What is known about fine stability properties of ODEs of the following kind? $$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$ where $d\geq 1$; $A$ is a constant matrix with all e.v. having real part $<0$; $b$ is a constant vector and $\phi:\mathbb{R}^d\to \mathbb{R}^d_{\geq 0}$ is a function, growing faster than linear on $+\infty$. A particular $\phi$ to think about is $\phi(x) = c\exp(x)$, for a fixed $c>0$ and $\exp(x)$ means coordinate-wise application of the exponential.

For example, for $d=1$ it is easy to see that there is a stable fixed point $x_s$, which is followed by an unstable one $x_u>x_s$, after which each solution goes to infinity. I would like to find out whether there are some qualitative differences in higher dimensions. A dream result would be a detailed description of the attraction basin for the stable fixed point.

Are there some classical (or not so) references covering this kind of equations?