Let $\mathcal{C}^1$ be the space of all differentiable vector fields $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. We consider the associated dynamical system $$\dot{x}=F(x)$$
A closed and invariant set $A \subset \mathbb{R}^n$ is called an attractor if there exists an neighborhood $U \subset \mathbb{R}^n$ such that
- $U$ is forward invariant: $\Phi(U,t) \subset U \quad t \geq 0$ (where $\Phi$ is the flow associated to $F$)
- The limiting set of $U$ is the attractor $A=\omega(U \cap \mathbb{R}^n)$
Let $\mathcal{M} \subset \mathcal{C}^1$ be the set of all vector fields which only have one attractor.
My question is the following: Can we prove that $\mathcal{M}$ is not dense in $\mathcal{C}^1$?