TITLE: A non vanishing vector field on $S^3$ with a periodic attractor
QUESTION [5 upvotes]: Is there a non vanishing  real analytic vector  field  $X$    on $S^3$ such that  $X$  has  an  attractor  periodic orbit(An asymptotically  stable periodic  orbit)   ? What  about the  smooth case?

REPLY [2 votes]: $S^3$ has the structure of a Lie group. Consider a left invariant vector field $X$. (Check that you can get closed orbits!)  In a tubular neighborhood of a small amount of time around a closed orbit $o$, the vector field looks like a constant field on $\mathbb{R}^3\simeq D^2 \times I$. Consider the functionr $f = \beta\cdot d(\cdot,o)^2$, where $\beta$ is a smooth compactly supported bump function. The choice of unit vector $\nu$ pointing towards the central orbit $o$ from within this neighborhood is clear. Now set $X' = X + \epsilon\cdot f\cdot \nu$ for some choice of $\epsilon$.
For $X'$ we have the old orbit $o$, unperturbed. Moreover, nearby orbits will veer towards $o$. Thus now $o$ is a stable attractor. One checks that with $\epsilon$ small enough, $X'$ is still nonvanishing.