For the system \begin{align*} x' &= f(x,y),\\ y' &= g(x,y), \end{align*} one would normally use the Jacobian to analyse the stability around equilibrium points. Let's assume that Linear stability doesn't provide enough information for us to conclude the nature of the fixed point (i.e. an equilibrium is non-hyperbolic). Can we use higher order terms in the taylor expansion to inform us about stability properties? For example, if the above system had equilibrium $(X,Y)$ then could one use the expansion: $$f(x,y) = f_x(X,Y)(x-X)+f_y(X,Y)(y-Y) +\frac{1}{2!}\big[ f_{xx}(X,Y)(x-X)^2 +2f_{xy}(x-X)(y-Y)+f_{yy}(y-Y)^2\big]$$ for the function $f$, with the same type of thing for $g$?
What I'm attempting to get at is that normally one would use only the first two terms of the above with regards to stability. If this didn't work, would a viable 'next thing to try' be to add more terms from the taylor series?