My goal is to proof the following:
Theorem: Let $(M,\mathbb{R},F)$ be a DS on a smooth (compact) manifold $M$, with dynamical law $\dot{x}=F(x)$. Assuming it has two attractors $A_1,A_2$ with basin of attraction $B(A_1), B(A_2)$. There are infinitely many DS with the same fixed attractor $A_1$ but an arbitrary dynamics on the attractor $A_2$.
proof
Both basins of attraction are by definition open sets. Hence, the complement $C_1=M \backslash B(A_1) \underset{closed}{\subset}M$. Since every closed subset of a compact set is compact and $B(A_2) \subset C_1$ we can choose a bump function $\sigma_{C_1}:C_1 \rightarrow \mathbb{R}$ which is 1 on $B(A_2)$ and zero outside of $C_1$.
Now we find that $$\dot{x}=F(x)+\sigma_{C_2} \cdot G(x)$$ fullfils $\dot{x}=F(x)$ on $B(A_1)$ but on $B(A_2)$ it fullfils $\dot{x}=G(x)+F(x)$ with an arbitrary function $G$. Since $G$ is an arbitrary function the theorem is proven. (We implicitly used that disjoint attractors have disjoint basins of attraction)
I am not sure if the proof is right. There are many DS with fractal basin of attractions. If $B(A_1)$ and $B(A_2)$ are both fractal, in my intuition the proof cannot work. Since there should be no smooth bump function on the complement of a fractal set. Additionally fractals are closed sets being in contradiction with the definition of a basin of attraction.