Singularities & Limit Cycles of C1 Vector Fields on S2

johnson123
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(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.
 
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johnson123 said:
(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.


(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.
 
S2 usually denotes a 2-sphere rather than a torus.
 
You're right... Johnson must have taken the liberty of denoting the Cartesian product S^2 := S\times S for the torus, which is OK set-theoretically, but goes against standard notation.
 
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