Is there a $C^1$ flow $\varphi\_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim\_{t\rightarrow +\infty}\varphi\_t(x) = 0,$$
$$\lim\_{t\rightarrow -\infty}\varphi\_t(x) = 0?$$


I think that this should not exist but I can't find a simple argument to rule out the existence of such a flow. Same question for a $C^0$ flow.



Planar flow with bounded orbits and a single equilibrium point
(*edited to include Willie Wong's idea for $C^0$ case.*)


This kind of flow can't exist in any dimension.


Let $S$ be the unit sphere and $B$ be the open unit ball. If the origin is a global attractor for $\varphi$, then $S \subset \bigcup\limits\_{t>0}{\varphi\_{-t}(B)}$.


By compactness, $S$ is covered by a union of a finite subset of the $\varphi\_{-t}(B)$. That implies that there is a constant $T$ such that no point on $S$ (or $\overline{B}$) flows for more than time $T$ outside $\overline{B}$.


Since the image of $[0, T] \times \overline{B}$ under $\varphi$ is compact, it can't cover the whole space, so the origin is not a global attractor for $\varphi^{-1}$.


