It is well known that for a system of ODEs, $\dot{\boldsymbol{y}} = \boldsymbol{Ay}$, the global stable equlibrium point is given by the eigenvector that correponds to the largest eigenvalue of $\boldsymbol{A}$, provided that the matrix $\boldsymbol{A}$ is primitive. This is according to the Perron-Frobenius theorem.

Then my question is that for a system $\dot{\boldsymbol{y}} = \boldsymbol{Ay} + \boldsymbol{B}\boldsymbol{y}\boldsymbol{y}^{T}$, do we have a similar theory for the global stability as that for $\dot{\boldsymbol{y}} = \boldsymbol{Ay}$?

We know that if we have $\boldsymbol{y}^{\ast}$ such that $\boldsymbol{0} = \boldsymbol{Ay}^{\ast} + \boldsymbol{B}\boldsymbol{y}^{\ast}\boldsymbol{y}^{\ast T}$, and the eigenvalues of the Jacobian matrix $J(\boldsymbol{y}^{\ast})$ are negative, we can conclude $\boldsymbol{y}^{\ast}$ is locally stable, but how about the global stablibility?

If all the entries of $\boldsymbol{y}(0)$ are between $0$ and $1$, do things change?

Thank you guys for helpful discussions!