Given a compact manifold $M$ and a vector field $F:M \rightarrow TM,F \in \mathcal{X}^1(M)$, one can define an assoicated dynamical system by $$\dot{x}=F(x)$$
We call two vector fields $F_1,F_2$ topologically equivalent if there exists a homemorphism $ h: M \rightarrow M$ mapping orbits of $F_1$ to orbits of $F_2$ preserving the temporal order. By $[F]$ we denote the equivalence classes.
My question is the following: Is there any known bound on the $\mathcal{C^1}$-norm between vector fields in the equivalence class, i.e. topologically equivalent vector fields?
Is there a known result on an $\varepsilon$ (depending on $F_1,F_2$) such that $$\forall F_1,F_2 \in [F]: \quad \|F_1-F_2\|_{\mathcal{C}^1(M) }\leq \varepsilon$$