Assume I have two $N$-dimensional systems ofODEs
$\frac{dX}{dt}=F(X)$ with $F: \mathbb{R}^N\rightarrow\mathbb{R}^N$
and
$\frac{dY}{dt}=G(Y)$ with $G: \mathbb{R}^N\rightarrow\mathbb{R}^N$.
Also, assume I have a function $V:\mathbb{R}^N\rightarrow \mathbb{R}$ such that $V$ is a complete Lyapunov function for both $F$ as well as $G$ (most importantly $V$ is strictly decreasing on parts of $\mathbb{R}^N$ which are not chain-recurrent, and iff $V(X)=V(Y)$ then $x$ and $y$ are part of the same chain-recurrent set).
What else can we say then about the relationship between $F$ and $G$?
Obviously $F$ and $G$ must have the same chain-recurrent sets, that is they have the same attractor structure. However, the flow of $F$ and $G$ can still be quite different. Intuitively it seems that $F$ and $G$ should be topologically equivalent, but I can't find any proof for that.