Dynamical systems

1. Oct 20, 2008

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.

2. Aug 21, 2009

Reb

(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.

3. Aug 21, 2009

Phrak

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

4. Aug 21, 2009

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.

Last edited: Aug 21, 2009