Suppose that $F :\mathbb{R}^3 \to \mathbb{R}^3$ is a smooth map with a fixed point $e \in \mathbb{R}^3$ and that the Jacobian $J_e(F)$ of $F$ at $e$ has three distinct real eigenvalues, two of which are in the open interval $(0 , 1)$ and the third equal to $1$.
I have read that in this situation it is possible that the resulting non-linear discrete dynamical system is Lyapunov stable, but this reference states that proving this requires "more advanced nonlinear analysis, which is beyond the scope of this textbook".
What methods are there to assess the Lyapunov (or any other kind of) stability of such a dynamical system?
Can anyone point me in the direction of some references that contain this type of stability theory, specifically in the non-linear case?