Consider a system of n nonlinear ODE's $\dot{x} = f(x)$, where $x \in \mathbb{R}^{n}$, with only a few equilibrium points scattered around (The one in my mind right now is the Kuramoto model ) Given an equilibrium point $x_{0} \in \mathbb{R}^{n}$, I want to find the largest (or sufficiently large), open ball $B_{r}(x_{0})$ around $x_{0}$, such that if $x(t_{0}) \in B_{r}(x_{0})$, $d(x(t), x_{0}) < r$. Such a set is also known to be a stable set
I know a very fast way to verify simultaneous stability of a system of linear ODEs: $\dot{x_{i}} = A_{i}x_{i}$, $1\leq i\leq m$, determine if there exist $P \succ 0$, s.t. $A_{i}P+PA_{i}^{\top} \prec 0$ for all i.
Now I heard that verifying whether a set(in this case an open ball) is stable or not can be turned into a simultaneous problem. However, I have thought about it for a while, but do not know how to proceed.
I think it has to do with the existence of a Lyapunov function V where $\{ x|V(x) < s \}$ is a stable set. However, I'm not exactly sure how to proceed.
Edit: I asked and heard that to determine if a neighborhood N around $x_0$ is a stable set, we can verify whether there exists $P \succ 0$ s.t. $A_xP+PA_x^T \prec 0$ for all $x \in N$, where $A_x$ is the linearized approximation of the ODE at x. Assuming the $f(x)$ is continuously differentiable, I guess this is only a sufficient but not necessary condition. However, I'm not sure of the proof for the sufficient direction nor whether it is necessary.