I am searching for a proof of the following theorem in the differential equations textbook I am TA-ing out of.
"Let $\lambda_1, \lambda_2$ be the eigenvalues of the linear system $\mathbf{x'} = A\mathbf{x}$ which corresponds to the almost linear system $\mathbf{x'} = A\mathbf{x} + \mathbf{g(x)}$. Assume that $\mathbf{x} = \mathbf{0}$ is an isolated critical point of both of these systems. Then the type and stability of $\mathbf{x} = \mathbf{0}$ for the linear system and for the almost linear system are as shown in [table in textbook]." - Brannan's Differential Equations: An Introduction to Modern Methods and Applications.
In the textbook it states that the proof is beyond the scope of the material, which is reasonable for an introductory class, but as a TA for it I would like to understand it. Can anyone point me to what this theorem is called or a paper/book where it is proven?
Thank you in advance!