Nexus between connectedness of invariant set and stability of dynamical system

Question

Let an autonomous dynamical system is characterized by the state equation

$$ \dot x(t) = f(x(t)),\quad x(0)=x_0 $$

with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, is something like:

A set of states $S \subseteq \mathbb R^n$ of above system is called an invariant set of the system if for all $x_0 \in S$ and for all $t \ge 0, x(t) \in S$.

Is there any (necessary and/or sufficient) condition on the connectedness of this invariant set for the system to be stable?

I want to know if there is any relation between the system being stable (unstable) based on the invariant set being a connected (not connected) set. I assume that the system does not have any limit-cycle.

Answer 1

I don't think stability and connectedness of an invariant set is related. Consider the linear system

$$ \dot{x} = A x, ~~ x(0) = x_0 $$

Obviously, all trajectories (invariant sets) are connected with the mapping $x(t) = e^{At} x_0$ regardless of the stability of the system.

However, I don't know if all of the invariant sets are connected for an arbitrary system $\dot{x} = f(x, t)$ It feels like to me that it is connected when $f(x, t)$ is smooth but I have no rigorous proof.

Answer 2

There is no any relation between stability and connectivity. Take two non intersecting invariant sets ( it might be any 2 trajectories of the system, which in regular autonomous case do not intersect). The union of these trajectories is also an invariant set, which is not connected. I assume you question is more relevant when the invariant set is minimal. Then I think yes it must be connected otherwise it can be split into subsets which are invariant. But again there is no any connection to stability here.