Singularities & Limit Cycles of C1 Vector Fields on S2

[johnson123]

(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.

 

[Reb]

(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.

 

[Phrak]

S² usually denotes a 2-sphere rather than a torus.

 

[Reb]

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.