I've stumbled accros an exercise that says.
Consider $x'=f(x),f\in C^1$ in the plane, $\phi(t,x)$ the corresponding dynamical system, defined for $t\in \mathbb{R},x\in\mathbb{R}^2$. Suppose there exists $X\subset\mathbb{R}^2$ such that $\omega(x)=X, \, \forall x\in X$.
Show that $X$ has only equilibrium points, that is $f\vert_X=0$.
At first, I cannot understand this intuitively. For instance, the simple oscillator has only closed orbits and an equilibrium point at $(0,0)$. Doesn't this have the same limit set?