Non-genericity of initial data with punctual \(\omega \)-limit set (Q2291680)

Consider the differential equation of gradient type \[ \begin{cases} x^\prime(t)=-\nabla f(x(t)),\\ x(0)=\;x_0\end{cases} \] and the set \[ \omega(x_0)=\bigcap_{t\geq 0}\overline{\{x(s):\;s\geq t\}}. \] The authors prove that there exists a function \(f:\mathbb{R}^2\to\mathbb{R}\) of class \(C^\infty\) such that there exists an open set \(\mathcal{O}\) satisfying \[ \omega(x_0)=\mathrm{S}, \text{ for all } x_0\in \mathcal{O}, \] where \(\mathrm{S}\) is the unit circle in \(\mathbb{R}^2\) (Theorem 1.1).