way to prove the statement? |