Consider a dynamical system of the form $$ \dot{x}=f(x), \quad x\in X, $$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ function whose derivative is non-decreasing along the trajectories of the system. By virtue of the LaSalle invariance principle we know that the system will converge to the largest invariance set of fixed points $\{x\in X\,:\, \dot{V}(x)=0\}$. Now, I know that there exist counter-examples showing that the system can converge to some trajectories in the invariant set without necessarily approaching to a point (see for instance this post). My question is whether there exist additional conditions (different from the fact that the set of fixed points is isolated) on the system and/or Lyapunov function $V(x)$ which guarantee asymptotic converge to fixed points and not to "moving" trajectories.

Thank you in advance.

Consider a dynamical system of the form $$ \dot{x}=f(x), \quad x\in X, $$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ function whose derivative is non-decreasing along the trajectories of the system. By virtue of the LaSalle invariance principle we know that the system will converge to the largest invariance set of fixed points $\{x\in X\,:\, \dot{V}(x)=0\}$. Now, I know that there exist counter-examples showing that the system can converge to some trajectories in the invariant set without necessarily approaching to a point (see for instance this post). My question is whether there exist additional conditions (different from the fact that the set of fixed points is isolated) on the system and/or Lyapunov function $V(x)$ which guarantee asymptotic converge to fixed points and not to "moving" trajectories.

Thank you in advance.