After being familiar with linear ODEs, the more general problems we face is non-linear kind. So we need to linearize nonlinear ODEs.

Taylor series for any function is applied here, so it’s worthy to go over Taylor expansion per wiki:

f(x) can be vectors of vectors for example

Another example on fluid motion study,

further solving

lastly the phase portrait looks like

It’s also worth noting the “Degenerate systems of equations”.

What if the two eigen values are nearly identical and eigen vectors are nearly parallel? It has profound application in real physical world.