Jacobain and consequence on Dynamics.

Hi,

When given a 2nd order autonomous system of linear ODEs. In lecture we've investigated the stability of fixed point by computing the matrix of partial derivatives (the Jacobian) evaluating at the fixed points, finding eigenvalues. Then, we usually form a new matrix J* which takes various forms based on what the nature of the eigenvalue is (real distinct, real degenerate, complex). From this we can then investigate the stability of any fixed points.

Unfortunately, this process was only covered as a revision (because it's material covered in previous courses). However, in previous courses the cases were simplified (for example the eigenvalues were never 0), but in this course that isn't the case. I'm having the most trouble in dealing with the stability when one eigenvalue is 0. (i think here that perhaps truncating to linear terms doesn't suffice)

So I'm a little lost for material from which I can study and learn the theory.

Could anyone recommend any books, online lecture notes or the like which cover this topic?

Thanks