I have a differential inclusion of the form $\dot{x}\in F(x)$, with $F(x)$ a set valued function that is also upper hemicontinuous.
I want to analyze if a critical point of the system, $x^*$, such that $0 \in F(x^*)$ is locally saddle-path stable.
How do I do that?
By Hartman–Grobman theorem, we know that if $F(x)$ were a differentiable function, one could linearize around $x^*$ and look at the eigenvalues of the Jacobian. Is there a similar procedure that applies when $F(x)$ is set valued around $x^*$?